item: #1 of 601 id: ejde-10 author: Ichida, Yu title: Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity date: 2023 words: 8775 flesch: 75 summary: In addition, the asymptotic behaviors are φ(ξ) ∼ C(ξ+ − ξ) 2 α+1 ψ(ξ) ∼ −C (ξ+ − ξ)− α−1 α+1 (2.3) as ξ → ξ+ − 0, and φ(ξ) ∼ C(ξ − ξ−) 2 α+1 ψ(ξ) ∼ C (ξ − ξ−) −α−1 α+1 (2.4) as ξ → ξ− + 0, with C > 0. In addition, the quenching rates are φ(ξ) ∼ −C(ξ+ − ξ) 2 α+1 ψ(ξ) ∼ C (ξ+ − ξ)− α−1 α+1 (2.1) as ξ → ξ+ − 0, and φ(ξ) ∼ −C(ξ − ξ−) 2 α+1 ψ(ξ) ∼ −C (ξ − ξ−) −α−1 α+1 (2.2) as ξ → ξ− + 0, with C > 0. keywords: dynamics; equation; ε2c2; φ(ξ cache: ejde-10.pdf plain text: ejde-10.txt item: #2 of 601 id: ejde-100 author: Gharehgazlouei, Fariba; Graef, John R.; Heidarkhani, Shapour; Kong, Lingju title: Existence and multiplicity of solutions to a fractional p-Laplacian elliptic Dirichlet problem date: 2023 words: 5751 flesch: 77 summary: Introduction In this article, we examine the nonlinear elliptic equation involving the fractional p-Laplacian and depending on a real parameter λ > 0, (−∆)spu = λf(x, u) + h(u), in Ω, u = 0, on RN\Ω, (1.1) where sp < N , Ω is a bounded open subset of RN with a Lipschitz boundary, the fractional p-Laplacian operator (−∆)sp is defined by (−∆)spu(x) = 2 lim ε↘0 ∫ RN\Bε(x) Here, |Ω| is the Lebesgue measure of Ω, ωN denotes the volume of the N -dimensional unit ball, and W̃ s,p(RN ) is the space of all u ∈ Xp s (Ω) such that ũ ∈W s,p(RN ), where ũ is the extension by zero of u. Remark 2.2. keywords: +1 n cache: ejde-100.pdf plain text: ejde-100.txt item: #3 of 601 id: ejde-1009 author: Llibre, Jaume; Zhao, Yulin title: Final evolutions for Lotka-Volterra systems in R^3 having a Darboux invariant date: 2025 words: 7763 flesch: 67 summary: For this class of Lotka-Volterra systems we can describe completely their phase portraits in the Poincaré ball. For more details on Lotka-Volterra systems see for instance keywords: invariant; lotka; phase; poincaré; singular; system; volterra cache: ejde-1009.pdf plain text: ejde-1009.txt item: #4 of 601 id: ejde-101 author: Liu, Kui; Feckan, Michal; O'Regan, Donal; Wang, Jinrong title: (omega, c)-periodic solutions for non-instantaneous impulsive systems with unbounded time-varying coefficients date: 2022 words: 8814 flesch: 79 summary: = Bi(ti)y(t−i ), i ∈ N+, y(t) = Bi(t)y(t−i ), t ∈ (ti, si], i ∈ N+, y(s+ i ) + bi, i ∈ N+, y(t) = Bi(t)y(t−i ) + bi, t ∈ (ti, si], i ∈ N+, y(s+ i ) keywords: j=1; s(t; s(ω; solutions cache: ejde-101.pdf plain text: ejde-101.txt item: #5 of 601 id: ejde-102 author: Zhou, Jundong title: k-Hessian curvature type equations in space forms date: 2022 words: 4954 flesch: 84 summary: (3.24) Applying (2.8) and (3.9), we obtain F ii∇iih11 = F ii∇11hii − h11F iih2 ii + F iihiih 2 11 −KF ii(h11δ 2 1i − h11δii + hii − hi1δi1) = F ii∇11hii − h11F iih2 ii + f̃h2 11 +Kh11 ∑ i F ii − f̃K. (3.25) Covariantly differentiating (3.11) twice yields F ii∇11hii = Gii∇11ηii ≥ −Gij,rs∇1ηij∇1ηrs+ ∑ i h11idν We prove that a 2(u− a) F iih2 ii + 1 2 (K + βφ′) ∑ i F ii ≥ C2h11. keywords: h11 cache: ejde-102.pdf plain text: ejde-102.txt item: #6 of 601 id: ejde-1036 author: Lan, Kunquan title: Existence and uniqueness of generalized normal solutions to first order fractional differential equations and applications date: 2024 words: 8055 flesch: 83 summary: Let C([a, b]; J) = {u ∈ C[a, b] : u(x) ∈ J for each x ∈ [a, b]}. If J = [c, d], then u(x) ∈ J for each x ∈ [a, b]. keywords: c([a; fractional cache: ejde-1036.pdf plain text: ejde-1036.txt item: #7 of 601 id: ejde-1037 author: Freitas, Mirelson M.; Santos, Mauro L.; Raposo, Carlos A.; Ramos, Anderson A.; Ferreira, Jorge title: Blow-up solutions for damped Rao-Nakra beams with source terms date: 2025 words: 6150 flesch: 89 summary: [16], ρ1h1utt − E1h1uxx − τ = 0, (1.4) ρ3h3vtt − E1I1ϕ1,xx − h1 2 τ +G1h1(wx + ϕ1) keywords: nakra; rao cache: ejde-1037.pdf plain text: ejde-1037.txt item: #8 of 601 id: ejde-104 author: Barreira, Luis; Valls, Claudia title: Some applications of Lyapunov regularity date: 2022 words: 9611 flesch: 79 summary: , q let αi = lim inf m→+∞ 1 m log m∏ l=1 |alii| and αi = lim sup m→+∞ 1 m log m∏ l=1 |alii|. , q let αi = lim inf m→+∞ 1 m log m∏ l=1 |alii| and αi = lim sup m→+∞ 1 m log m∏ l=1 |alii|. (3.1) keywords: lim; log; lyapunov; sequence cache: ejde-104.pdf plain text: ejde-104.txt item: #9 of 601 id: ejde-105 author: Kishimoto, Nobu title: Remarks on periodic Zakharov systems date: 2022 words: 8722 flesch: 81 summary: Let u ∈ C([0, T∞);H1) be the (forward-in-time) maximal-lifespan solution of i∂tu+ ∆u = − ( |u|2 − Pc(|u|2)− ν0 − ν1t ) u, t ∈ (0, T∞), x ∈ Tdλ, u ∣∣ t=0 = u∞0 . By the Hölder inequality, the Sobolev embedding, interpolation and the Duhamel formula, we see that, for t ∈ keywords: c([0; case; solution; uniqueness cache: ejde-105.pdf plain text: ejde-105.txt item: #10 of 601 id: ejde-1051 author: Wu, Chun title: Global boundedness in an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion date: 2025 words: 7365 flesch: 82 summary: Using wq−1 to test the third equation of (1.6) and integrating gives 1 q d dt ∫ Ω wq = −δ ∫ Ω wq + ∫ Ω uwq−1 ≤ −δ ∫ Ω wq + δ 2 ∫ Ω wq + C ∫ Ω uq (2.6) for all t ∈ (0, Tmax) which implies (2.5) with (2.3) and Lemma 2.4. □ Lemma 2.7. Applying integration by parts to the second equation in (1.6) and using the well-known equation 2∇v · ∇∆v = ∆|∇v|2 − 2|D2v|2, we find that d dt ∫ Ω v1−p|∇v|p = p ∫ Ω v1−p|∇v|p−2∇v · ∇(∆v − uvw)− (p− 1) ∫ Ω v−p|∇v|p(∆v − uvw) = p 2 ∫ Ω v1−p|∇v|p−2(∆|∇v|2 − 2|D2v|2)− p ∫ Ω v1−p|∇v|p−2∇v · ∇(uvw) − (p− 1) ∫ Ω v−p|∇v|p∆v + (p− 1) ∫ Ω wv1−p|∇v|p = p(p− 1) ∫ Ω v−p|∇v|p−2∇v · ∇|∇v|2 − p ∫ Ω v1−p|∇v|p−2|D2v|2 − p(p− 2) 4 ∫ Ω v1−p|∇v|p−4|∇|∇v|2|2 − p(p− 1) ∫ Ω v−p−1|∇v|p+2 + p 2 ∫ ∂Ω v1−p|∇v|p−2 · ∂|∇v|2 ∂ν + p(p− 2) 2 ∫ Ω wv−p+2|∇v|p−4∇v · ∇|∇v|2 + p ∫ Ω wv−p+2|∇v|p−2∆v − (p− 1)2 ∫ Ω wv1−p|∇v|p. (3.5) EJDE-2025/09 CHEMOTAXIS-CONSUMPTION MODEL 9 The pointwise identity [49, Lemma 3.2] and ∇|∇v|2 = 2D2v · ∇v imply that p(p− 1) ∫ Ω v−p|∇v|p−2∇v · keywords: chemotaxis; tmax; ∫ ω cache: ejde-1051.pdf plain text: ejde-1051.txt item: #11 of 601 id: ejde-106 author: Millla Miranda, Manuel; Louredo, Aldo Trajano; Clark, Marcondes Rodrigues; Gouveia, Giovana Siracusa title: Nonstationary Lame system without definite sign energy date: 2022 words: 8490 flesch: 87 summary: (4.6) Putting the above two expressions in (4.5) and then integrating on [0, t], 0 < t < tlm, we obtain 1 2 ‖u′lm(t)‖2H + 1 2 ‖ulm(t)‖2V + 1 ρ+ 1 (|ulm(t)|ρ, ulm(t))H + d0 ∫ t 0 ‖u′lm(s)‖2L2(Γ1)ds ≤ 1 2 ‖u1 l ‖2H + 1 2 ‖u0‖2V = 1, ∀x ∈ Ω, ϕ0 ∈ D(Ω), ϕl ∈ D(Ul), l = 1, 2, . . . keywords: div; i=1 cache: ejde-106.pdf plain text: ejde-106.txt item: #12 of 601 id: ejde-1060 author: Caraballo, Tomas; Ezzine, Faten; Hammami, Mohamed Ali title: Practical stability of stochastic differential delay equations driven by G-Brownian motion with general decay rate date: 2024 words: 9981 flesch: 72 summary: Now, we consider the nonlinear stochastic differential delay equations driven by a G-Brownian motion in the form dx(t) = f(t, xt)dt+ h(t, xt)d⟨B⟩t + g(t, xt)dBt, t ≥ t0, (3.1) where Bt is a one-dimensional G-Brownian motion, with Bt ∼ N (0, [σ2t, σ̄2t]), and (⟨B⟩)t≥0 is the quadratic variation process of the G-Brownian, and f : Based on this fact, we deduce that V (t, xt) ≤ V (0, x0) + φ1 ( K 2N )−1 ln K − 1 2N + ∫ t t0 LV (s, xs)ds + σ̄2 ∫ t t0 φ1(s)∥Vs(s, xs)g(s, xs)∥2ds, for t0 ≤ t ≤ K/2N and K ≤ K0(ε, ω). keywords: brownian; delay; differential; equations; lyapunov; motion; r(t; stability; stochastic; t t0 cache: ejde-1060.pdf plain text: ejde-1060.txt item: #13 of 601 id: ejde-1061 author: Lopera, Emer; Recova, Leandro; Rumbos, Adolfo title: Multiplicity results for Schrodinger type fractional p-Laplacian boundary value problems date: 2024 words: 10776 flesch: 80 summary: In this work, we study the existence and multiplicity of solutions to the problem −(∆)spu+ V (x)|u|p−2u = λf(u), x ∈ Ω; u = 0, x ∈ RN\Ω, where Ω ⊂ RN is an open bounded set with Lipschitz boundary ∂Ω, N ⩾ 2, V ∈ L∞(RN ), and (−∆)sp denotes the fractional p-Laplacian with s ∈ (0, 1), 1 < p, sp < N , λ > 0, and f : R → R is a continuous function. = 1, for x ∈ Ω; u = 0, in RN\Ω, (7.1) has a positive weak solution. keywords: problem; solution; theorem cache: ejde-1061.pdf plain text: ejde-1061.txt item: #14 of 601 id: ejde-108 author: Almeida, Adilson; Chemetov, Nikolai V.; Cipriano, Fernanda title: Uniqueness for optimal control problems of two-dimensional second grade fluids date: 2022 words: 5136 flesch: 68 summary: In this article, we study second grade fluids, which belong to the class of non- Newtonian complex viscoelastic fluids of differential type. From the mathematical point of view, the equations governing the evolution of second grade fluids are strongly nonlinear partial differential equations. keywords: control; fluids; problem; solution cache: ejde-108.pdf plain text: ejde-108.txt item: #15 of 601 id: ejde-1083 author: Luczak, Brian B. title: A priori estimates for the linearized relativistic Euler equations with a physical vacuum boundary and an ideal gas equation of state date: 2025 words: 26531 flesch: 76 summary: ∂iu i + ( (γ − 1) (u0)2 ũ0r − γ − 1 u0 r̃ ) ∂tr = − ui u0 ∂ir − γ − 1 u0 r∂iu i − γ − 1 u0 r ( Ci 1∂iu 0 + Ci 2∂ir + C3r∂iu i ) implies ∂tr̃ = − ui u0 ∂ir̃ − γ − 1 u0 r∂iũ i − γ − 1 u0 r ( ∂tũ 0 ) + uiũ0 − u0ũi (u0)2 ∂ir + ( (γ − 1) (u0)2 ũ0r − γ − 1 u0 r̃ ) keywords: + r; 2(γ−1; 2−γ; 2−γ γ−1; boundary; derivatives; ejde-2025/10; equations; estimates; following; h2k; lemma; norm; order; r 2−γ; r r; remark; section; subcritical; t r̃; terms; use; γ − cache: ejde-1083.pdf plain text: ejde-1083.txt item: #16 of 601 id: ejde-1084 author: Xin; Xinglong title: Persistence properties of solutions for multi-component Novikov equations date: 2025 words: 6234 flesch: 82 summary: For p > 1, if the initial data satisfies for some C > 0, ∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp ≤ C, then the solution satisfies ∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp ≤ C, EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 7 uniformly in the interval [0, T ], where for α ∈ [0,∞) and K ∈ R+, the weighted function I(x) is given by I(x) = { (ln(e2 + |x|))α, |x| ∈ [0, T ], Y (t) ≤ CY (0) ≤ C (∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp + ∥U0,xxI∥Lp) . keywords: equation cache: ejde-1084.pdf plain text: ejde-1084.txt item: #17 of 601 id: ejde-1087 author: Nunes, Ruikson; Nunez-Chavez, Miguel R. title: Exact boundary controllability for wave equations with fixed and moving boundaries in two-dimensional convex-complemented domains date: 2025 words: 6675 flesch: 77 summary: Making τ = T − t in latest inequality and observing that u(·, t) = v(·, T − t) is solution of (2.6) satisfying the estimate (2.7). Wave equation; energy decay; exact boundary controllability; non-cylindrical domains; moving boundary domains. keywords: boundary; decay; problem; wave cache: ejde-1087.pdf plain text: ejde-1087.txt item: #18 of 601 id: ejde-109 author: Burton, Theodore A.; Purnara, Ioannis K. title: Open mappings: The case for a new direction in fixed point theory date: 2022 words: 12917 flesch: 84 summary: Moreover, writing the second summand in the right-hand-side of (9.10) as β t2 + 1 sin ( t2 + x2(t) t2 + 1 )∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds t ≥ 0, it is not difficult to see that for any bounded function x it holds 0 ≤ β t2 + 1 ∣∣∣ sin( t2 + x2(t) t2 + 1 )∣∣∣ ∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds ≤ βte−t t2 + 1 ∫ t 0 sds+ β‖x‖p t2 + 1 ∫ t 0 s 1 + s2 + t2 ds ≤ βt3e−t 2(t2 + 1) + β‖x‖p t2 + 1 ∫ t 0 s 1 + t2 ds ≤ βte−t + β‖x‖pt2 2(t2 + 1)2 , and so lim t→∞ β t2 + 1 sin ( t2 + x2(t) t2 + 1 )∫ t 0 s (t2 + 1)te−t + |x(s)|p 1 + s2 + t2 ds = 0. + ∫ t 0 K(t, s)v(t, s, x(s))ds, for their importance in applied mathematics and the fact that the integral term un- der a wide set of conditions will define a compact map keywords: point; solution cache: ejde-109.pdf plain text: ejde-109.txt item: #19 of 601 id: ejde-1093 author: Gingolld, Harry; Quaintance, Jocelyn title: Spherical compactifications of central force equations date: 2025 words: 12549 flesch: 80 summary: = γ2 + (1− γ2)r2 = γ2 + (1− γ2)(q21 + q22 + · · ·+ q2n). − 2 [ γ2 − γ2 (γ2 + √ ω r2 + γ2 ) − γ2 (γ2 + √ ω̂ r̂2 + γ2 )] , (3.27) where r2 := QTQ, r̂2 := Q̂T Q̂, ω := γ2 + (1− γ2)r2, ω̂ := γ2 + (1− γ2)r̂2; see Figure 4. keywords: lim; m→∞; radius cache: ejde-1093.pdf plain text: ejde-1093.txt item: #20 of 601 id: ejde-1096 author: Yang, Minbo; Zhou, Fan title: Existence and multiplicity of solutions to quasilinear Dirac-Poisson systems date: 2025 words: 9515 flesch: 81 summary: (2.1) We will write A0 := iα · ∇ − aβ, Aω := A0 − ω denote the self-adjoint operator on L2 := L2(R3,C4) with domain D(Aω) ⊂ H1 := H1(R3,C4). = 1 2 (∥u+∥2 − ∥u−∥2)− Γε(u)− ∫ R3 F (x, |u|) dx ≤ ∥u+∥2 − 1 2 ∥u∥2 − ∫ R3 K1(x)G(|u|) dx ≤ 1 c2N |u+|2q − c0K1,inf |u+|qq − 1 2 ∥u∥2 ≤ q keywords: dirac; energy; lemma; solutions; system cache: ejde-1096.pdf plain text: ejde-1096.txt item: #21 of 601 id: ejde-110 author: Afrouzi, Ghasem A.; Chung, Nguyen Thanh; Naghizadeh, Zohreh title: Multiple solutions for p(x)-Kirchhoff type problems with Robin boundary conditions date: 2022 words: 6770 flesch: 82 summary: Hence, F (x, τt) ≥ τµF (x, t), ∀x ∈ Ω, t ∈ R, τ ≥ 1. (2.3) Let ϕ ∈ C∞0 (Ω) and ϕ 6≡ 0 such that ∫ Ω F (x, ϕ) dx > 0, by (A1) we have Jλ(τϕ) Introduction In this article, we study the existence of weak solutions for p(x)-Kirchhoff type problems with Robin boundary conditions −M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λg(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u = 0, x ∈ ∂Ω, (1.1) where Ω is a bounded domain in RN with smooth boundary ∂Ω, ∂u ∂ν is the outer normal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L∞(∂Ω), β− := infx∈∂Ω β(x) > 0, p ∈ C+(Ω), 1 < p− := infx∈Ω p(x) ≤ p+ := maxx∈Ω p(x) < keywords: dσx; p(x; β(x cache: ejde-110.pdf plain text: ejde-110.txt item: #22 of 601 id: ejde-1105 author: Xu, Tianyuan; Liu, Gege; Yin, Jingxue title: Existence and stability of forced waves for p-Laplace equations in a shifting habitat date: 2025 words: 7752 flesch: 81 summary: ∫∫ Qτ ( β 2 + u1 + u2 − r(ξ))u2αn(x)e −βtdxdt +D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtαn(x)dxdt = −D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1 − u2)e −βtα′ n(x)dxdt ≤ D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. (3.22) Noticing that α′ n(x) = 0 for |x| < n and |x| > n+ 1, we see that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ 2D ∫∫ Qτ (|u1x|2(p−1) + |u2x|2(p−1))|α′ n(x)|dxdt+ ∫∫ Qτ u2|α′ n(x)|dxdt ≤ C, where C is independent of n. Letting n → ∞ in (3.22), we obtain∫∫ Qτ u2e−βtdxdt+D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtdxdt ≤ C. (3.23) Recalling (3.22) and noticing that uix(i = 1, 2) are bounded, we infer that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ D(p− 1)(∥u1x∥L∞ + ∥u2x∥L∞)p−2 ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt +D ∫∫ Qτ u2|α′ n(x)|e−βtdtdx ≤ C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. From (3.23), the above inequality reduces to C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt → 0, as n → ∞. Now, (3.22) shows that 1 2 ∫ R e−βtu2(x, τ)dx+ ∫∫ Qτ e−βt(u2 + (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x))dxdt ≤ 0, which implies that u1 = u2 in Qτ . keywords: waves; ϕ(ξ cache: ejde-1105.pdf plain text: ejde-1105.txt item: #23 of 601 id: ejde-1109 author: Zeng, Lingzhong; Zhou, Ziyi title: Eigenvalue bounds for the clamped plate problem of L^2_xi operator date: 2025 words: 7287 flesch: 82 summary: G = k∑ i,j=1 (Γk+1 − Γj) aijtij + k∑ i,j=1 (Γj − Γi) aijtij = k∑ j,i=1 (Γk+1 − Γi) aijtji + k∑ i,j=1 sijtij = − k∑ j,i=1 (Γk+1 − Γi) aijtij + k∑ i,j=1 sijtij = −G+ k∑ [ Γi (Γk+1 − Γi) ]1/2 , (1.8) which is sharper than Γk+1 ≤ [ 1 + 8(n+ 2) n2 ]1 k k∑ i=1 Γi. (1.9) EJDE-2025/31 EIGENVALUE BOUNDS FOR THE CLAMPED PLATE PROBLEM 3 We note that, in fact, inequality (1.9) is better than inequality (1.5) given by Payne, Pólya and Weinberger. keywords: e⟨ξ; i=1; j=1; x⟩g0dv; γk+1 cache: ejde-1109.pdf plain text: ejde-1109.txt item: #24 of 601 id: ejde-111 author: Barkatou, Moulay; Carnicero, Félix Álvaro; Sanz, Fernando title: Turrittin's normal forms for linear systems of meromorphic ODEs over the real field date: 2023 words: 13537 flesch: 77 summary: We present versions of real (formal and polynomial) normal forms for any system, in such a way EJDE-2023/79 TURRITTIN’S THEOREM 3 that they can be obtained by transformations written in the base field K, without passing through the algebraic closure K = K( √ −1). Now, Turrittin’s results for the case where K = K can be stated in the two following theorems. keywords: block; case; diagonal; gauge; matrix; polynomial; proof; system; theorem; transformation cache: ejde-111.pdf plain text: ejde-111.txt item: #25 of 601 id: ejde-1110 author: Son, Dang Thanh title: Long-time behavior of solutions to the 2D magnetic B\'enard problem in porous media on unbounded domains date: 2025 words: 12225 flesch: 81 summary: In the next Lemma, we show that the map Λ(t; z0) is bounded and the semigroup {S(t)}t≥0 is uniformly differentiable on global attractor A, i.e., lim ε→0 sup 0<|z̄0−z0|<ε, z̄0,z0∈A |S(t)z̄0 − S(t)z0 − Λ(t; z0)(z̄0 − z0)| |z̄0 − z0| = 0. (5.2) Then for r = 1, 3, there exists a constantM(|z̄0|, |z0|) such that |S(t)z̄0 − S(t)z0 − Λ(t; z0)(z̄0 − z0)| ≤M |z̄0 − z0|, (5.3) where the linear operator Λ(t; z0) for t > 0 is the solution operator of the prob- lem (5.1). keywords: attractor; equations; existence; lemma; l̃r+1; problem; r−1; solution cache: ejde-1110.pdf plain text: ejde-1110.txt item: #26 of 601 id: ejde-1115 author: Achleitner, Franz ; Cuesta, Carlota M.; Diez-Izagirre, Xuban title: Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation date: 2025 words: 22418 flesch: 77 summary: Then, there exists an order one constant C > 0, such that ϕτ ∈ ϕ− +H2(Iτ,ε), ∥ϕτ − ϕ− − eλτξ∥H2(Iε) ≤ Cε2 . − ξ) near ξ∗, or in terms of the variable η |Φτ (η) + ϕc| > C √ τ (η∗ − η) . keywords: 2−α; lemma; non; proof; solutions; theorem; η 0; ηinflex cache: ejde-1115.pdf plain text: ejde-1115.txt item: #27 of 601 id: ejde-1119 author: Nie, Yuanyuan; Leng, Yan; Zhao, Xu; Zhou, Qian title: Critical Fujita exponents for a class of quasilinear coupled parabolic  equations date: 2025 words: 6314 flesch: 82 summary: Introduction In this article, we study the critical Fujita exponent for the Cauchy problem of quasilinear coupled parabolic equations ∂u ∂t = ∆um + (|x|+ 1)λvp, x ∈ Rn, t > 0, (1.1) ∂v ∂t = ∆vm + (|x|+ 1)µuq, x ∈ Rn, t > 0, (1.2) u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Rn, (1.3) where p, q > m > 1, λ ≥ 0, µ = λ(q −m) + 2(q − p) p−m ≥ 0 (1.4) and 0 ≤ u0, v0 ∈ C0(Rn) are nontrivial. It was demonstrated that the Cauchy problem of the heat equation ∂u ∂t = ∆u+ up, x ∈ Rn, t > 0 admits no nontrivial nonnegative global solution when 1 < p < pc = 1 + 2/n, otherwise, it admits both nontrivial global (with small initial data) and nonglobal nonnegative (with large initial data) solutions when p > pc. keywords: t)ψl(x)dx; u(x; |x|+ cache: ejde-1119.pdf plain text: ejde-1119.txt item: #28 of 601 id: ejde-112 author: Goyal, Sarika; Sharma, Tarun title: Fractional Kirchhoff Hardy problems with weighted Choquard and singular nonlinearity date: 2022 words: 11624 flesch: 86 summary: p ((uk(y)− u0(y))) p |x|α|x− y|µ|y|α dx dy = 0. (4.27) Using (4.27) in (4.26), we have 0 ≥ ( c+ dv2θ−2 ) lim k→∞ ‖uk − u0‖2 − γ lim k→∞ ‖uk − u0‖2H = c [ lim k→∞ ‖uk − u0‖2 − γ c lim k→∞ ‖uk − u0‖2H ] + dv2θ−2 lim k→∞ ‖uk − u0‖2. = (1 2 − 1 2p ) hc,γt 2 1−q − λ ( 1 1− q − 1 2p ) ‖l‖mS −(1−q) 2 t, which has minimum at tmin := (λ(2p+ q − 1)S −(1−q) 2 ‖l‖m (2p− 2)hc,γ ) 1−q 1+q . keywords: dx dy; q −; y|µ|y|α; y|µ|y|α dx; |x|α|x−; − ∫; ∫ ω cache: ejde-112.pdf plain text: ejde-112.txt item: #29 of 601 id: ejde-113 author: Fresneda Portillo, Carlos; Woldemicheal, Zenebe W. title: Boundary-domain integral equations for Dirichlet diffusion problems with non-smooth coefficient date: 2022 words: 6956 flesch: 66 summary: Let ρ ∈ D(Ω), the volume poten- tial and the remainder potential operator, corresponding to parametrix (3.1) and remainder (3.2) are defined as Pρ(y) := ∫ R3 P (x, y)ρ(x) dx, y ∈ R3, Pρ(y) := ∫ Ω P (x, y)ρ(x) dx, y ∈ Ω, Rρ(y) Domain and boundary integral operators After replacing the parametrix in the Green identities, we obtain an integral representation formula of the solution in terms of potential-type operators whose kernel is somehow related to the parametrix. keywords: boundary; domain; equations; hs−; operator; parametrix; solution; theorem cache: ejde-113.pdf plain text: ejde-113.txt item: #30 of 601 id: ejde-1137 author: Liu, Ling title: Global well-posedness to a multidimensional parabolic-elliptic-elliptic attraction-repulsion chemotaxis system date: 2025 words: 8020 flesch: 77 summary: 6 L. LIU EJDE-2025/26 Then, recalling ξγ = χα, (1.1), (1.6), (1.7) can be rewritten as ut = ∆u−∇ · (u∇s), x ∈ Ω, t ∈ (0, Tmax), 0 In the absence of chemorepulsive chemical (i.e. chemorepellent), namely ξ = 0, w is decoupled from the system (1.1) and the first two equations of (1.1) comprises a classical Keller-Segel model ut = ∆u− χ∇ · (u∇v), x ∈ Ω, t > 0, 0 = ∆v keywords: chemotaxis; system; tmax cache: ejde-1137.pdf plain text: ejde-1137.txt item: #31 of 601 id: ejde-1144 author: Xing, Zhaojun title: Single-component regularity criterion and inviscid limit for axially symmetric MHD-Boussinesq systems date: 2025 words: 6851 flesch: 82 summary: ∥pLp + ∫ t 0 ∫ R3 |∇H(s, x)|2|H(s, x)|p−2 dx ds ≤ ∥H0∥pLp , ∥H(t, ·)∥L∞ ≤ ∥H0∥L∞ , ∥ρ(t, ·)∥pLp + ∫ t 0 ∫ R3 |∇ρ(s, x)|2|ρ(s, x)|p−2 dx ds ≤ ∥ρ0∥pLp , ∥ρ(t, ·)∥L∞ ≤ ∥ρ0∥L∞ . (3.1) (ii) for (u0, h0, ρ0) ∈ L2 and t ∈ R+, ∥(u, h)(t, ·)∥2L2 + ∫ t 0 ∥∇h (s, ·) ∥2L2 ds ≤ C0(1 + t)2, (3.2) where C0 depends only on ∥(u0, h0, ρ0)∥L2 . Proof. This ends up with ∥∇H(t, ·)∥2L2 + ∫ t 0 ∥∇2H(s, ·)∥2L2 ds ≲ ∥∇H0∥2L2 + ∫ t 0 ∥Ω(s, ·)∥2L2∩L6∥∇h(s, ·)∥2L2 ds. (3.6) Then, we obtain the estimate of N . keywords: boussinesq; mhd; system; ∥2l2 cache: ejde-1144.pdf plain text: ejde-1144.txt item: #32 of 601 id: ejde-116 author: Zhao, Zhihong; Shaochun, Shaochun; Lu, Yulan title: Mathematical models for the transmission of malaria with seasonality and ivermectin date: 2022 words: 8846 flesch: 72 summary: In this section, we propose a seasonal effect of delay malaria transmission model taking into account the treatment and ivermectin. Recently, [18] mod- elled the effect of ivermectin on malaria transmission control by ordinary differential equations and the results showed that ivermectin was significantly more effective in malaria control compared to the no-intervention state. keywords: ivermectin; malaria; model; rate; transmission cache: ejde-116.pdf plain text: ejde-116.txt item: #33 of 601 id: ejde-1179 author: Ramos, Gustavo de Paula title: Asymptotic profile of least energy solutions to the nonlinear Schrodinger-Bopp-Podolsky system date: 2025 words: 4145 flesch: 74 summary: For instance, [3, 2, 10, 11, 15, 21, 25] addressed the existence of least energy solutions; [7, 12, 13] considered the mass-constrained problem; [8, 9, 17, 23, 22] obtained sign-changing solutions; and [4, 6] considered semiclassical states. For instance, [5, Theorem 1.3] proved such a result for radial solutions; [7, Theorem D] extended this conclusion for least energy solutions to the mass-constrained system for 2 < p < 14/5 and a sufficiently small mass ρ (notice that these solutions are also radial due to [7, Theorem C]); [20, Theorem 1.3] showed that solutions to the associated eigenvalue problem in a bounded smooth domain also have such an asymptotic profile and, more recently, [4, Theorem 1.7] verified such a behavior for the critical nonlinear SBP system in the semiclassical regime under the effect of an external effective potential V : R3 → [0,∞[ which vanishes at a point x0 ∈ R3. keywords: 2p−; energy; schrödinger; solutions cache: ejde-1179.pdf plain text: ejde-1179.txt item: #34 of 601 id: ejde-118 author: Barraza Martinez, Bienvenido; Hernandez Monzon, Jairo; Vergara Rolong, Gustavo title: Exponential stability of a damped beam-string-beam transmission problem date: 2022 words: 7475 flesch: 82 summary: − 〈i(l2 − x)g′1,n, λnv1,n〉L2(I2) − i(l2 − l1)λng1,n(l1)v1,n(l1) + 〈v′′1,n, (l2 − x)v′1,n〉L2(I2). − 〈i(l2 − x)g′1,n, λnv1,n〉L2(I2) − i(l2 − l1)g1,n(l1)λnv1(l1)− 〈βv2,n, (l2 − x)v′1,n〉L2(I2) + 〈ig1,n, λnv1,n〉L2(I2) } + ‖λnv1,n‖2L2(I2) + ‖v′1,n‖2L2(I2). keywords: beam; problem; stability; transmission cache: ejde-118.pdf plain text: ejde-118.txt item: #35 of 601 id: ejde-1183 author: Girg, Petr; Kotrla, Lukas title: Modeling of groundwater flow in porous medium layered over inclined impermeable beds date: 2025 words: 15629 flesch: 78 summary: Porous medium; filtration; nonlinear Darcy’s law; p-Laplacian; pressure-to-velocity power law. Mathematical model of water flow in porous medium layered over an inclined impermeable bed 2.1. keywords: cosφ+; flow; function; medium; problem; proof; sinφ; sinφ|p−2; solution; u(x; u′(1; water; ∥f∥l1(−1,1 cache: ejde-1183.pdf plain text: ejde-1183.txt item: #36 of 601 id: ejde-1189 author: Yang, Zhi-Jiao; Zhang, Guo-Bao; He, Juan title: Traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with nonlocal nonlinearities date: 2025 words: 8024 flesch: 82 summary: [32] further studied the stability of traveling wave solutions of system (1.3) with relatively large speed by the weighted energy method combining with the comparison principle. It is well known that traveling wave solutions can describe the transitions between different states of a physical system, propagation of patterns, and domain invasion of species in population biology (see, e.g., [6]). keywords: i∈z; system cache: ejde-1189.pdf plain text: ejde-1189.txt item: #37 of 601 id: ejde-119 author: Zhang, Xuping title: Lower and upper solutions for delay evolution equations with nonlocal and impulsive conditions date: 2002 words: 6301 flesch: 82 summary: We mention that in 2012, Chuong and Ke [10] studied the delay evolution inclu- sions involving nonlocal and impulsive conditions u′(t) +Au(t) ∈ F (t, u(t), ut), t ∈ Evidently, PC([−h, a], X) and B are also order Banach spaces with partial order “ ≤ ” reduced by the positive function cones KPC = {u ∈ PC([−h, a], X) : u(t) ≥ θ, t ∈ keywords: conditions; equations; v(0; w(0 cache: ejde-119.pdf plain text: ejde-119.txt item: #38 of 601 id: ejde-1193 author: He, Juhua; Wu, Ke; Zhou, Fen title: Existence of nontrivial solutions for biharmonic equations with critical growth date: 2025 words: 9619 flesch: 88 summary: Again by (2.6)-(2.11), there exists a small ε1 ∈ (0, ε2) such that I(twε) ≥ t2 2 ∫ R5 (∆wε) 2dx− t2 ∗∗ 2∗∗ ∫ R5 |wε|2 ∗∗ dx− α tp p ∫ R5 |wε|pdx ≥ t2 4 S 5/4 ∗∗ − t2 ∗∗ 2∗∗ S 5/4 ∗∗ − αCε5− p 2 tp 6 J. HE, K. WU, F. ZHOU EJDE-2025/69 for all ε ∈ (0, ε1). Hence, I(tεwε) ≥ max 0≤t≤1 { t 2 4 S 5/4 ∗∗ − t2 ∗∗ 2∗∗ S 5/4 ∗∗ − αCε5− p 2 tp} ≥ η 2 . keywords: 2∗∗ cache: ejde-1193.pdf plain text: ejde-1193.txt item: #39 of 601 id: ejde-1196 author: Agudelo, Oscar; Holubova, Gabriela; Kudlac, Martin title: Variational and numerical aspects of a system of ODEs with concave-convex nonlinearities date: 2025 words: 10281 flesch: 76 summary: We show multiplicity of nonnegative solutions for a range of the parameter λ and discuss the regularity and symmetry of nonnegative solutions. In [4], the authors study existence, nonexistence and multiplicity of nonnegative solutions of the single equation −v′′ keywords: du0; dv0; proof; solutions cache: ejde-1196.pdf plain text: ejde-1196.txt item: #40 of 601 id: ejde-12 author: Vidhyaa, Kumar S.; Thandapani, Ethiraju; Alzabut, Jehad; Ozbekler, Abdullah title: Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term date: 2023 words: 4354 flesch: 74 summary: [21] B. Kamaraj, R. Vasuki; Oscillation of second order difference equations with a superlinear neutral term, J. Adv. 12 K. S. VIDHYAA, E. THANDAPANI, J. ALZABUT, A. ÖZBEKLER EJDE-2023/45 [26] S. Meharbanu, S. Nalini; Oscillation of second order difference equations with several super- linear neutral terms, Adv. Differ. keywords: equations; oscillation cache: ejde-12.pdf plain text: ejde-12.txt item: #41 of 601 id: ejde-120 author: Li, Shangzhi; Guo, Shangjiang title: Dynamics of stochastic Lotka-Volterra predator-prey models driven by three independent Brownian motions date: 2022 words: 9824 flesch: 70 summary: For the parameters µ1 = µ2 = ρ1 = ρ2 = 0.1, we have λ2 ≈ 0.4485 and λ1 ≈ 0.8868. Similarly to [23, Proposition 3.2], we conclude that there are T = T (ς) and δ1 = δ1(ς) > 0 such that (4.5) holds, which implies that X2,x is away from zero for all x ∈ R2,◦ + . keywords: predator; prey; solution; system cache: ejde-120.pdf plain text: ejde-120.txt item: #42 of 601 id: ejde-1208 author: Jing, Zhao; Liu, Zhenhai; Papageorgiou, Nikolaos S. title: Weighted (p,q)-equations with gradient dependent reaction date: 2025 words: 5891 flesch: 84 summary: Introduction Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this article we study the parametric Dirichlet problem −∆a1 p u(z)−∆a2 q u(z) = f(z, u(z)) + λ|Du(z)|p−1 in Ω, u|∂Ω = 0, 1 < keywords: 1,p cache: ejde-1208.pdf plain text: ejde-1208.txt item: #43 of 601 id: ejde-1217 author: Borsuk, Mikhail title: Neumann-Robin problem for p(x)-Laplacian equations in a domain with the boundary edge date: 2025 words: 11379 flesch: 86 summary: (p− − 1)− 2 s( j⟨(p− − 1)κ + 3− p−⟩ − 1 )( j⟨(p+ − 1)κ − 2 s + 2⟩ − 1 ) . p−⟩ − 1 )( j⟨(p+ − 1)κ − 2 s + 2⟩ − 1 )} j j−1 . keywords: j−1; p+ −; − k)+; ∪ωr0 cache: ejde-1217.pdf plain text: ejde-1217.txt item: #44 of 601 id: ejde-122 author: Fan, Xiaoting; Wang, Wei title: Initial layer associated with Boussinesq systems for thermosolutal convection date: 2022 words: 6929 flesch: 83 summary: ∂tT λ In + (uλIn · ∇)TλIn = ∆TλIn +RλIn,T , (3.18) ∂tS λ In + (uλIn · ∇)SλIn = τ∆SλIn +RλIn,S , (3.19) uλIn|z=0,1 = 0, (3.20) (TλIn, S λ In)|z=0 = (1, 1), (TλIn, S λ In)|z=1 = (0, 0), (3.21) where the remainders RλIn,u, RλIn,T and RλIn,S are RλIn,u = − ∞∑ i=2 λi(λ[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i −∆uIn,i − √ Tak × uIn,i − (RTT In,i −RSSIn,i)k) + λ3uIn,1 · ∇uIn,1, RλIn,T = − ∞∑ i=2 λi ( ∂tT In,i + i∑ j=0 uIn,j · ∇T EJDE-2022/33 INITIAL LAYER ASSOCIATED WITH BOUSSINESQ SYSTEMS 13 According to (3.69), (3.72), and (4.7), I3 = − ∫ D uλa · ∇ ( (Tλe )2 2 ) dx dy dz = − ∫ D ∇ · ( uλa (Tλe )2 2 ) keywords: t λ cache: ejde-122.pdf plain text: ejde-122.txt item: #45 of 601 id: ejde-1221 author: Aparcana, Aldryn; Carhuas-Torre, Brandon; Castillo, Ricardo; Loayza, Miguel title: Existence and non-existence of solutions for Hardy parabolic equations with singular initial data date: 2025 words: 5461 flesch: 84 summary: At−α + ∫ t 0 (t− σ)−ζf(σ)φ(σ)dσ for t ∈ (0, T ). Let t ∈ (0, s) with s ∈ (0, T ) and 1/q = 1− ϵ > 0. keywords: non cache: ejde-1221.pdf plain text: ejde-1221.txt item: #46 of 601 id: ejde-123 author: Wang, Sainan; Su, Yu title: Existence of solution to critical Kirchhoff-type equation with dipole-type potential date: 2022 words: 6574 flesch: 89 summary: Specifically, the Schrödinger equation for the wave function of an electron interacting with a polar molecule can be written as H = − ~ 2m ∆ + e x ·D |x|3 − E, where D is the dipole moment of the molecule, e and m denote the charge and the mass of the electron, see [19]. We know that lim n→∞ ‖vn‖2Φ = ‖v‖2Φ. By Brézis-Lieb lemma again, we have lim n→∞ ‖vn‖2Φ − lim n→∞ ‖vn − v‖2Φ = ‖v‖2Φ, which implies lim n→∞ ‖vn − v‖2Φ = 0. keywords: d1,2; equation; lim; rad(rn cache: ejde-123.pdf plain text: ejde-123.txt item: #47 of 601 id: ejde-1235 author: Webb, Jeffrey R. L. title: Inequalities for fractional derivatives via the Marchaud derivative date: 2025 words: 13422 flesch: 83 summary: For ε > 0, the truncated fractional derivative is defined for t ∈ (0, T ] by Dα M,εf(t) = f(t) Γ(1− α)tα + α Γ(1− α) ψε(t), (2.11) where ψε(t) : Then Dα Mu(t) exists for t ∈ (0, T ]. keywords: continuous; derivative; fractional; u(t; γ(1− cache: ejde-1235.pdf plain text: ejde-1235.txt item: #48 of 601 id: ejde-124 author: Sun, Rui; Liu, Duchao title: Positive solution to quasilinear Schrodinger equations via Orlicz space framework date: 2022 words: 5001 flesch: 81 summary: [5] S. X. Chen; Existence of positive solutions for a class of quasilinear Schrödinger equations on RN , J. Math. [18] G. F. Li, Y. S. Huang, Z. Liu; Positive solutions for quasilinear Schrödinger equations with superlinear term, Complex Var. keywords: equations; quasilinear; schrödinger; solutions cache: ejde-124.pdf plain text: ejde-124.txt item: #49 of 601 id: ejde-1240 author: Qin, Jiali; Hao, Jianghao title: Asymptotic stability for thermodiffusion Timoshenko systems of type III date: 2025 words: 9114 flesch: 83 summary: Taking the derivative of L(t) with respect to t, using (3.2), (3.5), (3.7), (3.10), (3.14), (3.16) and (3.20), we have L′(t) ≤ − [ σ2N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) N4 − c ( 1 + 1 ε6 ) N5 − c ] ∫ 1 0 θ2xtdx − [ γ2N − cN1 − c ( 1 + 1 ε4 + 1 ε5 ) keywords: stability; system; timoshenko cache: ejde-1240.pdf plain text: ejde-1240.txt item: #50 of 601 id: ejde-1242 author: Feng, Meiqiang; Lu, Yichen title: Existence, uniqueness and multiplicity of nontrivial solutions for biharmonic equations date: 2025 words: 8094 flesch: 81 summary: Uniqueness of nontrivial solutions In this section, we use the following assumptions on f : (A1) f(x, u) Carathéodory conditions for x ∈ Ω and −∞ < u < +∞), and for fixed x ∈ Ω, f(x, u) is a decreasing function, that is f(x, u1) In this section, we suppose that f satisfies the following assumptions: (A3) f(x, u) satisfies Carathéodory conditions for x ∈ Ω and −∞ < u < +∞), and there exists 0 < σ ≤ N+4 N−4 such that |f(x, u)| ≤ a+ b|u|σ, a > 0, b > 0; (4.4) (A4) There exist 0 ≤ ξ < 1 2 and L > 0 such that F (x, u) = ∫ u 0 f(x, v)dv ≤ ξuf(x, u), ∀|u| ≥ L, x ∈ Ω; (4.5) (A5) keywords: f(x; h2(ω; solutions; ∩h1 cache: ejde-1242.pdf plain text: ejde-1242.txt item: #51 of 601 id: ejde-1245 author: Santos, Mauro L.; Freitas, Mirelson M.; Caljaro, Ronal Q. title: Long-time dynamics and upper-semicontinuity of attractors for a porous-elastic system with nonlinear localized damping date: 2025 words: 8383 flesch: 79 summary: [ ρ ∫ L 0 utuxhλ dx ]T 0 − [ J ∫ L 0 ϕtϕxhλ dx ]T 0 + ξ ∫ T 0 ∫ L 0 ϕ2h′λ dx dt+ b ∫ T 0 ∫ L 0 uxϕh ′ λ dx dt + ∫ T 0 ∫ keywords: attractors; system; − ∫ cache: ejde-1245.pdf plain text: ejde-1245.txt item: #52 of 601 id: ejde-125 author: Razani, Abdolrahman; Figueiredo, Giovany M. title: Weak solution by the sub-supersolution method for a nonlocal system involving Lebesgue generalized spaces date: 2022 words: 7313 flesch: 85 summary: = f2(x, u, v)|∇u|α2(x) Lq2(x) + g2(x, u, v)|∇u|γ2(x) Ls2(x) , with Dirichlet boundary condition, where Ω is a bounded domain in RN (N > 1) with C2 boundary. g1(x, u, v)|v|γ1(x) Ls1(x) in Ω, −A(x, |u|Lr2(x)) div ( a(|∇v|p2(x))|∇v|p2(x)−2∇v ) keywords: a(x cache: ejde-125.pdf plain text: ejde-125.txt item: #53 of 601 id: ejde-1250 author: Xu, Ling title: Asymptotic behavior of Kirchhoff type plate equations with nonlocal weak damping, anti-damping and subcritical nonlinearity date: 2025 words: 10466 flesch: 81 summary: + ∫ T 0 Im(t)dt ≤ C(R) {∫ T 0 ∥ιt(t)∥2dt+ k ∫ T 0 (∥wt∥pwt − ∥vt∥pvt, ιt)dt + k ∫ T 0 |(∥wt∥pwt − ∥vt∥pvt, ι)|dt+ ∫ T 0 ∥∇ι∥2dt+ ∫ T 0 dt ∫ T t ∥∇ι(τ)∥2dτ + ∫ T 0 dt ∫ T t ∥∇ι(τ)∥∥ιt(τ)∥dτ + ∣∣∣ ∫ T 0 (N (ιt), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (N (ιt), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (g(w)− g(v), ιt(τ))dτ ∣∣∣}, ∀T ≥ + ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ≤ C {∫ T 0 ∥ιt∥2dt+ k ∫ T 0 (∥wt∥pwt − ∥vt∥pvt, ιt)dt + k ∫ T 0 |(∥wt∥pwt − ∥vt∥pvt, ι)|dt+ ∣∣∣ ∫ T 0 ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt ∣∣∣ + ∣∣∣ ∫ T 0 ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)dt ∣∣∣+ ∣∣∣ ∫ T 0 m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ ∣∣∣ + ∣∣∣ ∫ T 0 (N (ιt), ιt)dt ∣∣∣+ ∣∣∣ ∫ T 0 (N (ιt), ι)dt ∣∣∣ + ∣∣∣ ∫ T 0 dt ∫ T t (N (ιt), ιt)dτ ∣∣∣+ ∣∣∣ ∫ T 0 (g(w)− g(v), ιt)dt ∣∣∣ + ∣∣∣ ∫ T 0 (g(w)− g(v), keywords: damping; inequality; t t; − ∫; ∣∣∣; ∥vt∥pvt; ∫ t; ∫ ω cache: ejde-1250.pdf plain text: ejde-1250.txt item: #54 of 601 id: ejde-126 author: Song, Yingwei; Zhang, Tie; Li, Jinpeng title: Dynamical behavior in a reaction-diffusion system with prey-taxis date: 2022 words: 5765 flesch: 75 summary: [(p(u) u v − p(u) u v∗ ) + (p(u) u v∗ − p(u∗) u∗ v∗ )]} dx + ∫ Ω [ Aσ(v − v∗) ( − v u + v∗ u − v∗ u + v∗ u∗ )] dx = ∫ Ω [ 1 p(u) ( gu(ξ)− v∗ d du (p(u) u ) Define a Lyapunov function E(t) = ∫ Ω (∫ u u∗ ξ − u∗ ξp(ξ) dξ +A ∫ v v∗ η − v∗ η dη ) keywords: prey cache: ejde-126.pdf plain text: ejde-126.txt item: #55 of 601 id: ejde-1261 author: Wang, Shuna; Liu, Jiang; Fang, Jun; Lin, Xiaojie title: Traveling waves of a diffusive modified Leslie-Gower model with chemotaxis date: 2025 words: 9017 flesch: 69 summary: Similarly, for r2V − U − k ≥ 0, if U ≥ U∗, we obtain r2V̇ − U̇ = −a1U(1− b1U − r1V ) = −a1U(1− b1U − r1 U + k r2 ) = a1U r2 r2 − r1k + k(r1 + b1r2) 1 + b1k r1 + b1r2 ] = −a1b1(r2 − r1k) r1 + b1r2 − a2 < 0. keywords: equilibrium; existence; leslie; model; r1+b1r2; r1k; subsystem cache: ejde-1261.pdf plain text: ejde-1261.txt item: #56 of 601 id: ejde-127 author: Ding, Hang; Zhou, Jun title: Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model date: 2022 words: 12073 flesch: 91 summary: − a‖u‖2 − b‖u‖4. dτ − 2(q + 1) (∫ t 0 (uτ , u) dτ )2 ≥ (q − 3)b 2C4 2 Q(t)(Q′(t))2 − (q + 1)‖u0‖22Q′(t)− 2(q + 1)J(u0)Q(t). (4.30) keywords: i(u0; j(u0; q+1; time; ‖u‖22 cache: ejde-127.pdf plain text: ejde-127.txt item: #57 of 601 id: ejde-1270 author: Conlon, Joseph G.; Dabkowski , Michael title: Properties of the Dirichlet Green's function for linear diffusions on a half line date: 2025 words: 31306 flesch: 88 summary: = −min τ>T τ − T 2τT [ y+ τx (T − τ) ]2 = − 1 2T [ −2xy+min α>1 {αx2+y2/α} ] . P ( τ∗ε,linear,x,T < νT )}] , (6.65) where we assume y ≥ C3T 2, εT ≤ y2, √ εδ = Λy. keywords: a(s; a(t; c3 t; c5 t; constant; function; solution; t f0(x; t s; t t; t τ; y t; τ∗ε cache: ejde-1270.pdf plain text: ejde-1270.txt item: #58 of 601 id: ejde-1273 author: Lee, Eun Kyoung; Sim, Inbo; Son, Byungjae title: Solutions to nonlinear elliptic problems with nonhomogeneous operators and mixed nonlocal boundary conditions date: 2025 words: 9167 flesch: 85 summary: Let σ ∈ (0, 1] and u ∈ C[0, 1] be a solution of u = σTλu and ∥u∥∞ ≥ rλ. Let τ ≥ 0 and u ∈ C[0, 1] be a solution of u = Tλu + τ . keywords: solution cache: ejde-1273.pdf plain text: ejde-1273.txt item: #59 of 601 id: ejde-1276 author: Kim, Yong-Cheol title: Holder regularity of weak solutions to nonlocal p-Laplacian type Schrodinger  equations with A_1^p-Muckenhoupt potentials date: 2025 words: 12738 flesch: 86 summary: (2.8) EJDE-2025/83 HÖLDER REGULARITY OF WEAK SOLUTIONS 7 For g ∈ W s,p(Rn), we consider the convex subsets of Xs,p(Ω) defined by Xs,p g (Ω)± = {v ∈ Xs,p(Ω) : (g − v)± ∈ Xs,p 0 (Ω)}, Xs,p g (Ω) := Xs,p g (Ω)+ ∩Xs,p g (Ω)− = {v ∈ Xs,p(Ω) : g − v ∈ Xs,p 0 (Ω)}. If the nonlocal equation mentioned just in the above is considered for 0 < s < 1, p ≥ 2 and a bounded domain Ω ⊂ Rn with C1,1 boundary, then they also established the first fine boundary regularity for its weak solutions in [18], i.e. there exist some α ∈ (0, s] and C > 0 depending only on n, p, s and Ω such that∥∥∥ u dsΩ ∥∥∥ Cα(Ω) ≤ C∥f∥ 1 p−1 L∞(Ω) for any weak solution u ∈ W s,p 0 (Ω) of the nonlocal equation, where dΩ(x) = dist(x, ∂Ω). keywords: p q; p−1; schrödinger; y s cache: ejde-1276.pdf plain text: ejde-1276.txt item: #60 of 601 id: ejde-1278 author: Huzak, Renato; Mardesic, Pavao; Resman, Maja; Zupanovic, Vesna title: Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits date: 2025 words: 11786 flesch: 76 summary: We consider an analytic germ of a system dx dt = F (x, ν), (1.1) with F real, analytic germ in x and in parameter ν, and with a non-hyperbolic singular point x = 0 at the bifurcation value ν = 0 (i.e. F (0, 0) = 0, Fx(0, 0) We use the name compensator for elementary expressions in variable x and parameter ν, i.e. expressions that cannot be further asymptotically expanded uniformly in ν. keywords: analytic; expansion; function; h(ν; point cache: ejde-1278.pdf plain text: ejde-1278.txt item: #61 of 601 id: ejde-128 author: Mouhcine, Zakariyae title: Resolvent kernel on H-type groups and a Green kernel for fractional powers of its sub-Laplacian date: 2022 words: 4286 flesch: 82 summary: = (ζ − L)−1 and the heat T (s) = esL operators [7, p.56] R(ζ,L) = ∫ ∞ 0 e−ζsT (s) ds, to find the resolvent kernel associated with the sub-Laplacian L. × Rm with the group law (x, u) · (y, v) = ( x+ y, u+ v + 1 2 〈x, Uy〉 ) , with x = (x1, . . . keywords: groups; kernel cache: ejde-128.pdf plain text: ejde-128.txt item: #62 of 601 id: ejde-1282 author: Chen, Xiao; Zhou, Wenxue title: Three-point integral boundary-value problems for piecewise fractional impulsivedifferential equations with p-Laplacian operator date: 2025 words: 8449 flesch: 82 summary: The significance of studying fractional impulsive differential equations lies in their extension of classical differential equation theory, their ability to reveal new characteristics of complex systems, their capacity to provide precise models for practical problems, and their promotion of innovation in related mathematical methodologies. [9] Z. Bai; Theory and application of fractional differential equation boundary value problem, Beijing: China Science and Technology Press, 2012. keywords: fractional; i=1; ti−1; γ(α; ψ(1; ψ(s cache: ejde-1282.pdf plain text: ejde-1282.txt item: #63 of 601 id: ejde-1285 author: Herron, Sigifredo; Lopera, Emer ; Sanchez, Diana title: Existence of three positive solutions for a p-sublinear problem involving a Schrodinger p-Laplacian type operator date: 2025 words: 6533 flesch: 83 summary: We prove the existence of three positive solutions for the problem −∆pu+ V (x)φp(u) = λf(u), x ∈ Ω, u(x) The case Ω = BR In this section we prove Theorem 1.3. keywords: problem; solutions; theorem cache: ejde-1285.pdf plain text: ejde-1285.txt item: #64 of 601 id: ejde-129 author: Tordecilla, Jesus Alberto Leon title: Existence of solutions to nonlocal elliptic problems with singular and combined nonlinearities date: 2022 words: 6514 flesch: 81 summary: m0 ≤M(‖vn‖2H1 0 (Ω) ) ≤ m∞ and since vn > 0, then by taking µ = λm−1 ∞ we find that −∆vn ≥ µvqn, x ∈ Ω, vn > 0, x ∈ Ω, vn = 0, x ∈ ∂Ω. Thus, by defining zn = µ 1 1−q vn we deduce that −∆ ( zn µ 1 1−q ) we mean a function u ∈ H1 0 (Ω) such that u > 0 in Ω and −M (∫ Ω |∇u|2 )∫ Ω ∇u∇φ = λ ∫ Ω (a(x)u−γ + uq)φ+ ∫ Ω f(u)φ = 0 for all φ ∈ H1 0 (Ω). keywords: problem cache: ejde-129.pdf plain text: ejde-129.txt item: #65 of 601 id: ejde-1293 author: Chen, Zilin; Yang, Yang title: Normalized solutions for fractional Schrodinger-Choquard systems with Sobolev critical coupled nonlinearity date: 2025 words: 11716 flesch: 85 summary: If N > 4s, it is easy to check that N + α + p(2sδp,s − N) < They mainly focused on the L2-subcritical case, and then obtained the existence of normalized positive ground state solution for any 0 < β < β0. keywords: critical; lemma cache: ejde-1293.pdf plain text: ejde-1293.txt item: #66 of 601 id: ejde-1298 author: Hong, Hakho title: Local solutions for a Brinkman equation coupled with heat-convective and concentration-diffusive equations and a volumetric mass source date: 2025 words: 6223 flesch: 78 summary: Assume (1.3), (1.4) and Q0(w̄, θ̄) = 0, S(w̄, θ̄) = 0, Q1(w̄, θ̄) = 0. (1.9) 4 H. HONG EJDE-2025/23 Suppose that the initial data ρ0,u0, φi0 satisfy (ρ0 − ρ̄,u0,w0 − w̄, θ0 − θ̄) ∈ HN (R3), inf x∈R3 ρ0(x) > 0, inf x∈R3 w0(x) > 0, inf x∈R3 θ0(x) > 0 (1.10) for an integer N ≥ 3. (1.16) Setting φ = ρ− ρ̄, m = w − w̄, ζ = θ − θ̄, and using assumption (1.9), we rewrite system (1.1)1, (1.15)1, (1.1)3, (1.16) as follows: φt + ρ̄divu+ u · ∇φ−∇wQ0(w̄, θ̄) ·m−Q′ 0θ(w̄, θ̄)ζ = G1(φ,u,m, ζ), ut − µ ρ̄ ∆u− µ+ λ ρ̄ ∇divu+ 1 αρ̄ u+ Pρ(ρ̄, θ̄; w̄) ρ̄ ∇φ + Pθ(ρ̄, θ̄; w̄) ρ̄ ∇ζ + n∑ i=1 Pwi(ρ̄, θ̄; w̄) ρ̄ ∇mi = G2(φ,u,m, ζ), mt − df∆m+ w̄ divu−DwS(w̄, θ̄)m− S′ θ(w̄, θ̄)ζ = G3(φ,u,m, ζ), ζt + θ̄Pθ(ρ̄, θ̄; w̄) ρ̄eθ(ρ̄, θ̄) divu = κ ρ̄eθ(ρ̄, θ̄) ∆ζ + ∇wQ1(w̄, θ̄) ·m+Q′ 1θ(w̄, θ̄)ζ ρ̄eθ(ρ̄, θ̄) + e(ρ̄, θ̄) ρ̄eθ(ρ̄, θ̄) ( ∇wQ0(w̄, θ̄) ·m+Q′ 0θ(w̄, θ̄)ζ ) keywords: brinkman; system cache: ejde-1298.pdf plain text: ejde-1298.txt item: #67 of 601 id: ejde-1299 author: Bensalem, Abdelhamid; Salim, Abdelkrim; Benchohra, Mouffak; N'Guerekata, Gaston M. title: Optimal control and  approximate controllability for second-order integro-differential equations with state-dependent delay and non-instantaneous impulses date: 2025 words: 8625 flesch: 79 summary: if θ ∈ Ik, k ∈ Nm 0 , ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(θ) = Θk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(0) = ζ0 ∈ H, ϑ′(0) = ζ1 ∈ H. The concept of controllability has long been recognized as having a significant role in engi- neering and mathematical control theory. + ∫ θ 0 Υ(θ, ε)ϑ(ε)dε+K(θ, ϑℑ(θ,ϑθ), (Ψϑ)(θ)) + Pu(θ), if θ ∈ Ik, k ∈ Nm 0 , ϑ(θ) = Υk(θ, ϑ(θ − k )), if θ ∈ Jk, k ∈ Nm 1 , ϑ′(θ) = Θk(θ, ϑ(θ − k )), keywords: differential; equations; q(θ; θ ∈; ϑ(θ cache: ejde-1299.pdf plain text: ejde-1299.txt item: #68 of 601 id: ejde-130 author: Hasil, Petr; Sisolakova, Jirina; Vesely, Michal title: Oscillation of modified Euler type half-linear differential equations via averaging technique date: 2022 words: 6933 flesch: 78 summary: = 0, (3.1) lim t→∞ f(t)g2(t) t log t = 0, (3.2) lim t→∞ f(t)g(t) t = 0, (3.3) ginf := lim inf t→∞ g(t) > 0. (3.4) Especially, (3.2) and (3.4) give lim t→∞ f(t)g(t) t log t = 0, (3.5) 6 P. HASIL, J. ŠIŠOLÁKOVÁ, M. VESELÝ EJDE-2022/41 i.e., lim t→∞ t log t f(t)g(t) =∞. (3.6) − 1 f(t) ∫ t+f(t) t r(τ)| cosp ϕ(τ)|p dτ ∣∣∣ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t r(τ) ∣∣ |cosp (ave[ϕ, f ](t))|p − | cosp ϕ(τ)|p ∣∣dτ ≤ lim sup t→∞ 1 f(t) ∫ t+f(t) t r(τ)C |ave[ϕ, f ](t)− ϕ(τ)|dτ ≤ lim sup t→∞ C ( t log t f(t)g(t) max τ∈[t,t+f(t)] |ϕ(τ)− ave[ϕ, f ](t)| ) × f(t)g2(t) t log t · 1 f(t)g(t) ∫ t+f(t) t r(τ) dτ = lim sup t→∞ C ·∆ (ϕ, ave[ϕ, f ]) · f(t)g2(t) t log t · r[f, g], i.e., lim t→∞ ∣∣∣ |cosp (ave[ϕ, f ](t))|p ave[r, f ](t) keywords: equations; lim; linear; log t; t t cache: ejde-130.pdf plain text: ejde-130.txt item: #69 of 601 id: ejde-131 author: Chang, Caihong; Zhang, Zhengce title: Asymptotic behavior of blowup solutions for Henon type parabolic equations with exponential nonlinearity date: 2023 words: 8152 flesch: 82 summary: If N > 10 + 4σ, Ω(ξ) = C1ξ q+ + C2ξ q− , (3.8) where C1, C2 > 0, and q± = 1 2 [ 4−N ± √ (N − 2)(N − 10− 4σ) ] , which are the roots of the quadratic equation q2 + (N − 4)q + (Nσ − 2σ +N − 1) = 0. = (2 + σ)λ1 |γ+| t+O(1), t→∞, (1.11) where γ+ = 1 2 [ 2−N + √ (N − 2)(N − 10− 4σ) ] < 0, and λ1 is defined in Lemma 2.5 as the first eigenvalue of an associated linearized problem. keywords: 2+σ; equations; log; solutions cache: ejde-131.pdf plain text: ejde-131.txt item: #70 of 601 id: ejde-132 author: Alves, Maria Jose; Assuncao, Ronaldo B. title: Existence of solutions for a problem with multiple singular weighted p-Laplacians and vanishing potentials date: 2022 words: 9457 flesch: 81 summary: The Euler-Lagrange energy functional I : E → R associated with problem (1.1) is defined by I(u) := 1 p ∫ RN |∇u|p |x|ap dx+ 1 p ∫ RN P (x)|u|p |x|ap∗(a,b) dx + 1 q ∫ RN |∇u|q |x|cq dx+ 1 q ∫ RN Q(x)|u|q |x|cq∗(c,d) dx− ∫ RN F (u) dx. + 1 q ‖u‖q1,q − c0 θ ∫ RN |u|p∗(a,b) |x|bp∗(a,b) dx− 1 kp ‖u‖p1,p = (1 p − 1 kp ) ‖u‖p1,p keywords: problem; rn p; |x|ap; |x|ap∗(a; |x|cq; |x|cq dx; |x|cq∗(c; ∫ rn; ∫ |x|>r cache: ejde-132.pdf plain text: ejde-132.txt item: #71 of 601 id: ejde-133 author: Sapagovas, Mifodijus; Novickij, Jurij; Ciupaila, Regimantas title: Stability analysis of the Peaceman-Rachford method for parabolic equations with nonlocal conditions date: 2022 words: 6473 flesch: 69 summary: The main task of this article is to construct efficient FDM for the two-dimensional parabolic equation (1.1) with nonlocal boundary condition (1.2). We consider an efficient finite difference method solving of two- dimensional parabolic equations with nonlocal conditions. keywords: difference; matrix; method; n+1/2 cache: ejde-133.pdf plain text: ejde-133.txt item: #72 of 601 id: ejde-1332 author: Nam, Bui Duc; Nghia, Bui Dai; Tuan, Nguyen Anh title: Mild solutions to Love-type equations on R^2 date: 2025 words: 13038 flesch: 85 summary: Let s ≤ θ and η, d ≥ 0 such that 0 ≤ d + s − θ ≤ η ≤ d. Suppose that G(0) = 0 and ∥G(w1)−G(w2)∥Hη(R2) ≤ C∥w1 − w2∥Hd(R2), for all w1, w2 ∈ Hd(R2). = 1 2π ∫∫ R2 cos (√ ξ2 + η2 1 + k(ξ2 + η2) t ) â(ξ, η)eixξ+iyη dξ dη. keywords: equation; k(ξ2; solution; η)|2; η2)s; η2)s t cache: ejde-1332.pdf plain text: ejde-1332.txt item: #73 of 601 id: ejde-134 author: Kostic, Marko title: Multi-dimensional c-almost periodic type functions and applications date: 2022 words: 11632 flesch: 75 summary: We will always assume henceforth that BX = X, i.e., that for each x ∈ X there exists B ∈ B such that x ∈ B. Further on, there exists B ∈ B such that x ∈ B keywords: bohr; c)-almost; function; periodic; recurrent cache: ejde-134.pdf plain text: ejde-134.txt item: #74 of 601 id: ejde-1344 author: Zhou, Yao; Liu, Hongliang title: Upper semicontinuity of uniform attractors for singular perturbed second order nonautonomous delay lattice systems date: 2025 words: 10893 flesch: 82 summary: + ∫ t t−ϑ ∥u̇ϵ(r)∥2dr ≤ ϑK̃3(ϵ, q1), t ∈ R. EJDE-2025/64 ATTRACTORS FOR DELAY LATTICE SYSTEMS 15 Then for t ∈ R, ϵ ∈ (0, ϵ], we have∫ t+1 t ∥u̇ϵ(r)∥2dr ≤ K̃3(ϵ, q1) + ϵ∥u̇ϵ(t)∥2 ≤ K̃3(ϵ, q1) + C1(q1) +M1 . In view of [3, 4, 27], for any g ∈ H(g0), t ∈ R, {Ag 0,t}t∈R is the pullback attractor of {Ug0 (t, τ)}t≥τ and Ag 0,t = {ut|{ut(·), t ∈ R} is a complete bounded trajectory of {Ug0 (t, τ)}t≥τ} = ∩r≥0∪s≥rUg0 (t, t− s)B0 ⊂ B0 ⊂ ℓ2ϑ. that is, for all g ∈ H(g0), t ∈ R, Ag 0,t is compact in ℓ2ϑ; for all t ≥ τ ∈ R, Ug0 (t, τ)A g 0,τ = Ag0,t; for all B ⊂ B(ℓ2ϑ), lims→+∞ dh(U g 0 (t, t − s)B,Ag0,t) keywords: attractors; lattice; sup; systems cache: ejde-1344.pdf plain text: ejde-1344.txt item: #75 of 601 id: ejde-1351 author: Zhang, Zujin; Yuan, Weijun; Yao, Zhengan title: Stability of Leray weak solutions to 3D Navier-Stokes equations date: 2025 words: 10251 flesch: 81 summary: [(u · ∇)u] · vn} dxdτ = ∫ t 0 ∫ R3 u · ∂τvndxdτ + ∫ R3 u0 · vn(0)dx+ ∫ t 0 ∫ R3 f · vndxdτ, as well as ∫ R3 v(t) · un(t)dx+ ∫ t 0 ∫ R3 {∇v : ∇un + = ∫ t 0 ηn(|τ − σ|)u(σ)dσ, vn(τ) = ∫ t 0 ηn(|τ − σ|)v(σ)dσ (0 ≤ τ ≤ t). then un,vn ∈ C1((0, t); Ḣ1(R3)), and we may test (1.1)1 and (1.3)1 by vn and un respectively, and obtain ∫ R3 u(t) · vn(t)dx+ ∫ t 0 ∫ R3 {∇u : ∇vn + keywords: ḃ0; c ∫; t s; theorem; ∫ r3; ∫ t cache: ejde-1351.pdf plain text: ejde-1351.txt item: #76 of 601 id: ejde-136 author: Qu, Siqi; He, Xiaoming title: Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency date: 2022 words: 8426 flesch: 85 summary: In this article we study the fractional Schrödinger-Poisson system ε2s(−∆)su+ V (x)u = φ|u|2 ∗ s−3u, x ∈ R3, (−∆)sφ = |u|2 ∗ s−1, x ∈ R3, where s ∈ (1/2, 1), ε > 0 is a parameter, 2∗s = 6/(3−2s) is the critical Sobolev exponent, V ∈ L 3 2s (R3) is a nonnegative function which may be zero in some region of R3. Introduction In the past decades, the nonlinear Schrödinger-Poisson system −∆u+ V (x)u+K(x)φu = f(x, u), x ∈ R3, −∆φ = K(x)u2, x ∈ R3, (1.1) has been the interesting object for many researcher. keywords: ds,2(r3; fractional; on(1; poisson; schrödinger cache: ejde-136.pdf plain text: ejde-136.txt item: #77 of 601 id: ejde-1361 author: Zhang, Lei; Liu, Lintao; Chen, Haibo title: Minimizers for fractional Schrodinger equations with inhomogeneous perturbation date: 2025 words: 9508 flesch: 84 summary: M − C5A p+1 M α̃2s M α̃ (t+1)[N−(N+2s)(p+1)] M + 4s N(p− 1) Ap+1 M α̃2s M a∗ (M a∗ ) M a∗ ∫ RN |ξ|s|Q̌(ξ)|2dξ ≤ C 1 + C̃α̃ −(t+1)(N+4s) M a∗ α̃ s(1−t) M ∫ RN (1 + |ξ|2s)|Q̌(ξ)|2dξ ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M (3.24) EJDE-2025/59 MINIMIZERS FOR FRACTIONAL SCHRÖDINGER EQUATIONS 11 as M → ∞, where Q̌ denote the Fourier transform of Q. By the Hölder inequality, (1.7), (3.18) and (3.20), we obtain that |T4| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2 ≤ C A2 M α̃ 2s M a∗ ( C3α̃ −2s(t+1) M ∫ RN Q2(x)dx )1/2(∫ RN |(−∆)s/2Q|2dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s(1−t) M as M → ∞. (3.25) From the Hölder inequality, (1.7), (3.18), (3.20) and (3.21), it follows that |T5| ≤ C A2 M α̃ 2s M a∗ (∫ RN Q2(x)|(−∆)s/2φ(α̃−t−1 M x)|2dx )1/2 × (∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ −s(t+1) M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 1 2− 3 2 t) M asM → ∞. (3.26) Similarly, |T6| ≤ C A2 M α̃ 2s M a∗ (∫ RN φ2(α̃−t−1 M x)|(−∆)s/2Q|2dx )1/2(∫ RN B2(φ(α̃−t−1 M x), Q(x))dx )1/2 ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃2s M α̃ − s(t+1) 2 M ≤ C(1 + C̃α̃ −(t+1)(N+4s) M )α̃ s( 3 2− 1 2 t) M asM → ∞. (3.27) keywords: fractional; lemma; m p−1; p−1 cache: ejde-1361.pdf plain text: ejde-1361.txt item: #78 of 601 id: ejde-1369 author: Shen, Liejun; Squassin, Marco title: Concentrating normalized solutions for 2D nonlocal Schrodinger equations with critical exponential growth date: 2025 words: 13209 flesch: 81 summary: In the spirit of [34], when W̄ (x) ≡ 0 for all x ∈ RN in (1.5), the authors in [39] deduced the existence of nontrivial solutions solutions to the nonlocal problem of Choquard type −∆u+ ωu = = |s|p−2s for all s ∈ R, (1.5) is of the form −∆u+ u = (|x|−µ ∗ |u|p)|u|p−2u, x ∈ RN . keywords: equations; lemma; proof; r(a; solutions; theorem cache: ejde-1369.pdf plain text: ejde-1369.txt item: #79 of 601 id: ejde-137 author: Lagha, Aesha; Hattori, Hattori title: Cauchy problems for chemotaxis systems with chemo attractant and repellent date: 2022 words: 7766 flesch: 81 summary: There exists a positive number ε0 such that if ‖(n0 − n∞, u0, c1,0, c2,0)‖N ≤ ε0, the Cauchy problem (1.3)-(1.4) has a unique solution (n, u, c1, c2)(t) globally in time which satisfies (n− n∞, u)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−1(R3)), (c1, c2)(t) ∈ C([0,∞);HN (R3)) ∩ C1([0,∞);HN−2(R3)) and there are constants λ1 > 0, λ2 > 0, λ3 > 0 and C0 > 0 such that ‖(n− n∞, u, c1, c2)‖2N + λ1 ∫ t 0 ‖∇(n− n∞)‖2N−1 + λ2 ∫ t 0 ‖∇(c1, c2)‖2N + λ3 ∫ t 0 ‖(u, c1, c2)‖2N ≤ C0‖(n0 − n∞, u0, c1,0, c2,0)‖2HN . C‖∂αρ0‖+ C‖ρ‖N ∫ t 0 (‖∂αu‖2 + ‖∂αρ‖2)ds + C‖u‖N ∫ t 0 ‖∂αρ‖2ds+ C‖u‖N ∫ t 0 ‖∂αu‖2ds + C‖c1‖N ∫ t 0 (‖∂αu‖2 + ‖∂α∇c1‖2)ds + C‖c2‖N ∫ t 0 (‖∂αu‖2 + ‖∂α∇c2‖2)ds. keywords: cauchy; chemotaxis; time; u(t cache: ejde-137.pdf plain text: ejde-137.txt item: #80 of 601 id: ejde-138 author: Hao, Yu-Cai; Zhang, Guo-Bao title: Stability of bistable traveling wavefronts for a nonlocal dispersal epidemic system date: 2022 words: 9106 flesch: 88 summary: By Lemma 3.1 and the comparison principle, one has max { φi(η −(x, t))− δpie−β0t, 0 } ≤ ui(x, t;ϕi) ≤ min { φi(η +(x, t)) + δpie −β0t, ki } , i = 1, 2, (3.4) where x ∈ R, t ≥ 0, and η±(x, t) : Hence, combining with (3.3) and (3.4), we obtain φi(x+ ct)− ε 2 ≤ ui(x, t;ϕi) ≤ φi(x+ ct) + ε 2 , ∀x ∈ R, t ≥ 0, i = 1, 2, which implies that ‖u(·, t;ϕ)− φ(·+ ct)‖ < ε, ∀t ≥ 0. keywords: stability; system; u(x; wavefronts cache: ejde-138.pdf plain text: ejde-138.txt item: #81 of 601 id: ejde-1383 author: Llibre, Jaume; Valls, Claudia title: Global asymptotic stability in quadratic systems date: 2025 words: 6762 flesch: 75 summary: c < (−a01 + b10)/a10 + 2 √ −(a210 + a01b10)/a210, (4) a01 > 0, −2a01 < a10 < 0, −a01 + 2a10 − 2 √ −2a01a10 − a210 < b10 < −a01 + 2a10 + 2 √ −2a01a10 − a210, and (b10 − a01)/a10 − 2 √ −(a210 + a01b10)/a210 < 2)XY − d(b2 − bcd+ d2)Y 2, Ẏ = −(a2b10 − a(a10 − b01)c1 − a01c 2 1)X + (c1(a10b+ a01d) − a(bb10 + b01d))Y + c1(a 2 − acc1 + c21)X 2 + (a2d+ 2c21d+ ac1(b− 2cd))XY + d(c1d+ a(b− cd))Y 2. (3.13) We first impose that system (3.13) can be written in the form (a) of Theorem 2.4, that is, that satisfies abb10 + bb01c1 − aa10d− a01c1d = c1(ab− bcc1 + c1d) keywords: a01; a10; b01; condition; equilibrium; system; theorem cache: ejde-1383.pdf plain text: ejde-1383.txt item: #82 of 601 id: ejde-1385 author: Zheng, Feng-Xia; Li, Hong-Xu title: Almost automorphic solutions to non-autonomous dynamic equations with Stepanov-like almost automorphic forcing terms on time scales date: 2025 words: 7978 flesch: 85 summary: ∈ K for t ∈ T. By Lemma 3.2, we have f ∈ SpAAK(T×X,Y ), and a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and a function f̃ such that lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f(s+ ξn, x)− f̃(s, x)∥p∆s )1/p = 0 for t ∈ T, lim n→∞ ( 1 K ∫ [t,t+K)T sup x∈K ∥f̃(s− ξn, x)− f(s, x)∥p∆s )1/p = 0 for t ∈ T. (3.9) and g is Sp-a.a., for any sequence {ξ′n}∞n=1 ⊂ Π, there exist a subsequence {ξn}∞n=1 of {ξ′n}∞n=1 and two functions f̃ , g̃ such that lim n→∞ f(t+ ξn) = f̃(t) for t ∈ T, lim n→∞ f̃(t− ξn) = f(t) for t ∈ T, (2.2) lim n→∞ ( 1 K ∫ [t,t+K)T ∥g(s+ ξn)− g̃(s)∥p∆s )1/p = 0 for t ∈ T, lim n→∞ ( 1 K ∫ [t,t+K)T ∥g̃(s− ξn)− g(s)∥p∆s )1/p = 0 for t ∈ T. (2.3) keywords: t+k)t; time cache: ejde-1385.pdf plain text: ejde-1385.txt item: #83 of 601 id: ejde-1388 author: Zhang, Mingbo title: Approximations of Euler-Peano scheme for reflected stochastic differential equations with non-Lipschitz coefficients date: 2025 words: 10661 flesch: 83 summary: = ∫ t 0 b(s, x(s, x0))ds+ ∫ t 0 σ(s, x(s, x0))dB(s) + ϕ(t, x0), t ≤ T. The uniqueness and continuous dependence with respect to x0 of the solution of equation (1.1) will be presented in Theorem 3.10. In this article we are concerned with the reflected stochastic differential equations (RSDEs) x(t) = x0 + ∫ t 0 σ(s, x(s))dB(s) + ∫ t 0 b(s, x(s))ds+ ϕ(t), x0 ∈ D̄, ϕ(t) keywords: solution; stochastic; sup; |x|≤r; σ(s; ∫ t cache: ejde-1388.pdf plain text: ejde-1388.txt item: #84 of 601 id: ejde-139 author: Alvarez-Caudevilla, Pablo title: Asymptotic behavior of cooperative systems involving p-Laplacian operators date: 2022 words: 8884 flesch: 70 summary: Moreover, Ω is a smooth bounded domain of RN , N ≥ 1, with smooth boundary ∂Ω, for example of class C2 or Lipschitz. and we denote the open sets/subdomains of Ω where the potentials a and d vanish, as Ωa0 := {x ∈ Ω : a(x) keywords: 1,p; eigenvalue; problem; − ∫ cache: ejde-139.pdf plain text: ejde-139.txt item: #85 of 601 id: ejde-1395 author: El Idrissi, Ahmed; Srhiri, Halima; El Boukari, Brahim; El Ghordaf, Jalila title: Local and global solvability of fractional porous medium equations in critical Besov-Morrey spaces date: 2025 words: 8159 flesch: 76 summary: In fact, if we have p = h, then Ṅ s p,p,r = Ḃs p,r. 3. It is important to note that replacing the Lp-norm by the Mp h-norm is not sufficient to ensure a direct transition from Besov spaces to Besov-Morrey spaces. keywords: besov; p p; spaces; −2m+n p cache: ejde-1395.pdf plain text: ejde-1395.txt item: #86 of 601 id: ejde-140 author: Wang, Ru; Chang, Xiaojun title: Existence of global solutions and blow-up for p-Laplacian parabolic equations with logarithmic nonlinearity on metric graphs date: 2022 words: 7409 flesch: 83 summary: Integrating on (0, t) with t ∈ (0, T ), by 1 < p < 2 it follows that 1 2 ∫ G |w(t)|2dx ≤ ∫ t 0 ∫ G ((p− 1) log |w̃(s)|+ 1)|w̃(s)|p−2|w(s)|2 dx ds ≤ C ∫ t 0 ∫ G |w(s)|2 dx ds for some constant C > 0 independent of u1(t) and u2(t). A function u := u(x, t) is called a weak solution of problem (1.1) on G × (0, T∗), if u ∈ L∞(0, T∗;W 1,p(G)) with ut ∈ L2(0, T∗;L 2(G)) satisfies (1.1) in the distribution sense, i.e.,∫ G utvdx+ ∫ G |u′|p−2u′v′dx+ ∫ G |u|p−2uvdx = ∫ G |u|p−2uv log |u|dx, for all v ∈W 1,p(G), a.e. t ∈ (0, T∗), where u(x, 0) = u0(x) ∈ X0. 4 R. WANG, X. J. CHANG EJDE-2022/51 Our first result is about the existence of a local solution. keywords: equations; lemma; log; logarithmic cache: ejde-140.pdf plain text: ejde-140.txt item: #87 of 601 id: ejde-1400 author: Begout, Pascal; Diaz, Jesus Ildefonso title: Title: Solutions with expanding compact support of saturated Schrodinger equations: self-similar solutions date: 2025 words: 8217 flesch: 81 summary: If ∥F∥L2(RN ) ≤ δ and ∥F∥L∞(Kc) ≤ 1 M , then there exists a self-similar solution (u, U) to (1.2) such that u ∈ C ( (0,∞);H2(RN ) ) ∩ C1 ( (0,∞);H1(RN ) ) ∩ C2 ( (0,∞);L2(RN ) ) (2.10) and for any t > 0, suppu(t) is compact. On the other hand, if for some 0 < q ≤ ∞, u ∈ C ( (0,∞);Lq(RN ) ) then φ ∈ Lq(RN ) and it follows from (2.5) that ∀t > 0, ∥u(t)∥Lq(RN ) keywords: solution cache: ejde-1400.pdf plain text: ejde-1400.txt item: #88 of 601 id: ejde-1401 author: Molica Bisci, Giovanni; De Lima, Henrique; Leite, Ary V. F. ARY V.F.; Velasquez, Marco A. L. title: Complete noncompact and stochastically complete m-quasi Yamabe gradient solitons date: 2025 words: 4982 flesch: 75 summary: = 1 2 ∆|∇u|2 ≥ |∇2u|2 = n m2 (R− ρ)2u2 ≥ 0, implying that R = ρ. □ Proceeding, we will deal with complete noncompact m-quasi Yamabe gradient solitons having exponential volume growth. Now, we are in a position to present our first characterization result related to complete non- compact m-quasi Yamabe gradient soliton. keywords: gradient; quasi; riemannian; soliton; yamabe cache: ejde-1401.pdf plain text: ejde-1401.txt item: #89 of 601 id: ejde-141 author: Diaz, Jesus Ildefonso; Shaposhnikova, Tatiana A.; Zubova, Maria N. title: A strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shape date: 2022 words: 14087 flesch: 76 summary: = −C0g(x, t) + C0∂tφ(x, t), y ∈ ∂G0, t ∈ (0, T ), wφ(x, y, 0) = φ(x, 0), y ∈ ∂G0, wφ → 0, |y| → ∞. (6.13) We consider the sequence of solutions {wφ,R} to the problems ∆ywφ,R = 0, y ∈ T 0 R \G0, t ∈ (0, T ), C0∂twφ,R + ∂νwφ,R − C0σ(φ− wφ,R) Using the maximum principle, we derive the estimate |wφ,R| ≤ K0R0 |y|n−2 , y ∈ T 0 R \G0, t ∈ (0, T ). keywords: l2(0; operator; problem; solution; t 0; t j; ε−γ ∫; ∫ t cache: ejde-141.pdf plain text: ejde-141.txt item: #90 of 601 id: ejde-1411 author: Wang, Tao; Tian, Xiaoyu; He, Wenling title: New type of multi-bump solutions for Schrodinger-Poisson systems date: 2025 words: 6587 flesch: 88 summary: U pe−x1 ; • H1(R3) is the usual Sobolev space endowed with inner product (u, v) = ∫ R3(∇u∇v+uv)dx and norm ∥u∥2 = ∫ R3(|∇u|2 + u2)dx; • D1,2(R3) is the completion of C∞ 0 (R3) with respect to the norm ∥u∥2D1,2 = ∫ R3 |∇u|2dx; • Hk and Dk are symmetric Sobolev subspaces defined by Hk By using Hölder inequality and Sobolev inequality, we obtain ∥Φu∥2D1,2 = ∫ R3 Φuu 2dx ≤ keywords: solutions; |p+ cache: ejde-1411.pdf plain text: ejde-1411.txt item: #91 of 601 id: ejde-142 author: de Paiva, Francisco Odair; Lima, Sandra Machado de Souza; Miyagaki, Olimpio Hiroshi title: Existence of at least four solutions for Schrodinger equations with magnetic potential involving and sign-changing weight function date: 2023 words: 6593 flesch: 85 summary: [15] Gidas, B.; Nirenberg, L.; Symmetry of positive solutions of nonlinear elliptic equations in RN . [21] Kwong, M. K.; Uniqueness of positive solutions of ∆u − u + up = 0 in Rn, Archive for Rational Mechanics and Analysis, 105.3 (1989): 243-266. keywords: problem; solutions cache: ejde-142.pdf plain text: ejde-142.txt item: #92 of 601 id: ejde-1420 author: Yang, Xin; Zhou, Jueliang title: Classification of boundary-equilibria for two-dimensional continuous piecewise linear systems with two intersecting switching lines date: 2025 words: 12965 flesch: 78 summary: −a3/a2 > 0, −(a1 − a4)/a2 > 0 the solution on the x-axis (j) −a3/a2 = 0, −(a1 − a4)/a2 = 0 the solution outside Q1 (k) −a3/a2 > 0, −(a1 − a4)/a2 < 0 the solution on the y-axis (l) −a2/a3 = 0, (a1 − a4)/a3 = 0 D1 1 = ( −2a2, a1 − a4 − √ (a1 − a4)2 + 4a2a3 ) and D2 1 = ( −2a2, a1 − a4 + √ (a1 − a4)2 + 4a2a3 ) . When the separatrix with the characteristic direction D2 1 is located in Q1 except the x-axis (resp. on the x-axis, outside Q1), we obtain H2 1 := −2a2 ( a1 − a4 + √ (a1 − a4)2 + 4a2a3 ) > (resp. keywords: resp; κ ∈ cache: ejde-1420.pdf plain text: ejde-1420.txt item: #93 of 601 id: ejde-143 author: Wu, Yingzhu; Yu, Yuanhong; Xiao , Jinsen title: Oscillation for second order nonlinear differential equations with a sub-linear neutral term date: 2022 words: 4962 flesch: 81 summary: Then a1/γ(t)(−z′(t)) is an increasing function and thus z′(s) ≤ ( a(t) a(s) )1/γz′(t), s ≥ t ≥ t2. > 0 for all t ≥ t1. keywords: differential; oscillation cache: ejde-143.pdf plain text: ejde-143.txt item: #94 of 601 id: ejde-1438 author: Zheng, Yanzhi; Yin, Jingxue; Ji, Shanming title: Complete classification of self-similar solutions for singular polytropic filtration equations date: 2025 words: 19255 flesch: 81 summary: ≤ −m1−p ( a−M − N − α N (a+M) ) = 2m(1− 2m)(mα− δ + 3mδ) 3(1−m)(mα− δ + 2mδ)2 · C2 2,0u(α) v(α∗) , C2,1 = − (1−m)(1− 2m) 2m3(p− 1) δ2−pℓ−1+ 1 m−p = − 2m2(1− 2m) (1−m)(mα− δ + 2mδ)2 C2 2,0u 2(α) v(α) , C1,2 = − (2− p)(1−m) 2m2(p− 1)2 δ3−2pℓ1+ 1 m−2p, δC1,2 = − 2(2− p)m3 (1−m)(mα− δ + 2mδ)2 C2 2,0u 3(α) v(α) , C0,3 = − (2− p)(3− 2p) 6m(p− 1)3 δ4−3pℓ3+ 1 m−3p, δ2C0,3 = − 2m4(2− p)(3− 2p) 3(1−m)2(mα− δ + 2mδ)2 C2 2,0u 4(α) v(α) , δB2 = − (2− p)m2 (1−m)(mα− δ + 2mδ) C2,0u 3(α) v(α) . keywords: equation; filtration; m(p−; near; solutions; δ m cache: ejde-1438.pdf plain text: ejde-1438.txt item: #95 of 601 id: ejde-144 author: Chen, Feng title: Periodic solutions of stochastic Volterra equations date: 2022 words: 4897 flesch: 79 summary: We prove the existence of periodic solutions in distribution of stochastic Volterra equations. This paper concerns the existence of periodic solutions in the distribution of stochastic Volterra equations. keywords: equations; k(t; periodic; solutions cache: ejde-144.pdf plain text: ejde-144.txt item: #96 of 601 id: ejde-1440 author: Tu, Kun; Ding, Hui-Sheng title: Shadowing properties of evolution equations with exponential trichotomy on Banach spaces date: 2025 words: 5515 flesch: 83 summary: + ∫ t 0 T (t, r)(y′(r)−A(r)y(r))dr, t ∈ R+. + ∫ t 0 T (t, r)f(r, x(r))dr, t ∈ R+, and x(t)− y(t) keywords: + ∞; s t; t t; y(r; ∫ + cache: ejde-1440.pdf plain text: ejde-1440.txt item: #97 of 601 id: ejde-145 author: Liu, Zhenhai; Papageorgiou, Nikolaos S. title: Double phase equations with an indefinite concave term date: 2022 words: 4125 flesch: 80 summary: In (3.4) we choose h = un − u ∈ W 1,η 0 (Ω), pass to the limit as n → +∞ and use (3.16). Introduction Let Ω ⊆ RN be a bounded domain with a Lipschitz boundary ∂Ω. In this paper we study the double phase problem −∆a pu(z)−∆qu(z) keywords: 1,η cache: ejde-145.pdf plain text: ejde-145.txt item: #98 of 601 id: ejde-1452 author: Zhang, Xiaohui; Xu, Xian title: Multiple solutions for parametric weighted (p,q)-equations date: 2025 words: 11460 flesch: 91 summary: 0textfor a.a.z ∈ Ω, all x ⩾ 0. If hypotheses (H0), (H1) hold, λ ∈ L −, vλ ∈ S− λ , and µ ∈ (0, λ), then µ ∈ L − and there exists vµ ∈ S− µ ⊆ intY (−P1) such that vλ ⩽ vµ. Lemma 2.13. keywords: 1,p; f(z; inty; lemma; ∥v∥x cache: ejde-1452.pdf plain text: ejde-1452.txt item: #99 of 601 id: ejde-1454 author: Chavez, Alan; Aragones, Nelson; Zavaleta, Ulices; Pinto, Manuel title: Compact almost automorphic dynamics of linear non-autonomous differential equationswith exponential dichotomy and of delayed biological models date: 2025 words: 9601 flesch: 68 summary: If x(·) is defined on the interval [t0 − τ, σ] with t0, σ ∈ R, then the function xt ∈ C([−τ, 0],R) is defined by xt(θ) := x(t + θ) for all θ ∈ [−τ, 0] and t0 ≤ t ≤ σ. Let C+ be the cone of non-negative functions in C([−τ, 0],R), i.e., C+ = {ϕ ∈ C([−τ, 0],R) : ϕ(t) ≥ 0}, and define the set C+ 0 keywords: function; lim; n→+∞; solution cache: ejde-1454.pdf plain text: ejde-1454.txt item: #100 of 601 id: ejde-1457 author: Yayla, Sema title: Structure and stability of global attractors for a Cahn-Hilliard tumor growth model with chemotaxis date: 2025 words: 10727 flesch: 79 summary: − χσ̄χ, (4.8) ⟨σ̄χ t, ξ⟩ + ⟨∇σ̄χ,∇ξ⟩ = χ⟨∇ϕ̄χ,∇ξ⟩ − ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), η⟩, (4.9) for all η, ξ ∈ H1(Ω). = ϕχ − ϕ∗, σ̄χ := σχ − σ∗ and µ̄χ := µ− µ0, we obtain from (2.9) that ⟨ϕ̄χ t, η⟩ + ⟨∇µ̄χ,∇η⟩ = ⟨p(ϕχ)(σ̄χ − χϕ̄χ − µ̄χ), η⟩, (4.7) µ̄χ = −∆ϕ̄χ + Ψ′(ϕχ) − Ψ′(ϕ∗) keywords: lemma; theorem cache: ejde-1457.pdf plain text: ejde-1457.txt item: #101 of 601 id: ejde-146 author: Idczak, Dariusz title: A parabolic bipolynomial fractional Dirichlet-Laplace problem date: 2022 words: 7861 flesch: 82 summary: We know that d dt (f(t), ψ(t))X = (f ′(t), ψ(t))X + (f(t), ψ′(t))X (2.6) for t ∈ = f ′(t)ϕ(t) + f(t)ϕ′(t) for t ∈ (a, b), and any function ϕ ∈ C∞c (a, b;R). keywords: b;x; function cache: ejde-146.pdf plain text: ejde-146.txt item: #102 of 601 id: ejde-1469 author: Almutairi, Sarah; Saoudi, Kamel title: Combined effects of critical Hardy-Sobolev exponent and singular nonlinearities in nonlocal problems with variable weights date: 2025 words: 6208 flesch: 72 summary: □ References [1] K. Kefi, M. Kratou, K. Saoudi; Combined effects of critical and singular nonlinearities in fractional problems, submitted (2025). The third result concerns the regularity of weak solutions of problem (1.1). keywords: problem; solution; |x|t cache: ejde-1469.pdf plain text: ejde-1469.txt item: #103 of 601 id: ejde-147 author: Wang, Lixia title: Localized nodal solutions for semiclassical nonlinear Kirchhoff equations date: 2022 words: 9864 flesch: 85 summary: For every 1 ≤ i ≤ mj, ṽi is a nontrivial solution of − (a+ bAj)∆v + V (yij)v = |v|p−2v, v ∈ H1(R3), (4.13) where yij = limε→0 εy i j,ε ∈ Λ̄; (iii) For any 2 < q < 6, lim ε→0 ‖vj,ε − ṽ0 − mj∑ i=1 ṽi(· − yij,ε)‖Lq(R3) = 0. (4.14) Proof. Since vj,ε, ṽ0 and ṽ1 solves (2.4), (4.12), and (4.13) with i = 1 respectively, we have − a∆v2 j,ε − b ∫ R3 |∇vj,ε|2dx∆v2 j,ε − b (∫ R3 |∇vj,ε|2 −Aj ) ∆ṽ0 − b (∫ R3 |∇vj,ε|2 −Aj ) ∆ṽ1 + ξεχεv 2 j,ε + ξεχεṽ0 + ξεχεṽ1 + V (εx)v2 j,ε + (V (εx)− V (0))ṽ0 + (V (εx)− V (yij))ṽ1(· − y1 j,ε) = |vj,ε|p−2vj,ε − |ṽ0|p−2ṽ0 − |ṽ1|p−2ṽ1(· − y1 j,ε). keywords: h1(r3; lemma; solutions; |∇vj; ∫ r3 cache: ejde-147.pdf plain text: ejde-147.txt item: #104 of 601 id: ejde-1474 author: Chen, Zhuoru; Wang, Taige; Xie, Xiangfei title: Bilinear estimates posed in finite domains in 2D and 3D date: 2025 words: 2912 flesch: 77 summary: ∇v(s)∥ds ≤ C ∫ T 0 ∥∇u(s)∥(∥∇v(s)∥+ ∥∇v(s)∥1/2∥Av(s)∥1/2)ds whence ∫ T 0 ∥u(s) · Second, on ∫ T 0 ∥∇(u(s) · keywords: estimates cache: ejde-1474.pdf plain text: ejde-1474.txt item: #105 of 601 id: ejde-1475 author: Girg, Petr; Kotrla, Lukas; Svandova, Anezka title: p-Laplacian in phenomenological modeling of flow in porous media and CFD simulations date: 2025 words: 9766 flesch: 67 summary: In Section 2, we present several mathemat- ical models of groundwater flow in phreatic aquifers and related models used in EJDE-2022/2025/CONF/26 FLOW IN POROUS MEDIA AND CFD SIMULATIONS 181 irrigation and drainage. [24, 50], despite their limited well yields due to water flow occurring only in cracks and fractures. keywords: flow; fracture; groundwater; law; media; model; network; rock; solution; water cache: ejde-1475.pdf plain text: ejde-1475.txt item: #106 of 601 id: ejde-1476 author: Tumanyan, Ani title: Normal solvability and Fredholm properties for special classes of hypoelliptic operators date: 2025 words: 7906 flesch: 72 summary: We also provide applications to the smoothness of solutions, index invariance on the scale, and spectral properties of such operators. Isomorphic characteristics for quasielliptic op- erators with constant coefficients on a special scale of weighted spaces have been derived in the works of Demidenko (see [10, 11]), and such operators have been 2020 Mathematics Subject Classification. keywords: operator; p q; q ∈ cache: ejde-1476.pdf plain text: ejde-1476.txt item: #107 of 601 id: ejde-1477 author: Zhang, Zhenbu title: Traveling wave solutions for an epidemic model date: 2025 words: 9351 flesch: 84 summary: The differential susceptibility epidemic model ∂I ∂t = dIIxx + ηβI l∑ j=1 αjSj − (µ+ γ)I x ∈ R, t > 0, ∂Si ∂t = diSixx + µpiS 0 − ηβαiISi − µSi, x ∈ R, t > 0, i = 1, 2, . . u = (u1, u2, . . . keywords: j=1; m+1; model; speed; ui(z; wave cache: ejde-1477.pdf plain text: ejde-1477.txt item: #108 of 601 id: ejde-148 author: Ye, Xiaobing; Wang, Liangchen title: Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source date: 2022 words: 7586 flesch: 82 summary: Testing the first equation in (1.1) by lnu+ 1 and integrating we have d dt ∫ Ω u lnu = ∫ Ω (lnu+ 1)∆u− ∫ Ω (lnu+ 1)∇ · (u∇v) + µ ∫ Ω (lnu+ 1)u(1− u) = − ∫ Ω |∇u|2 u + ∫ Ω ∇u · ∇v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ − ∫ Ω u∆v + µ ∫ Ω (lnu+ 1)u(1− u) ≤ ∫ Ω uw + µ ∫ Ω u+ µ ∫ Ω u lnu− µ ∫ Ω u2 − µ ∫ Ω u2 lnu (3.2) for all t ∈ (0, Tmax). [1, 8], testing the first equation of (1.1) by up−1(p ≥ 2) and integrating by parts over Ω, using (4.22) and Young’s inequality we have 1 p d dt ∫ Ω up + (p− 1) ∫ Ω up−2|∇u|2 + µ ∫ Ω up+1 = (p− 1) ∫ Ω up−1∇u · ∇v + µ ∫ Ω up ≤ c2(p− 1) ∫ Ω up−1|∇u|+ µ(p− 1) ∫ Ω up ≤ p− 1 2 ∫ Ω up−2|∇u|2 + (c22 2 + µ ) (p− 1) ∫ Ω up (4.23) for all t ∈ (0, Tmax). keywords: tmax; ∫ ω cache: ejde-148.pdf plain text: ejde-148.txt item: #109 of 601 id: ejde-1487 author: Chen, Jiaxue; Li, Yeping; Yin, Rong title: Zero-viscosity-capillarity limit for the contact discontinuity for the 1-D full compressible  Navier-Stokes-Korteweg equations date: 2025 words: 9222 flesch: 79 summary: y ζy − ν ( 1 ΘCD ) y ζy (ζyyv − ζyϕy)− (ζyV CD y + ϕyΘ CD y ) + (ΘCD yy v −ΘCD y V CD y ) v2 + ν ( 1 ΘCD ) y ζy ΘCD yy V CD −ΘCD y V CD y (V CD)2 − ( ν 1 ΘCD ΘCD yy V CD −ΘCD y V CD y (V CD)2 ) y ζy − (ψy + UCD y )2 v ( 1 ΘCD ) y ζy + (UCD y )2 V CD ( 1 ΘCD ) y ζy − ψ2 y + 2ψyU CD y vΘCD ζyy − λ(ψy + UCD y ) (5(ϕy + V CD y )2 2v6 − ϕyy + V CD yy v5 )( 1 ΘCD ) |ψyϕyyζyy|+ |ϕ3yψyy|+ |ψyϕ 2 yζyy|, J2 = |ϕyψy(Θ CD y + V CD y )|+ |ζyψy(V CD y +ΘCD y )|+ |ϕ2yUCD y |+ |ζ2yUCD y | + |ζyϕyUCD y |+ |ϕyψy(U CD y V CD y + UCD yy )|+ |ζ2yΘCD y V CD y | + |ζyϕy((ΘCD y )2 +ΘCD yy +ΘCD y V CD y + (UCD y )2)|+ |ψyζyU CD y ΘCD y | + |ϕyψy(V CD y V CD yy + (V CD y )3 + V CD yyy )|+ |ψyζyΘ CD y ((V CD y )2 + V CD yy )| + |ϕyζy(UCD y V CD y ΘCD y + UCD y (V CD y )2 + UCD y V CD yy )|, J3 = |ϕψy(V CD y ΘCD y +ΘCD yy + (V CD y )2 + V CD yy )|+ |ζψy((V CD y )2 + V CD yy )| + |(ϕ+ ζ)ζyU CD yy |+ |ϕψy(V CD y UCD yy + UCD yyy + (V CD y )2UCD y + V CD yy keywords: c ∫; cd y; sup; ucd; v cd; y +; y v; θcd; τ τ0; τ0 ∫; τ0≤τ≤τ; ∫ τ cache: ejde-1487.pdf plain text: ejde-1487.txt item: #110 of 601 id: ejde-149 author: Fu, Song-Ren; Ning, Zhen-Hu title: Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R^3 date: 2022 words: 6933 flesch: 84 summary: dt+ ∫ T 0 ∫ Ω a(x)utH(u) dx dt+ ∫ T 0 ∫ R3 a(x)utH(u) dx dt + 3 2 ∫ T 0 ∫ R3 (u2 t − |∇gu|2g − u2 − 1 3 u6) dx keywords: equation; ∫ r3; ∫ t cache: ejde-149.pdf plain text: ejde-149.txt item: #111 of 601 id: ejde-1499 author: Zhai, Xiaoping title: Linear stability of the Couette flow for non-isentropic compressible fluids date: 2025 words: 6238 flesch: 73 summary: Integration in x equations in (1.15), one infer that ∂tρ0 = −α0, ∂tα0 = − 1 γM2 ∂yyρ0 − 1 γM2 ∂yyθ0 + ν∂yyα0, ∂tω0 = α0, ∂tθ0 = −(γ − 1)α0. (2.1) From (2.1), we can further get α0 satisfies the damped wave equations ∂ttα0 − ν∂t∂yyα0 − 1 M2 ∂yyα0 = 0, in R, (2.2) and ρ0 + θ0 satisfies the wave equation ∂tt(ρ0 + θ0)− 1 M2 ∂yy(ρ0 + θ0) (2.15) 8 X. ZHAI EJDE-2025/107 From the equations in (2.10) and definitions of Z1 and Z2, a simple computations gives ∂tZ1 = −∂tm m Z1 − 1 4 ∂tp p Z1 − 1 M p1/2Z2, ∂tZ2 = − (∂tm m + νp ) Z2 keywords: compressible; couette; flow; linear; stability; ∂tp cache: ejde-1499.pdf plain text: ejde-1499.txt item: #112 of 601 id: ejde-15 author: Remy, Pascal title: Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations date: 2023 words: 12143 flesch: 78 summary: Then, denoting by C(a, b) the domain C(a, b) = {(x, y) ∈ R2 : x ≤ a and y ≥ b} for any (a, b) ∈ R2, the Newton polygon at t = 0 of the operator (3.7) is defined as the convex hull of C(κ,−κ) ⋃⋃ i∈K ⋃ q∈Qi C(λ(q) + i, vi,q,p∗ − i), where λ(q) = q1 + · · ·+ qn denotes the length of q = (q1, . . . Let i ∈ K, q ∈ Qi, p ∈ Pi,q, j ≥ vi,q,p, and `0, `1, . . . keywords: gevrey; γ(1 cache: ejde-15.pdf plain text: ejde-15.txt item: #113 of 601 id: ejde-150 author: Marcial, Marcos Roberto; Miyagaki, Olimpio H.; Pereira, Gilberto A. title: Topological structure of the solution set for a fractional p-Laplacian problem with singular nonlinearity date: 2022 words: 7466 flesch: 85 summary: We consider the convex, closed subset of IΛ × C(Ω) given by GΛ := { (λ, u) ∈ IΛ × C(Ω) : λ ∈ IΛ, u ≤ u ≤ u and u = 0 on Ωc } . In ×BRn : u ≤ u ≤ u, u = 0 on ∂Ω } , where Rn = RΛn . keywords: solution cache: ejde-150.pdf plain text: ejde-150.txt item: #114 of 601 id: ejde-151 author: Wang, Lixiong; Chen, Haibo; Yang, Liu title: Ground state solutions for fractional p-Kirchhoff equation date: 2022 words: 5538 flesch: 83 summary: 14 L. WANG, H. CHEN, L. YANG EJDE-2022/61 [11] X. Huang, Y. Zhang; Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation. [16] Z. Liu, M. Squassina, J. Zhang; Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 4, Paper No. 50, 32. keywords: p2s; y|n+ps; |x−; ∫ rn cache: ejde-151.pdf plain text: ejde-151.txt item: #115 of 601 id: ejde-1515 author: Nunez-Chavez, Miguel R. title: Controllability under positivity constraints for non-linear and non-local parabolic PDEs date: 2025 words: 16085 flesch: 87 summary: − y ∈ C2+ 1 2 ,1+ 1 4 (QT ) ⊂ C([0, T ];L2(Ω)) and as (z(0)− y(0), φ0) Multiplying by −∆(y − y) in (3.3) and integrating in Ω, we obtain∫ Ω (y − y)t(−∆(y − y)) keywords: control; controllability; dx′; function; lemma; linear; proof; solution; system; t t; z(x; − y cache: ejde-1515.pdf plain text: ejde-1515.txt item: #116 of 601 id: ejde-152 author: Wang, Zhenqiang title: Higher differentiability for solutions to nonhomogeneous obstacle problems with 1 date: 2022 words: 10339 flesch: 83 summary: [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ ( ∫ B2R |Du|pdx) n n−2α ]n−2α n + c ∫ BR |τhVp(Dψ)|2 |h|2α dx+ c|h|(α+1)(1−β) (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|p( ∫ BR |DF | np n−2α dx) Suppose that there exists ρ ∈ (0, R) and M > 0 such that n∑ s=1 ∫ Bρ |τs,hF (x)|pdx ≤Mp|h|p, for all h with |h| < R−ρ 2 . keywords: c ∫; n−2α; n−2α dx; n−2α n; ∫ b2r; ∫ br; ∫ ω cache: ejde-152.pdf plain text: ejde-152.txt item: #117 of 601 id: ejde-1521 author: Song, Xiao; Wang, Chenhua; Wang, Xiaojie; Xu, Fuyi title: Global unique solution for 3D incompressible inhomogeneous magneto-micropolar equations with discontinuous density date: 2025 words: 13496 flesch: 81 summary: [Dt; curl]DtujDtwj dx. (3.38) 14 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Multiplying (??) by t2 and then integrating over [0, t] give rise to 2∥t∇Dt(ωj , Hj)∥2L∞ t (L2) + 2∥tdivDtωj∥2L∞ t (L2) + ∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2) + ∥t∇Dtuj∥2L∞ t (L2) + ∥tdivDtuj∥2L∞ t (L2) + ∥t(curlDtuj − 2Dtwj)∥2L∞ t (L2) ≲ ∥ √ t∇Dt(uj , ωj , Hj)∥2L2 t (L 2) + ∥ √ tdivDtωj , Dtωj∥2L2 t (L 2) + ∫ t 0 τ2 ∫ R3 ∇DtπjD 2 t uj dx dτ + ∫ t 0 τ2 ∫ R3 ∇DtHj · [Dt;∇]DtHj dx dτ + ∫ t 0 τ2 ∫ R3 ∇Dtuj · keywords: global; hj)∥l2 t; l2 t; t ḃ; t uj; − ∫; ∫ r3 cache: ejde-1521.pdf plain text: ejde-1521.txt item: #118 of 601 id: ejde-1529 author: Fan, Zian title: Normalized ground state solutions for Kirchhoff equation with subcritical or critical perturbation date: 2025 words: 6160 flesch: 86 summary: Lately in the aid of subcritical approximation method, we prove the existence of normalized ground state solutions for q = 2∗ for any η > 0 . If q = 2∗, by using the definition of S0 we obtain∫ R3 |u|2 ∗ dx ≤ S −2∗/2 0 ∥∇u∥2 ∗ 2 . (2.2) Since the embedding H1(R3) → Lj(R3)(2 < j ≤ 2∗) is continuous, then we deduce that Iq ∈ C1(E,R). keywords: m(c; pγp cache: ejde-1529.pdf plain text: ejde-1529.txt item: #119 of 601 id: ejde-153 author: Adhikari, Dhruba R.; Aryal, Ashok; Bhatt, Ghanshyam; Kunwar, Ishwari J.; Puri, Rajan; Ranabhat, Min title: Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators date: 2022 words: 12742 flesch: 80 summary: For a sequence {xn} in X and x0 ∈ X, we denote by xn → x0 and xn ⇀ x0 the strong convergence and weak convergence, respectively. We may assume that there exist x0 ∈ X and w0 ∈ X∗ such that xn ⇀ x0 in X and Aϕtnxn ⇀ w0 in X∗. keywords: degree; monotone; operator cache: ejde-153.pdf plain text: ejde-153.txt item: #120 of 601 id: ejde-1531 author: Ha, Tae Gab title: Global solutions and blow-up for wave equations with variable coefficients and boundary supercritical source date: 2025 words: 13930 flesch: 84 summary: Indeed, considering w = uηm in (3.1) and then integrating over (0, T ), we have∫ T 0 ⟨uηmtt , uηm⟩ dt+ ∫ T 0 µ(t)∥ |∇gu ηm|g∥22 dt+ η ∫ T 0 ⟨uηmt , uηm⟩Γ1 dt 10 T. G. HA EJDE-2025/104 + ∫ T 0 ⟨q(uηmt ), uηm⟩Γ1 dt− ∫ T 0 ⟨h(uηm), uηm⟩Γ1 dt = ∫ T 0 ⟨f, uηm⟩ dt. Next, considering w = uηmt in (3.1) and then integrating over (0, T ), we have∫ T 0 ⟨uηmtt , u ηm t ⟩ dt+ ∫ T 0 µ(t) ∫ Ω ⟨∇gu ηm,∇gu ηm t ⟩g dx dt+ η ∫ T 0 ∥uηmt ∥22,Γ1 dt + ∫ T 0 ⟨q(uηmt ), uηmt ⟩Γ1 dt− ∫ T 0 ⟨h(uηm), uηmt ⟩Γ1 dt = ∫ T 0 ⟨f, uηmt ⟩ dt. From (3.14)-(3.18) and (3.22), we arrive at lim m→∞,η→0 ∫ T 0 ⟨q(uηmt ), uηmt ⟩Γ1 dt = ∫ T 0 ⟨ψ, ut⟩Γ1 dt. keywords: energy; equation; source; t s; wave; γ+2,γ1; ρ+1; ρ+2; ∫ t; ∫ γ1 cache: ejde-1531.pdf plain text: ejde-1531.txt item: #121 of 601 id: ejde-1534 author: Lu, Can; Wang, Liangwei; Yin, Jingxue; Zhou, Meiling title: Complicated asymptotic behavior of solutions to doubly nonlinear diffusionequation in unbounded spaces date: 2025 words: 7543 flesch: 85 summary: Let p p−1 ≤ σ < p m(p−1)−1 , if 0 ≤ u0 ∈ L∞(ρσ). (3.10) EJDE-2025/108 DOUBLY NONLINEAR DIFFUSION EQUATIONS 7 For any ε > 0, and the following assumptions are satisfied 0 ≤ u0 ∈ Yσ(RN ), then there exists a constant R1 > 1 > 0, It follows that for |x| > R1, (1 + |x|2)−σ/2 u0(x) < ε 2 . keywords: equation; m(p−1)−1; yσ(rn cache: ejde-1534.pdf plain text: ejde-1534.txt item: #122 of 601 id: ejde-154 author: Qin, Liuna; Xiao, Changguo; Zhang, Yinghui title: Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force date: 2022 words: 6063 flesch: 82 summary: For T > 0, let (ρ − ρ̃, u, φ − φ̃)(x, t) be a solution of (1.1) in [0, T ] and introduce E(T ) Then there exists δ > 0 such that if E(T ) + ε1 ≤ δ, (1.6) then the following a-priori estimate holds ‖(ρ− ρ̃, u,∇φ−∇φ̃)(·, t)‖2H2 + ∫ t 0 ‖(ρ− ρ̃,∇u,∇2φ−∇2φ̃)(·, s)‖2H2ds ≤ C‖(ρ0 − ρ̃, u0)‖2H2 , (1.7) for any t ∈ keywords: decay; equations; navier; poisson; stokes; system cache: ejde-154.pdf plain text: ejde-154.txt item: #123 of 601 id: ejde-1543 author: Fernandes, Juliana; Maia, Liliane title: Nehari manifold for degenerate logistic parabolic equations date: 2025 words: 8062 flesch: 79 summary: u|t=0 = u0(x), x ∈ Ω, (1.1) where Ω is an open smooth bounded domain in RN , N ≥ 2, λ is a real positive parameter, 1 < ν < 2∗ − 1, where 2∗ = +∞ if N = 2, or 2∗ = 2N/(N − 2) if N ≥ 3, and b is a continuous function satisfying b(x) ≤ 0 and b(x) = 0 in a smooth proper subdomain Ω0 of Ω, with positive Lebesgue measure and smooth boundary. keywords: solutions cache: ejde-1543.pdf plain text: ejde-1543.txt item: #124 of 601 id: ejde-155 author: Di Fazio, Giuseppe; Fanciullo, Maria Stella; Zamboni, Piero title: Boundary regularity for strongly degenerate operators of Grushin type date: 2022 words: 6051 flesch: 78 summary: In this way we can define infS u, supS u and oscS u. Now, let Br be a ball centered at x0 ∈ ∂Ω and u ∈ H1,p v (Ω ∩B4r, w) we set ũ(x) = { min{u,m} if x ∈ Ω ∩B4r m if x ∈ Rn \ (Ω ∩B4r) where m = inf∂Ω∩B4r This is a kind of generalization of the [1, 4] to quasilinear elliptic equations. keywords: b3r; p−1 cache: ejde-155.pdf plain text: ejde-155.txt item: #125 of 601 id: ejde-157 author: Guo, Cuiping; Guo, Shangjiang title: Stationary and oscillatory dynamics of Nicholson's blowflies equation with Allee effect date: 2022 words: 9378 flesch: 78 summary: Nonconstant steady states Steady state solutions of (1.2) satisfy d∆u(x)− u(x) + f(u(x)) In particular, we present the bifurcation direction for each branch of steady state solutions and periodic solutions. keywords: solutions; state; steady; u(t; u∗2(p cache: ejde-157.pdf plain text: ejde-157.txt item: #126 of 601 id: ejde-1570 author: Wei, Qifan; Zhang, Xuemei title: Existence and multiplicity of solutions to triharmonic problems date: 2025 words: 9361 flesch: 81 summary: In particular, by using a variant version of the mountain pass lemma, Hu-Wang [29] obtained the existence of nontrivial solutions for the following fourth-order problem ∆2u+ α∆u = f(x, u) in Ω, u = ∆u = 0 on ∂Ω, (1.3) where ∆2(u) = ∆(∆u) stands for the biharmonic operator, Ω ⊂ RN (N > 4) is a smooth bounded domain, and α < µ1 is a parameter, where µ1 is the first eigenvalue of (−∆) in H1 0 (Ω). [37] studied the fourth-order problem ∆2u+ β∆u = a(x)|u|s−2u+ f(x, u) in Ω, u = ∆u = 0 on ∂Ω, (1.4) where Ω ⊂ RN (N > 4) is a smooth bounded domain, β < µ1, a(x) ∈ L∞(Ω), s ∈ (1, 2) and f ∈ C(Ω̄ × R,R). keywords: g(x; order; solutions; theorem cache: ejde-1570.pdf plain text: ejde-1570.txt item: #127 of 601 id: ejde-1576 author: Neto, Antonio Francisco title: New approach to the Lagrange-Burmann theorem via omega calculus and applications date: 2025 words: 13702 flesch: 83 summary: = τ1 − λ Ω = λ ln ( 1− ζ G(λ) ) , (7.7) In this case α = 0, β = 1, and γ = 0 = τ1 (and hence δ = 0). keywords: function; omega; proof; theorem; λ ω cache: ejde-1576.pdf plain text: ejde-1576.txt item: #128 of 601 id: ejde-158 author: Pinelas, Sandra; Tunc, Osman title: Solution estimates and stability tests for nonlinear delay integro-differential equations date: 2022 words: 4325 flesch: 77 summary: , l. Hence, W ′(·) ≤ −2y m∑ i=1 fi(t, x, y)− 2yg(x, y) + [ n∑ i=1 (αiτi) + l∑ i=1 (diriτi) + n∑ i=1 (γiτi) ] y2 + (α1 + d1r1 − γ1) ∫ t t−τ1 y2(s) ds+ (α2 + d2r2 = ∫ t t− 1 4 1 1 + t4 + s2 x′(s) keywords: differential; equations; i=1; integro; stability cache: ejde-158.pdf plain text: ejde-158.txt item: #129 of 601 id: ejde-1586 author: Meng, Fanmeng; Zhou, Xian-Feng title: Decay estimates and extinction properties of parabolic equations with classical and fractional time derivatives date: 2025 words: 9664 flesch: 83 summary: We say u(x, t) vanishes in finite time if there exists a constant T > 0 such that u(x, t) ≡ 0 in Ω for t ≥ T . = ( Y (0)− m0 2λ1C∗ t ) > 0, 0 < t < T, Y (t) ≡ 0, t ≥ T, (5.16) where T = 2λ1C⋆Y (0) m0 . keywords: fractional; |u(x cache: ejde-1586.pdf plain text: ejde-1586.txt item: #130 of 601 id: ejde-1588 author: Lee, Mikyoung; Ok, Jihoon title: L^q-regularity estimates for double obstacle problems  with quasilinear operators and Schrodinger-type lower order terms date: 2025 words: 9900 flesch: 78 summary: We deal with the function u ∈ A0(Ω) that satisfies the variational inequality∫ Ω a(x,Du) ·D(u− φ) dx+ ∫ Ω V |u|s−2u(u− φ) dx ≤ ∫ Ω |F |p−2F ·D(u− φ) dx (1.7) for all φ ∈ A0(Ω). ·D(v1 − v2)+ dx ≤ 0. keywords: estimates; obstacle; ωr v; − ∫; ∫ ωr cache: ejde-1588.pdf plain text: ejde-1588.txt item: #131 of 601 id: ejde-159 author: Lou, Zhaowei; Sun, Yingnan title: A KAM theorem for higher dimensional reversible nonlinear Schrodinger equations date: 2022 words: 9910 flesch: 87 summary: We introduce the Banach space `ρI of all 4 Z. LOU, Y. SUN EJDE-2022/69 complex sequences z = (zj)j∈Zd\I with ‖z‖ρ = ∑ j∈Zd\I e|j|ρ|zj | <∞, where |j| = √ |j1|2 + · · ·+ |jd|2. = r 2ν+3 , rν+1 = rν − 2δν , r0 = r, εν+1 = cγ−5δ−1 ν K5τ+19 ν ε5/3 ν + ε7/6 ν , ε0 = ε, e−Kνδν = ε1/2 ν , ην = ε1/3 ν , sν+1 = 1 4 ηνsν , s0 = s, Lν+1 = Lν + εν , L0 = L, ρν = ρ(1− ν+1∑ i=2 2−i). 5.5.1. keywords: j k; kam; zd1; zd2; σ j cache: ejde-159.pdf plain text: ejde-159.txt item: #132 of 601 id: ejde-1590 author: Nghia, Le Trung title: Hardy operators and commutators on generalized central function spaces date: 2025 words: 7684 flesch: 85 summary: In this article, we study the boundedness of operators of Hardy type on generalized central function spaces, such as the generalized central Hardy space HAp,r φ (Rn), the generalized central Morrey space Ṁp,r φ (Rn), and the generalized central Campanato space ˙CMO p,r φ (Rn), with p ∈ (1,∞), and φ(t) : (0,∞) → (0,∞). Furthermore, there exists a constant C > 0 depend- ing on n, p such that ∥b∥ ˙CMO p,r ≤ C∥[b,H∗]∥Ṁp,r φ →Ṁp,r φ . keywords: hap′,r′; hardy; r φ; spaces; ˙cmo cache: ejde-1590.pdf plain text: ejde-1590.txt item: #133 of 601 id: ejde-1597 author: Sun, Jinyi; Mai, Yuanwei; Yang, Minghua title: Well-posedness of generalized magnetohydrodynamic equations in variable Lebesgue spaces date: 2025 words: 7391 flesch: 79 summary: This article concerns the well-posedness of the generalized magnetohydrodynamic equations in variable Lebesgue spaces. By using some basic properties of variable Lebesgue spaces and decay estimates of the fractional heat kernel, we prove the existence of local and global solutions to the generalized magnetohydrodynamic equations in two different types of variable Lebesgue spaces. keywords: 2α−1; spaces; variable cache: ejde-1597.pdf plain text: ejde-1597.txt item: #134 of 601 id: ejde-1599 author: Wang, Tiantian; Fan, Hongxia title: Topological properties of the solution set and T-controllability for second-order neutral evolution equations with delay date: 2025 words: 6899 flesch: 68 summary: = I; (iii) C(t)x is continuous in t on R for each fixed x ∈ X. Define the associated sine family {S(t) : t ∈ R} by S(t)x = ∫ t 0 C(s)xds, x ∈ X, t ∈ R. The infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} is the operator A : D(A) ⊂ X → X defined by Ax = d2 dt2 C(t)x |t=0, x ∈ D(A), where D(A) = {x ∈ X : C(·)x ∈ C2(R,X)}. +Bu(t) + f(t, x(a(t)), x[h(x(t), t)]), t ∈ J keywords: controllability; equations; set; solution cache: ejde-1599.pdf plain text: ejde-1599.txt item: #135 of 601 id: ejde-160 author: Chen, Shaohua; Xu, Runzhang; Yang, Chao title: Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy date: 2022 words: 5143 flesch: 83 summary: (2.3) Then Φ(t) blows up in finite time T , where T <  1 2θ−γ ln ( (2θ−γ)Φ(0) (α−1)Φ′(0)−θΦ(0) + 1 ) if 2θ > γ, Φ(0) (α−1)Φ′(0)−γΦ(0)/2 if 2θ ≤ γ. (2.4) Proof. = 2.4 and µ = 0.3, then (2.2) is satisfied and the solution blows up with blowup time T ∗ < 2.37 based on (2.4). keywords: blowup; finite; time cache: ejde-160.pdf plain text: ejde-160.txt item: #136 of 601 id: ejde-1606 author: Zhu, Xi; Zhu, Min; Wang, Ying; Wang, Ke title: Wave-breaking for two-component Fornberg-Whitham systems with dissipation date: 2025 words: 11510 flesch: 76 summary: By Cantor’s diagonalization ar- gument, for any test function φ ∈ C∞ c (R), the quantities ∥φun − φu∥Bs−1 q,r and ∥φηn − φη∥Bs−2 q,r converge uniformly to 0 on [0, T ] as n → ∞. Using the Fatou property of Besov spaces from Lemma 2.3(vi), for all t ∈ Let s ∈ R and 1 ≤ q, r ≤ ∞. keywords: blow; bs−1 q; equation; system; wave; ∥ρ0∥l1 cache: ejde-1606.pdf plain text: ejde-1606.txt item: #137 of 601 id: ejde-161 author: Zhao, Qiulan; Cheng, Hongbiao; Li, Xinyue; Li, Chuanzhong title: Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions date: 2022 words: 12535 flesch: 82 summary: (3.6) When m = 2, setting ε = 1, we obtain pt = αqxx − αhxq − 2αhpx − αh2q + αq(p2 + q2) + αq((qpx − pqx) + 2αh(p2 + q2)), qt = −αpxx − αhxp− 2αhpx + αh2p− αp(p2 + q2) 2α(qrx − rqx) + 4αh(p2 + q2)− βh(q2 − p2) + 2h2), st = −αrxx − αpxx + βpxx + 2αqhx + 4αhqx − αshx − αhsβqhx − 2βhqx − αhsx − 4αh2p+ 2αh2r + 2βh2p+ 2αp2r keywords: darboux; ejde-2022/71; equation; hierarchy; i=0; mkdv; nls; nonlinear; n−1; solutions; transformation cache: ejde-161.pdf plain text: ejde-161.txt item: #138 of 601 id: ejde-162 author: Lopes, Juliana Honda; Planas, Gabriela title: Existence of solutions for a non-isothermal Navier-Stokes-Allen-Cahn system with thermo-induced coefficients date: 2022 words: 8645 flesch: 77 summary: We note that (2.3) implies the existence of some positive constants Ci, i = 1, 2, 3 such that − C1 ≤ F ′′(s), −C2 ≤ F (s) ≤ F ′(s)s+ C3 for all s ∈ R, (2.5) where F (s) = ∫ Then φ1 − φ2 solves (φ1 − φ2)t + u · ∇(φ1 − φ2) = ∇ · (ε(θ)∇(φ1 − φ2))− 1 ε(θ) (F ′(φ1)− F ′(φ2)) together with ∂ ∂η (φ1 − φ2) = 0 on ∂Ω× (0, T ) and (φ1 − φ2)(0) = 0 in Ω. Multiplying the above equation by φ1 − φ2 and integrating in Ω, we see 1 2 d dt ‖φ1 − φ2‖2 + ε0‖∇(φ1 − φ2)‖2 ≤ − ( 1 ε(θ) (F ′(φ1)− F ′(φ2)), φ1 − φ2 ) ≤ C‖F ′(φ1)− F ′(φ2)‖‖φ1 − φ2‖ ≤ C‖φ1 − φ2‖2, EJDE-2022/72 NON-ISOTHERMAL NAVIER-STOKES-ALLEN-CAHN SYSTEM 9 where we used that ∇ · u = 0, the Mean Value Theorem for F ′ and the fact that F ′′ is bounded. keywords: cahn; existence; l2(0; navier; stokes; system cache: ejde-162.pdf plain text: ejde-162.txt item: #139 of 601 id: ejde-163 author: Zhao, Xutong; Zhou, Mingjun; Zhou, Qian title: Asymptotic behavior of solutions to coupled porous medium systems with boundary degeneracy date: 2022 words: 7823 flesch: 78 summary: that ωpλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) = ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ × ( 1 + (1−m)µ (2− λ)m ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−(2−λ)µ x2−λ )(p−1)/(m−1) × (1 + (1−m)µ (2−λ)m ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−(2−λ)µ x2−λ 1 + (1−m)µ (2−λ)m ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(2−λ)µ x2−λ )1/(m−1) ≤ ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ , x ≥ 0, t ≥ 0, 14 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 and similarly, ωqλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) ≤ ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m)( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) for x ≥ 0 and t ≥ 0. = Θ 1/m 1 (t)ωλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) , x ≥ 0, t ≥ 0, (4.23) v̂(x, t) keywords: m−1)/m; s)ds cache: ejde-163.pdf plain text: ejde-163.txt item: #140 of 601 id: ejde-1633 author: Wu, Xinrui; Liang, Xingyu title: Global well-posedness of 3D inhomogeneous incompressible nematic liquid crystal systems in critical Besov spaces with initial density perturbed around the equilibrium date: 2025 words: 7508 flesch: 79 summary: Their result require 0 < c0 ≤ ρ0 ≤ C0 < +∞ and small norm ∥(u0,∇d0)∥Ḃ1/2 2,1 (R3) . ≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇2d∥L∞(0,T ;L∞(R3)) ≤ C∥∇d∥Ls,1(0,T ;Lm(R3))∥t∇d∥ L∞(0,T ;Ḃ 1+ 3 m m,1 (R3)) . keywords: ḃ −1; estimate; global; liquid; lp(r3; l∞(0,t; p p,1; spaces; system; −1 + cache: ejde-1633.pdf plain text: ejde-1633.txt item: #141 of 601 id: ejde-1648 author: Unlu, Mehmet title: Inverse scattering method for an integrable system of derivative nonlinear Schrodinger equations date: 2025 words: 14427 flesch: 74 summary: = K1(x, x)K̄2(x, x). − rxx − iqrrx = 0, (1.1) where x and t are the independent variables taking values on the real axis R, the subscripts denote the respective partial derivatives, the dependent variables q and r are complex-valued functions of x and t. keywords: matrix; scattering; solutions; system cache: ejde-1648.pdf plain text: ejde-1648.txt item: #142 of 601 id: ejde-1658 author: Karakostas, George L. title: Existence of solutions for a n-dimensional systems of nonlocal boundary value problems date: 2025 words: 15320 flesch: 83 summary: + ∫ t 0 Θ(s)−1dsΘ(0)x′(0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds, (2.4) which will be used to express the solution as a fixed point of an operator equation. The quantity ρ1 := V (A−1 0 Ψ0) + ∥(A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 1 ∥E ( V (A0Ψ1) + V (A1Ψ0) ) satisfies the condition ρ1 < 1 and moreover condition (H0) is satisfied with ρ1 and the constants M1 := ∥(A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1( ∫ 1 u Θ(s)−1ds+B1Θ(1)−1)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds, K1 := |A−1 0 keywords: + a−1; a0 ∫; a1 ∫; b1θ(1)−1 ∫; ds−b1θ(1)−1 ∫; nx)(u; nx)(u)du; value; θ(s)−1 ∫; ψ0[x; ψ1[x; − ∫; ∫ t cache: ejde-1658.pdf plain text: ejde-1658.txt item: #143 of 601 id: ejde-166 author: Barbatis, Gerassimos; Branikas, Panagiotis title: Heat kernel estimates for fourth-order non-uniformly elliptic operators with non-strongly convex symbols date: 2022 words: 4925 flesch: 74 summary: This implies [1, Theorem 7.12] an analogous inequality for the symbol A(x, ξ) of H, namely ReA(x, ξ) ≥ cw(x)|ξ|4 , x ∈ Ω , ξ ∈ R2. We define the weighted Sobolev space W 1,∞ w (Ω) = {u ∈W 1,∞ loc (Ω) : ∃c ≥ 0 : |u(x)| ≤ cw(x), |∇u(x)| ≤ cw(x)3/4, x ∈ Ω}. keywords: q(x cache: ejde-166.pdf plain text: ejde-166.txt item: #144 of 601 id: ejde-1668 author: Zheng, Jiashan; Wang, Yuying title: Blow-up prevention and rate of convergence of solutions for N-dimensional parabolic-parabolic systems with consumption of chemoattractant date: 2025 words: 10636 flesch: 62 summary: + ∫ t 0 ∥∇e(t−s)∆u(·, s)v(·, s)∥Lq(Ω)ds ≤ C1 + C2 ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1(t−s)∥u(·, s)v(·, s)∥Lk(Ω) (4.22) for all t ∈ (0, Tmax), where λ1 is the first positive eigenvalue of −∆ under homogeneous Neumann boundary conditions. From (4.21)and the identity −1 2 − N 2 (1 k − 1 q ) > −1 (4.23) we obtain ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1(t−s)ds ≤ ∫ ∞ 0 ( 1 + σ− 1 2− N 2 ( 1 k− 1 q ) ) e−λ1σdσ < +∞ for all t ∈ (0, Tmax), (4.24) and that for some C3 > 0, the latter on the right-hand side of (4.22) bringing together (4.23)-(4.24) also trivially provides C2 ∫ t 0 ( 1 + (t− s)− 1 2− N 2 ( 1 k− 1 q ) ) keywords: chemotaxis; lemma; system cache: ejde-1668.pdf plain text: ejde-1668.txt item: #145 of 601 id: ejde-167 author: Kratou, Mouna title: Kirchhoff systems involving fractional p-Laplacian and singular nonlinearity date: 2022 words: 6618 flesch: 83 summary: β < 1, 2− α− β < p ≤ pθ < q < p∗s, then, there exists a number Λ0 = (q + α+ β − 2 ‖c‖∞k(q − p) ) p p+α+β−2 ( 2− α− β − q k(2− α− β − p) |Ω| p∗s−q p∗s )− p p−q S 2−α−β p+α+β−2 , such that for 0 < (λ‖a‖∞) p p−q + (µ‖b‖∞) p p−q < Λ0, problem (1.1) has at least two nontrivial positive solutions. There exists Λ0 = (q + α+ β − 2 ‖c‖∞k(q − p) ) p p+α+β−2 ( 2− α− β − q k(2− α− β − p) |Ω| 2∗s−q 2∗s ) keywords: p−q cache: ejde-167.pdf plain text: ejde-167.txt item: #146 of 601 id: ejde-168 author: Dutta, Prerona title: Extending Lagrangian transformations to nonconvex scalar conservation laws date: 2022 words: 5505 flesch: 76 summary: � In conclusion, we observe that ρ is an admissible weak solution to the Cauchy problem for the scalar conservation law (2.4) where ρ = σ − L and σ is given by (2.20). Finally we establish that entropy admissible weak solutions to this system are equivalent to those of the scalar equation. keywords: conservation; system cache: ejde-168.pdf plain text: ejde-168.txt item: #147 of 601 id: ejde-1684 author: Liu, Qingbo; Zhao, Lan title: Global bifurcation for semilinear eigenvalue problems involving nonlocal terms date: 2025 words: 9096 flesch: 79 summary: In this situation, ψ′(|α|2/2) = λ1 − λ1 = 0. Case 2: λ = λ1. keywords: eigenvalue; theorem cache: ejde-1684.pdf plain text: ejde-1684.txt item: #148 of 601 id: ejde-169 author: Correia, Jeziel N.; Oliveira, Claudionei P. title: Existence of positive solutions for fractional Laplacian systems with critical growth date: 2022 words: 13968 flesch: 89 summary: × Lqloc(RN ), (un, vn)→ (u, v) a.e. in RN × RN . Arguing in the same way, we have∫ RN b(x)|Φδ,b|2dx ≤ |b|q|Φδ,0|22t, ∀b ∈ RN . From Lemma 4.1(iii), given ε > 0 there exists δ = δ(ε) > 0 such that sup b∈RN f(`0Φδ,b, t0Φδ,b) ≤ SH + ε 2 + ε 2 ≤ SH + ε, ∀δ ∈ (0, δ]. keywords: ds,2(rn; lemma; on(1; y|n+2s; |x− cache: ejde-169.pdf plain text: ejde-169.txt item: #149 of 601 id: ejde-17 author: Anh, Nguyen Thi Van; Yen, Bui Thi Hai title: Existence and controllability for neutral partial differential inclusions nondenselly defined on a half-line date: 2023 words: 8739 flesch: 78 summary: Pyt + limλ→∞ ∫ t 0 S′(t− s)RλLysds + limλ→∞ ∫ t 0 S′(t− s)Rλg(s)ds, g ∈ SF,y, if t ∈ J. It is clear that the fixed points of the operator N are integral solutions of the problem (1.1)-(1.2). To be precise, the following equation was considered on the whole line d dt D(xt) = AD(xt) + L(xt) + f(t), t ∈ R. The authors proved the existence of an almost automorphic solution for above equation and applied abstract results to a neutral wave equation with delay. keywords: d(a; differential; lim; s′(t−; ‖p‖ cache: ejde-17.pdf plain text: ejde-17.txt item: #150 of 601 id: ejde-171 author: Baroni, Paolo; Coscia, Alessandra title: Gradient regularity for non-autonomous functionals with Dini or non-Dini continuous coefficients date: 2022 words: 13378 flesch: 77 summary: In particular, when p ≥ 2, |z1 − z2|p ≤ c|Vp(z1)− Vp(z2)|2 holds, while for 1 < p ≤ 2 (see [38, Lemma 2]) we will use that |z1 − z2| ≤ c ∣∣Vp(z1)− Vp(z2) ∣∣2/p + c|z1|(2−p)/2 ∣∣Vp(z1)− Vp(z2) ∣∣ (2.24) EJDE-2022/80 REGULARITY FOR DOUBLE PHASE FUNCTIONALS 9 both for a suitable constant c ≡ c(p). |z2|)|z1 − z2|2 ≤ ∣∣Vϕ(z1)− Vϕ(z2) ∣∣2 ≤ cϕ′′(|z1|+ keywords: b2r; estimate; lemma; regularity; − ∫; ∣∣2 cache: ejde-171.pdf plain text: ejde-171.txt item: #151 of 601 id: ejde-1712 author: Dias, Fabio Scalco; Oliveira, Regilene; Valls, Claudia title: Dynamics of a May-Leonard asymmetric system of ordinary differential equations date: 2025 words: 9203 flesch: 75 summary: Regions r0 r1 r2 r3 s0 s1 w0 PP R1 UN SN S S UN SN S R1 R2 UN S S SN UN S UN R1 R3 UN S S S UN SN UN R3 R4 UN S SN S S UN UN R1 R5 UN S SN SN S S UN R5 R6 UN SN SN S S UN S R5 R7 UN SN S SN UN S S R5 L1,2 UN S-N S r1 UN ∄ S-N L1,2 L2,3 UN S S ∄ UN S-N UN L2,3 L2,4 UN S S-N r2 S-N s0 UN L1,2 L4,5 UN S SN ∄ p0 p1 p2 p3 q1 q2 w0 PP R1 S SN UN S S SN UN R1 R2 S S UN SN UN S UN R2 R3 S SN UN S UN S UN R3 R4 S SN UN S S UN UN R1 R5 UN SN S S S UN UN R3 R6 UN S S SN S UN UN R1 R7 S UN SN S S UN UN R1 R8 S UN SN S UN S UN R3 R9 UN S S SN UN S UN R3 R10 UN SN S S UN S UN R3 R11 S S SN UN UN S UN R2 R12 S UN SN S S SN UN R1 R13 UN S S SN S SN UN R1 R14 UN SN S S S SN UN R3 P1 S-N ∄ keywords: s s; sn s; system; un s cache: ejde-1712.pdf plain text: ejde-1712.txt item: #152 of 601 id: ejde-172 author: Lin, Xiaolu; Zheng, Shenzhou title: Mixed local and nonlocal Schrodinger-Poisson type system involving variable exponents date: 2022 words: 6975 flesch: 79 summary: = α|u|p(x)−2u+ β|u|q(x)−2u in Ω, −∆φ = up in Ω, u = φ = 0 in RN \ Ω, (1.1) where λ is a positive parameter, and V (x) ∈ C(RN ) is a potential function. = u(x)e−ıt to the time-dependent Schrödinger-Poisson system −i∂ψ ∂t = −∆ψ + φ(x)ψ − f(ψ) in Ω, −∆φ = |ψ|2 in Ω, ψ = φ = 0 on ∂Ω. (1.2) keywords: lemma; schrödinger; solutions; theorem cache: ejde-172.pdf plain text: ejde-172.txt item: #153 of 601 id: ejde-174 author: Zhao, Haiqin; Wu, Shi-Liang title: Regular traveling waves for a reaction-diffusion equation with two nonlocal delays date: 2022 words: 6763 flesch: 87 summary: f(U2(ξ1 − y − cτ1))]dy + ω2 ∫ R Γ2(D2τ2, y)[f(U1(ξ1 − y − cτ2))− f(U2(ξ1 − y − cτ2))]dy ≤ −µΠ(ξ1)eν1ξ1 + ω1f ′(0) ∫ R Γ1(D1τ1, y) ×max{0, U1(ξ1 − y − cτ1)− U2(ξ1 − y − cτ1)}dy + ω2f ′(0) ∫ R Γ2(D2τ2, y) max{0, U1(ξ1 − y − cτ1))dy + ω1 ∫ R Γ2(D2τ2, y)f(U(ξ − y − cτ2))dy = 0, (2.2) where ω1 := pe−d1τ1 and ω2 := (1− p)e−d2τ2 . keywords: waves cache: ejde-174.pdf plain text: ejde-174.txt item: #154 of 601 id: ejde-175 author: Han, Xiao; He, Yujing; Wei, Hui title: Existence of positive periodic solutions for a nonlinear system of second-order ordinary differential equations date: 2022 words: 4352 flesch: 79 summary: By using the fixed point theorem of cone expansion and compression we obtain the existence of positive periodic solutions. Existence ofsolutions; positive periodic solutions; fixed point theorem. keywords: solutions cache: ejde-175.pdf plain text: ejde-175.txt item: #155 of 601 id: ejde-176 author: Ortegon Gallego, Francisco; Ouyahya, Hakima; Rhoudaf, Mohamed title: Existence of a solution and its numerical approximation for a strongly nonlinear coupled system in anisotropic Orlicz-Sobolev spaces date: 2022 words: 11920 flesch: 77 summary: It consists of two coupled nonlinear elliptic equations governing the temperature, u, and the electric potential, ϕ, inside a semiconductor device, namely, −A(u) = ρ(u)|∇ϕ|2 in Ω div(ρ(u)∇ϕ) = 0 in Ω, ϕ = ϕ0 on ∂Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ Rd (the thermistor geometry) is a bounded domain, d ≥ 2 is an integer, and the operator A, given by A(u) , d, the function ai(x, s, ζ) : Ω×R×R 7→ R is a Carathéodory function, that is, measurable with respect to x in Ω for all (s, ζ) ∈ R2, and continuous with respect to (s, ζ) for keywords: ai(x; d∑ i=1; i=1; solution; ω ai(x cache: ejde-176.pdf plain text: ejde-176.txt item: #156 of 601 id: ejde-177 author: Molica Bisci, Giovanni; Servadei, Raffaella; Zhang, Binlin title: Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications date: 2022 words: 8698 flesch: 76 summary: The space Xs 0(Ω) is defined as Xs 0(Ω) := { g ∈ X : g = 0 a.e. in Rn \ Ω } , EJDE-2022/85 MONOTONICITY PROPERTIES OF EIGENVALUES 5 where X denotes the linear space of Lebesgue measurable functions from Rn to R such that the restriction to Ω of any function g in X belongs to L2(Ω) and the map (x, y) 7→ (g(x)− g(y)) ∣∣ 6 qν(x) a.e. x ∈ Rn (4.43) for all j ∈ N. By (1.5), (4.42), (4.43), and the Lebesgue Dominated Convergence Theorem, we obtain that ∫ Ω f(x, uj(x))uj(x)dx→ ∫ Ω f(x, u∞(x))u∞(x) dx∫ Ω f(x, uj(x))u∞(x) dx→ ∫ Ω f(x, u∞(x))u∞(x) dx (4.44) as j → +∞, while, by (1.7) and (4.42) we obtain∫ Ω g(x)uj(x) dx→ ∫ Ω g(x)u∞(x) dx (4.45) as j → +∞. keywords: a.e; f(x; problem; theorem cache: ejde-177.pdf plain text: ejde-177.txt item: #157 of 601 id: ejde-178 author: Xu, Jiaohui; Caraballo, Tomas title: Well-posedness of stochastic time fractional 2D-Stokes models with finite and infinite delay date: 2022 words: 11245 flesch: 82 summary: + ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)F (s, us)ds + ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)G(s, us)dW (s), t ∈ (0, T ], P-a.s. (3.5) On the other hand, for t ∈ (0, T ], we have E‖(Nu)(t)‖2 ≤ 3E‖Eα(−tαA)ϕ(0)‖2 + 3E ∥∥ ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)F (s, us)ds ∥∥2 + 3E ∥∥ ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)G(s, us)dW (s) ∥∥2 := I1 + I2 + I3. (3.6) Now we estimate each term on the right-hand side of (3.6). keywords: 2α−; f t; fractional; lemma; sup; time cache: ejde-178.pdf plain text: ejde-178.txt item: #158 of 601 id: ejde-179 author: Camasta, Alessandro; Fragnelli, Genni title: Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions date: 2022 words: 9200 flesch: 78 summary: Thus, we introduce Y := { u ∈ H2 1/a(0, 1) : u(x0) = (au′)(x0) = 0 } and, proceeding as in [9] and [21] (if x0 ∈ {0, 1}) or as in [7] (if x0 ∈ (0, 1)), one can prove the following result. If u0 ∈ Xµ and h ∈ L2(0, T ;Xµ), a function u is said to be a weak solution of (3.7) if u ∈ C ( [0, T ];Xµ ) ∩ L2 ( 0, T ;H2 a(0, 1) ) and∫ 1 0 u(T, x)ϕ(T, x) dx− ∫ 1 0 u0(x)ϕ(0, x) dx− ∫ (0,T )×(0,1) u(t, x)ϕt(t, x) dx dt + a(1)u(T, 1)ϕ(T, 1) β1 − a(1)u0(1)ϕ(0, 1) β1 − a(1) β1 ∫ T 0 u(t, 1)ϕt(t, 1)dt + a(0)u(T, 0)ϕ(T, 0) β0 − a(0)u0(0)ϕ(0, 0) β0 − a(0) β0 ∫ T 0 u(t, 0)ϕt(t, 0)dt = − ∫ (0,T )×(0,1) a(x)uxx(t, x)ϕxx(t, x) dx dt− γ1 β1 ∫ T 0 a(1)u(t, 1)ϕ(t, 1)dt − γ0 β0 ∫ T 0 a(0)u(t, 0)ϕ(t, 0)dt+ ∫ (0,T )×(0,1) h(t, x)ϕ(t, x) dx dt + ∫ T 0 a(1)h(t, 1)ϕ(t, 1) keywords: a(0; boundary; conditions; h2 a(0 cache: ejde-179.pdf plain text: ejde-179.txt item: #159 of 601 id: ejde-1792 author: Munoz Rivera, Jaime; Ochoa Ochoa, Elena; Quintanilla, Ramon title: Thermoelastic plates with type I heat conduction with second gradient date: 2025 words: 5138 flesch: 73 summary: d iω ∫ Ω ∆θ(iρωv − η∆θ − g2)dΩ = dρ ∫ Ω ∆θ v dΩ+ d iω ∫ Ω η|∆θ|2dΩ+ d iω ∫ Ω ∆θg2dΩ. Finally, multiplying equation (2.6) by u we find that iρ ∫ Ω ωvu dΩ+ c ∫ Ω |∆u|2 dΩ− η ∫ Ω ∆θu dΩ = ∫ Ω g2u dΩ. Using equation (2.5) we obtain ρ ∫ Ω |v|2 dΩ = −ρ ∫ Ω vg1 dΩ+ c ∫ Ω |∆u|2 dΩ− η ∫ Ω ∆θu dΩ− ∫ Ω g2u dΩ. The above inequality implies∫ Ω |v|2 dΩ ≤ c ∫ Ω |∆u|2 dΩ+ c ∫ Ω |∆θ|2 dΩ+ c̃ϵ∥U∥H∥G∥H. Using (3.3) and (2.8) we obtain∫ Ω |v|2 dΩ ≤ c̃ϵ∥U∥H∥G∥H + c̃∥G∥2H, for ϵ small. keywords: equation; semigroup; solutions cache: ejde-1792.pdf plain text: ejde-1792.txt item: #160 of 601 id: ejde-1794 author: Almeida, Wendy F.; Figueiredo, Giovany M. title: Solutions to magnetic Schrodinger equations with arbitrary growth at infinity date: 2025 words: 5491 flesch: 74 summary: Recently, the study of magnetic Schrödinger equations has been approached from various per- spectives, however only a limited number of works have addressed this topic. Magnetic Schrödinger equations; arbitrary growth at infinity. keywords: g(x; magnetic cache: ejde-1794.pdf plain text: ejde-1794.txt item: #161 of 601 id: ejde-180 author: Xie, Zheng; Chen, Jing title: Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2 date: 2023 words: 7278 flesch: 82 summary: • for x ∈ R2 and r > 0, Br(x) := {y ∈ R2 : |y − x| < r}. In view of (A4), (A5), (A7), and (A10), it is easy to deduce that g is a Carathéodory function and satisfying the following properties: (A11) g(x, y, t) ≤ δt+ f(t) for any t ≥ 0 and δ ≥ 0; (A12) limt→0 g(x,y,t) t = 0 uniformly in (x, y) ∈ R2; (A13) 0 < 2G(x, y, t) keywords: problem; solutions cache: ejde-180.pdf plain text: ejde-180.txt item: #162 of 601 id: ejde-181 author: Duan, Yubo; Jiang, Yiming; Tian, Yang; Wei, Yawei title: Stochastic Burgers equations with fractional derivative driven by fractional noise date: 2023 words: 7192 flesch: 78 summary: Lαβ,β(t2 − s)B(u(s))ds‖p Ḣγ = E‖ ∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 ) AγL α β,β(t2 − s)B(u(s))ds‖p 6 C(α, β, γ)E (∫ t1 0 ‖((t2 − s)β−1 − (t1 − s)β−1)(t2 − s)− βγ α ‖‖B(u(s))‖ds )p 6 C(α, β, γ,M, p) (∫ t1 0 ( (t1 − s)β−1 − (t2 − s)β−1 ) p p−1 (t2 − s)− pβγ α(p−1) ds ) B(u(s))ds‖p 6 E (∫ t1 0 (t1 − s)β−1‖Aγ ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) B(u(s))‖ds )p 6 C(α, β, γ)(t2 − t1) pβγ α E (∫ t1 0 (t1 − s)β−1‖B(u(s))‖ds )p . keywords: c(α; ∫ ∞ cache: ejde-181.pdf plain text: ejde-181.txt item: #163 of 601 id: ejde-1812 author: Liang, Jin; Mu, Yunyi; Xiao, Ti-Jun title: Evolution psi-Hilfer fractional differential equations in Banach spaces date: 2025 words: 7228 flesch: 78 summary: respectively:( Dα,β;ψx ) (t) = Ax(t) + f ( t, x(t), ∫ t 0 ρ(t, s)x(s)ds ) , 0 < α < 1, 0 ≤ β ≤ 1, t ∈ (0, b], I1−γ;ψx(0) = x0, α ≤ γ = α+ β − αβ < 1, (4.4) and( Dα′,β′;ψy ) (t) = Ay(t) + f ( t, y(t), ∫ t 0 ρ(t, s)y(s)ds ) , 0 < α′ < 1, 0 ≤ β′ As a matter of fact, for each x1, x2 ∈ C1−α−β(1−α);ψ(J,X) and t ∈ J , we can obtain ψ1−α−β(1−α)(t)∥(Tx1)(t)− (Tx2)(t)∥ ≤ ψ1−α−β(1−α)(t) ∫ t 0 ∥Kα(ψ(t)− ψ(s))[f(s, x1(s))− f(s, x2(s))]∥ψ′(s)ds ≤ Mℓ2 Γ(α) ψ1−α−β(1−α)(t) ∫ t 0 (ψ(t)− ψ(s))α−1ψα+β(1−α)−1(s) 6 J. LIANG, Y. MU, T.-J. XIAO EJDE-2025/109 × ψ1−α−β(1−α)(s)∥x1(s)− x2(s)∥ψ′(s)ds ≤ Γ(α+ β(1− α))Mℓ2ψ α(t) Γ(2α+ β(1− α)) ∥x1 keywords: differential; equations; fractional cache: ejde-1812.pdf plain text: ejde-1812.txt item: #164 of 601 id: ejde-1820 author: Mishra, Shivam Kumar; Abbas, Syed; Nieto, Juan Jose title: Periodic solution and stationary distribution of a stochastic epidemic modelwith two different epidemics and different epidemiological frameworks date: 2025 words: 9484 flesch: 66 summary: − lnI1 − lnI2 + κ(t). Therefore, in recent times, many studies have been done on stochastic epidemic models and researchers have demonstrated how environmental noise affects population model dynamics (see [11, 22, 7]). keywords: epidemic; model; stochastic; system cache: ejde-1820.pdf plain text: ejde-1820.txt item: #165 of 601 id: ejde-183 author: Abebe, Abraham; Chhetri, Maya title: A nonexistence result for p-Laplacian systems in a ball date: 2023 words: 3894 flesch: 83 summary: See also [10], where nonexistence of positive solutions is established when a weight function is large for a semipositone superlinearproblem in a ball. In this case, however, the nonexistence result for positive solution for λ large has been extended to the case when Ω is a smooth bounded domain in RN (N ≥ 2) in [4]. keywords: p−1 cache: ejde-183.pdf plain text: ejde-183.txt item: #166 of 601 id: ejde-184 author: Bao, Qinglan; Wei, Guangsheng; Zettl, Anton title: Friedrichs extension of singular symmetric differential operators date: 2023 words: 11944 flesch: 82 summary: C−1 = −C = C∗, (1.1) and let Z2n(I) := {(qr,s)2nr,s=1 ∈M2n(L1 loc(I)), qr,r+1 6= 0 We have Ĝ = ( 0 G1 −G∗1 0 ) , G1 = ( Ĉda−n 0 0 −Ĉdb−n ) . keywords: differential; friedrichs; matrix; operator; solution; theorem cache: ejde-184.pdf plain text: ejde-184.txt item: #167 of 601 id: ejde-185 author: Behncke, Horst; Hinton, Don title: Spectral theory of C-symmetric non-selfadjoint differential operators of order 2n date: 2023 words: 12051 flesch: 78 summary: Since the Fredholm index of Tmin − z is constant in K0 and dimN(Tmax− z) = dimN(T+ max− z̄), it follows that (H1) holds in K0. = { Φ̃(x, z)χ∗(t, z), a ≤ x ≤ t, χ(x, z)Φ∗(t, z), a ≤ t < x, (3.10) are the integral kernels or Green’s functions of the resolvents Rz = (Tα − z)−1, respectively R̃z = (T+ α − z), i.e., (Rzf)(x) = ∫ ∞ a G(z, x, t)A(t)F (t)dt. where F is as in (4.5) below. keywords: case; coefficients; conditions; differential; exp; operators; selfadjoint; spectrum; theorem cache: ejde-185.pdf plain text: ejde-185.txt item: #168 of 601 id: ejde-1857 author: Boussetouan, Imane; Amrouche, Cherif title: Existence and regularity of solutions for elliptic systems with mixed boundary conditions date: 2025 words: 13892 flesch: 83 summary: (iii) Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), a× n ∈ W2−1/p,p(ΓN ), g ∈W 1−1/p,p(ΓN ), b ∈W 2−1/p,p(ΓD), and h× n ∈ W1−1/p,p(ΓD), then u belongs to W2,p(Ω) and ∥u∥W2,p(Ω) ≤ C ( ∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h× n∥W1−1/p,p(ΓD) + ∥a× n∥W2−1/p,p(ΓN ) + ∥b∥W 2−1/p,p(ΓD) ) . Furthermore, if Ω is of class C 2,1, f ∈ Lp(Ω), g ∈ W 1−1/p,p(ΓN ), and h ∈ W1−1/p,p(ΓD) with F = 0, then u belongs to W2,p(Ω) and ∥u∥W2,p(Ω) ≤ C ( ∥f∥Lp(Ω) + ∥g∥W 1−1/p,p(ΓN ) + ∥h∥W1−1/p,p(ΓD) ) . keywords: problem; solution cache: ejde-1857.pdf plain text: ejde-1857.txt item: #169 of 601 id: ejde-186 author: Benedikt, Jiri; Pulpan, Jan title: Numeric estimates of the principal eigenvalue of the p-Laplacian using interval arithmetic date: 2023 words: 4092 flesch: 65 summary: t = LinRange(0, 1, n-1) tI = [@interval(i) for i in t] U1 = [u[1] for u in sol(t).u] U1 I = [@interval(u[1]) for u in sol(t).u] U2 = [u[2] for u in sol(t).u] U2 I = [@interval(u[2]) for u in sol(t).u] return t, tI, U1, U1 I, U2, U2 I, Λ1 end The function plaplace solve returns vectors U1 and U2 of numerical approxi- mations of the values of u1 = ϕ1,p and u2 at n (a parameter) equidistant division points t of the interval [0, 1]. Returns spline coefficients ‘csc V‘ as well as interval values ‘V‘ of the spline function. keywords: end; function; interval cache: ejde-186.pdf plain text: ejde-186.txt item: #170 of 601 id: ejde-188 author: Castro, Alfonso; Jacobsen, Jon title: Regular solutions to elliptic equations date: 2023 words: 6490 flesch: 76 summary: Nonlinear elliptic equation; radial solution; regular radial solution; singular radial solution; bifurcation analysis; Pohozaev identity; shooting method; superlinear nonlinearity; subcritical nonlinearity; sub-super critical nonlinearity; jumping nonlinearity. Similarly, when Ω is a ball or an annulus and g is radial in x, the existence of radial solutions to (1.2) is largely determined by the relation between g(x, u)/u and the set σrad(−∆) = {ρi : i = 1, 2, . . .}. keywords: equations; solutions cache: ejde-188.pdf plain text: ejde-188.txt item: #171 of 601 id: ejde-1891 author: Wu , Jingpeng; Zhu, Min title: Initial-boundary value problem of plasma-charge model in the half space date: 2025 words: 9509 flesch: 86 summary: Then by (3.4) and take K2 large enough (depending only on K1,K) such that we have Ï ≥ h(Y (s),W (s))−K1 − |Y − ξ̊| ( |Fρ(Y )|+ δ−2 0 ) ≥ 1 4 R2 −K1 − C(l + 1)Q 4/3 t,δ ≥ 1 8 R2. Then by (3.4) and take K2 large enough (depending only on K1,K) such that we have Ï ≥ h(Y (s),W (s))−K1 − |Y − ξ̊| ( |Fρ(Y )|+ δ−2 0 ) ≥ 1 16 Q2 t,δ −K1 − C(l + 1)Q 4/3 t,δ ≥ 1 32 Q2 t,δ. keywords: plasma cache: ejde-1891.pdf plain text: ejde-1891.txt item: #172 of 601 id: ejde-19 author: Levandosky, Julie L.; Vera, Octavio title: Smoothing properties for a coupled Zakharov-Kuznetsov system date: 2023 words: 15874 flesch: 86 summary: For ξν as defined in (6.3), α = (α1, α2) such that |α| = β, 4 ≤ β ≤ K, the following holds:∑ |α|=β ∣∣ ∫ t 0 ∫ ξν(∂αu)∂α(uux) ∣∣+ ∣∣ ∫ t 0 ∫ ξν(∂αv)∂α(vvx) ∣∣ ≤ C + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αu)2 ) + C ∑ |α|=β (∫ t 0 ∫ ξν(∂αv)2 ) (6.21) for 0 ≤ t ≤ T , where C depends only on sup 0≤t≤T ∫ ξν(∂γu)2, sup 0≤t≤T ∫ ξν(∂γv)2, (6.22)∫ T 0 ∫ (ξν)x(∂γux)2, ∫ T 0 ∫ (ξν)x(∂γvx)2, (6.23)∫ T 0 ∫ (ξν)x(∂γuy)2, ∫ T 0 ∫ (ξν)x(∂γvy)2 (6.24) for γ = (γ1, γ2) where |γ| ≤ β − 1. EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 35 The proof uses the same ideas as in the proof of Lemma 3.2. In this case, the remainder terms satisfy∣∣ ∫ t 0 ∫ ξuxy(uux)xy ∣∣ EJDE-2023/11 SMOOTHING PROPERTIES FOR ZAKHAROV-KUZNETSOV SYSTEMS 9 = ∣∣ ∫ t 0 ∫ ξuxy(2uxuxy + uyuxx + uuxxy) ∣∣ ≤ |ux|L∞ ∫ t 0 ∫ ξu2xy + C|uy|L∞ ∫ t 0 ∫ ξu2xx + |uy|L∞ ∫ t 0 ∫ ξu2xy ≤ C ∫ t 0 ∫ ξ(u2xx + u2xy) where C depends only on ‖u‖H3 . keywords: c ∫; case; solution; sup; term; |α| t; ∂αuxxx; ∫ b; ∫ r2; ∫ t; ∫ ξν cache: ejde-19.pdf plain text: ejde-19.txt item: #173 of 601 id: ejde-190 author: Delgado, Briceyda B.; Pardo, Rosa title: Resonant solutions for elliptic systems with Neumann boundary conditions date: 2023 words: 6745 flesch: 76 summary: By the maximum principle [2, Theorem 4.1], for all h ≥ 0, h 6= 0, we have v = K h ∈ P̊ , where P̊ = {u ∈ C(Ω): u > 0 in Ω}. The nonlinearity f = (f1, f2), where fi : Ω × R2 → R, i = 1, 2 are Carathéodory functions, that is, fi = fi ( x, s ) are measurable in x ∈ Ω and continuous with respect to s = (s1, s2) ∈ R2. keywords: 1/µ+; lim; solutions cache: ejde-190.pdf plain text: ejde-190.txt item: #174 of 601 id: ejde-191 author: Ghimenti, Marco G.; Micheletti, Anna Maria title: Yamabe boundary problem with scalar-flat manifolds target date: 2023 words: 6745 flesch: 76 summary: By the maximum principle [2, Theorem 4.1], for all h ≥ 0, h 6= 0, we have v = K h ∈ P̊ , where P̊ = {u ∈ C(Ω): u > 0 in Ω}. The nonlinearity f = (f1, f2), where fi : Ω × R2 → R, i = 1, 2 are Carathéodory functions, that is, fi = fi ( x, s ) are measurable in x ∈ Ω and continuous with respect to s = (s1, s2) ∈ R2. keywords: 1/µ+; lim; solutions cache: ejde-191.pdf plain text: ejde-191.txt item: #175 of 601 id: ejde-192 author: Hollifield, Elliott title: Positive solutions for nonlinear fractional Laplacian problems date: 2023 words: 5805 flesch: 71 summary: This charac- terization allowed them to prove several regularity results by using local techniques and provides a framework for interested researchers to further the study of the still emerging field of fractional Laplacian problems. [7] Maya Chhetri, Petr Girg; Some bifurcation results for fractional Laplacian problems. keywords: fractional; laplacian cache: ejde-192.pdf plain text: ejde-192.txt item: #176 of 601 id: ejde-195 author: Knowles, Ian; Tamang, Sundar title: Inverse volatility problem for currency options date: 2023 words: 4882 flesch: 76 summary: − wc,λ)2. [ [(λ+ rF )(w2 λ − w2 c,λ)− 2β(wλ − wc,λ)] 2 K2ρ2 + (w′2λ − w′2c,λ) ] , (3.7) and n(K) keywords: functional; volatility cache: ejde-195.pdf plain text: ejde-195.txt item: #177 of 601 id: ejde-196 author: Li, Meiqin; Ji, Bingbing; Zhou, Jianxin title: Local min-orthogonal principle and its applications for solving multiple solution problems date: 2023 words: 7575 flesch: 80 summary: → p(v) leads to J ′(p(vk))→ J ′(p(v)) Locally M-type saddles with J > 0 J = 46.1140, ‖u‖∞ = 4.5370 at (0.0208,−0.0104) J = 29.4731, ‖u‖∞ = 5.7561 at (0.6510,−0.0052) J = 17.6390, ‖u‖∞ = 4.6441 at (0.6458,−0.6354) (a)NMO = 15 NMO = 90 NMO = 110 Figure 8. keywords: max; method; min; saddle; type cache: ejde-196.pdf plain text: ejde-196.txt item: #178 of 601 id: ejde-1962 author: Wei, Yawei; Zhou, Xiaodong title: Construction of single-peak solutions for Grushin equations via reduction method date: 2025 words: 7234 flesch: 84 summary: = ( 1 (1 + γ)2 |x|2+2γ + |y|2 ) 1 2+2γ (1.15) for z = (x, y) ∈ RN+l, and set B̃r(0) : For example, in [2], for u ∈ D1,2 γ (RN+l) and ρ > 0, a rescaled sequence of functions of the form ue,ρ(z) := ρ Nγ−2 2 u(ρx, ρ1+γy+ e) is also defined, where z = (x, y) ∈ RN+l and e ∈ Rl. keywords: equation; rn+l; solutions; zε(z; ∂uε; ∂yj cache: ejde-1962.pdf plain text: ejde-1962.txt item: #179 of 601 id: ejde-197 author: Mariani, Maria C.; Asante, Peter K.; Kubin, William; Tweneboah, Osei K.; Beccar-Varela, Maria title: Determining the background driving process of the Ornstein-Uhlenbeck model date: 2023 words: 6693 flesch: 61 summary: [38] proposed the Detrended Fluc- tuation Analysis (DFA) while examining a sequence of DNA nucleotides to study the self-similarity [35] and long-range dependence of time series. The reader is invited to read [3, 32, 42, 43] for further information on the Shan- non entropy, transformation of time series into diffusion processes and the derivation of the shannon entropy for the stationary and non-stationary series. keywords: analysis; data; differential; equation; lévy; model; ornstein; process; series; stochastic; time; uhlenbeck cache: ejde-197.pdf plain text: ejde-197.txt item: #180 of 601 id: ejde-1971 author: Kim, Tujin title: Non-steady magneto-hydrodynamics-heat system with joule and buoyancy effects under mixed boundary conditions date: 2025 words: 17906 flesch: 83 summary: (4.38) EJDE-2025/119 NON-STEADY MAGNETOHYDRODYNAMICS-HEAT SYSTEMS 19 Taking into account (4.37), (4.38) and applying the inequality |a + b|p ≤ 2p(|a|p + |b|p), p ∈ (1,∞), we have I2 ≡ 1 ∥u∥L6(0,T ;V) ∣∣∣ ∫ T 0 [ ek1t⟨curl(ŵ + v0)× (ŵ + v0), u⟩ ] dt ∣∣∣ ≤ c ∥u∥L6(0,T ;V) ∫ T 0 ∥ curl(ŵ + v0)∥L2∥(ŵ + v0)∥1/2L2 ∥(ŵ + v0)∥1/2V ∥u∥L6 dt ≤ ∥ŵ + v0∥1/2C(0,T ;L2) ( c ∥u∥L6(0,T ;V) ∥ŵ + v0∥3/2L9/5(0,T ;V) ∥u∥L6(0,T ;V) ) ≤ ∥ŵ + v0∥C([0,T ];HV) + c∥(ŵ + v0)∥3L9/5(0.T ;V) ≤ c∥ŵ′∥1/2 L6/5(0,T ;V∗) ∥ŵ∥1/2L6(0,T ;V) + ∥v0∥+ c∥ŵ + v0∥3L6(0,T ; Let us estimate∣∣∣ ∫ T 0 ek1t⟨(ŵ + v0)θ0,∇θ̂⟩ dt ∣∣∣ = ∣∣∣ ∫ T 0 [ ek1t⟨ŵθ0,∇θ̂⟩+ ek1t⟨v0θ0,∇θ̂⟩ ] dt ∣∣∣. First, we have∣∣∣ ∫ T 0 ek1t⟨ŵθ0,∇θ̂⟩ dt ∣∣∣ ≤ ∫ T 0 ek1t∥ŵ∥L3∥θ0∥L6∥∇θ̂∥ dt ≤ κ0 12 ∥θ̂∥2 L2(0,T ;W 1,2 ΓD ) + c′Te4k1T ε ∥θ0∥4W 1,2 + ε 6 ∥ŵ∥6L6(0,T ;V). keywords: + k; boundary; conditions; curl; ek1 t; h̄0; h̊(t; problem; t curl; taking; − ∫; ∣∣∣; ∫ t cache: ejde-1971.pdf plain text: ejde-1971.txt item: #181 of 601 id: ejde-198 author: Mavinga, Nsoki; Morris, Quinn A.; Robinson, Stephen B. title: Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions date: 2023 words: 9836 flesch: 88 summary: First, we establish that the functional J is uniformly Lipschitz in α, β, and x. Note that |J(α2, β2, x)− J(α1, β1, x)| = 1 2 ∣∣∣(α2 − α1)‖x+‖2(m,ρ) + (β2 − β1)‖x−‖2(m,ρ) ∣∣∣ ≤ 1 2µ1 ‖x‖2(c,σ) (|α2 − α1|+ |β2 − β1|) ≤ 1 2µ1 K (|α2 − α1|+ |β2 − β1|) . = ‖x2 − x1‖2(c,σ) − 〈α2x2 − α1x1, x2 − x1〉(m,ρ) + 〈s2(x2 + y2)− − s1(x1 + y1)−, x2 − x1〉(m,ρ) = ‖x2 − x1‖2(c,σ) − α2‖x2 − x1‖2(m,ρ) − (α2 − α1)〈x1, x2 − x1〉(m,ρ) + s2〈(x2 + y2)− − (x1 + y1)−, x2 − x1〉(m,ρ) + (s2 − s1)〈(x1 + y1)−, x2 − x1〉(m,ρ) (2.4) keywords: lemma; β(y cache: ejde-198.pdf plain text: ejde-198.txt item: #182 of 601 id: ejde-2 author: Lan, Kunquan title: Linear higher-order fractional differential and integral equations date: 2023 words: 9827 flesch: 85 summary: If u ∈ L1(a, b) satisfies (Iαa+(u − Pn−1))(n−1) ∈ AC[a, b], then for each x ∈ (2) If u ∈ L1(a, b) satisfies (Iαa+(u−Pn−1))(n−1) ∈ AC[a, b] and u is a solution of (3.1)-(3.3), then u is a solution of (3.4)-(3.3). keywords: solution; ∈ ac[a; ∈ c[a cache: ejde-2.pdf plain text: ejde-2.txt item: #183 of 601 id: ejde-20 author: Emamirad, Hassan; Rougirel, Arnaud title: De Bruijn identities in different Markovian channels date: 2023 words: 3756 flesch: 83 summary: First we remark that for t = (e2r − 1)/2 = 0, we have e2τ − 1 = 0, so τ should be equal zero. ︸ =A2(τ) , ∂ ∂τ B(τ, y, ξ) = yeτ (eτy − ξ) keywords: ϕ(x cache: ejde-20.pdf plain text: ejde-20.txt item: #184 of 601 id: ejde-200 author: Takac, Peter title: Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model date: 2023 words: 7794 flesch: 74 summary: = u(log S, t) on the stock price S ∈ (0,∞) for large negative / positive values of the logarithmic stock price x = log S ∈ R1, i.e., for S → 0+ and S → +∞, respectively. = log S ∈ R1. keywords: nonlinear; space cache: ejde-200.pdf plain text: ejde-200.txt item: #185 of 601 id: ejde-201 author: Webb, Glenn title: Nonlocal advection diffusion equations and the two-slit experiment in quantum mechanics date: 2023 words: 4379 flesch: 74 summary: The interpretation of the solution is that ∫ x2 x1 ρ(x, t) dx is the probability of finding a single particle in the interval (x1, x2) at time t, 2. Schrödinger equation model The one-dimensional time-dependent complex-valued Schrödinger equation is the foundational phenomenological model of quantum mechanics: ∂ ∂t ψ(x, t) keywords: equation; ρ(x cache: ejde-201.pdf plain text: ejde-201.txt item: #186 of 601 id: ejde-202 author: Amster, Pablo title: A third look at the first result of Landesman-Lazer type date: 2021 words: 4779 flesch: 73 summary: Summarizing, we have proven that if U := BR(0)× (−M,M)n ⊂ R2N then the homotopy h(x, y, s) := (∫ T 0 g(x+ s(uxy(t)− x)) dt, s (uxy(T )− x) + (1− s)y ) does not vanish on ∂U . u‖L2 because ∫ T 0 〈 keywords: landesman; lazer; result cache: ejde-202.pdf plain text: ejde-202.txt item: #187 of 601 id: ejde-203 author: Arango, Jaime title: Oscillation time and damping coefficients in a nonlinear pendulum date: 2021 words: 3477 flesch: 74 summary: Analogously, for V (t) we obtain V (t) =x0t sin t+ 3ax20 ∫ t 0 cos(t− s) cos2 sX1(s) ds+O(|x0|4) ≡V1(t) + V2(t) Notice that x̂0 ≤ x0 and the equality holds in the conservative case α = 0 only. keywords: damping; oscillation; time cache: ejde-203.pdf plain text: ejde-203.txt item: #188 of 601 id: ejde-204 author: Acharya, Ananta; Das, Ujjal; Shivaji, Ratnasingham title: Existence and multiplicity results for p-q-Laplacian boundary value problems date: 2022 words: 3256 flesch: 80 summary: We study positive solutions to the boundary value problem −∆pu−∆qu = λf(u) in Ω, u = 0 on ∂Ω, where q ∈ (1, p) and Ω is a bounded domain in RN , N > 1 with smooth bound- ary, λ is a positive parameter, and f : Bifurcation diagram for positive solutions to (5.1) keywords: solution cache: ejde-204.pdf plain text: ejde-204.txt item: #189 of 601 id: ejde-205 author: Baustian, Falko; Takac, Peter title: Space-time analyticity of weak solutions to semilinear parabolic systems with variable coefficients date: 2021 words: 37869 flesch: 81 summary: ∈ RN (or CN ); its coefficients are M ×M matrices (real or complex) which are assumed to be real analytic (jointly) in both variables x ∈ RN and t ∈ (0, T ). [0, T ); thus, each Xβ(·, t) (|β| ≤ m) belongs to L∞(RN ) at every time t ∈ [0, T ). keywords: analyticity; cauchy; complex; function; holomorphic; p t; p(rn; problem; r ∈; solution; space; t ∈; theorem; time t; u0 ∈; y t; y ∈; z ∈; z0 ∈; ∈ bs;p; ∈ c; ∈ e1−; ∈ mrp(e; ∈ rn; ∈ u cache: ejde-205.pdf plain text: ejde-205.txt item: #190 of 601 id: ejde-208 author: Chen, Yutong; Su, Jiabao; Sun, Mingzheng; Tian, Rushun title: An elliptic equation involving the square root of the Laplacian without asymptotic limits date: 2021 words: 8548 flesch: 83 summary: − ( ‖z‖2 − µm ∫ Ω |z(x, 0)|2dx ) − (∫ {|v(x,0)|M} ) f̃(x, v(x, 0))ṽ(x, 0)dx > ( ‖w‖2 − µm ∫ Ω |w(x, 0)|2dx ) − ( ‖z‖2 − µm ∫ Ω |z(x, 0)|2dx ) − ∫ Ω (µm+1 − µm − ε)|w(x, 0)|2dx− C‖ṽ‖ > ( µm µm−1 − 1 ) ‖z‖2 + ε µm+1 ‖w‖2 − C‖ṽ‖. (4.36) By (4.32) and (4.36), we obtain o(‖vn‖) = 〈J ′(vn), ṽn〉 > ( µm µm−1 − 1 ) 0)|2dx ] − (∫ {|v(x,0)|M} ) f̃(x, v(x, 0))ṽ(x, 0)dx > ( 1− µm+1 µm+2 ) ‖w‖2 − [ ‖z‖2 − µm+1 ∫ Ω |z(x, 0)|2dx ] − ∫ Ω (µm+1 − µm − ε)|z(x, 0)|2dx− C‖ṽ‖ > ( 1− µm+1 µm+2 ) ‖w‖2 + ε µm ‖z‖2 − C‖ṽ‖. (4.18) keywords: proposition; v(x; µm+1 cache: ejde-208.pdf plain text: ejde-208.txt item: #191 of 601 id: ejde-209 author: Calanchi, Marta; Ruf, Bernhard title: Eigenvalues and bifurcation for Neumann problems with indefinite weights date: 2021 words: 6010 flesch: 77 summary: φ∗, φ ∗ two associated eigenvectors, then φ∗, φ ∗ are orthogonal∫ Ω ∇φ∗∇φ∗ dx = 0, ∫ Ω a(x)φ∗φ ∗ dx = 0. (b) (First eigenvalues) λ+ 1 = inf u∈B+ ∫ Ω |∇u|2dx ≥ 0, λ−1 = − inf u∈B− ∫ Ω |∇u|2dx ≤ 0 are simple, with associated positive eigenfunctions φ+ 1 and φ−1 . If λ+ 1 := infu∈B+ ∫ Ω |∇u|2dx = 0, there is a sequence un = wn + sn, with ∫ Ω wn = 0 and sn ∈ R such that∫ Ω a(x)u2 n = 1, ∫ Ω |∇wn|2dx→ 0, as n→ +∞. Therefore wn → 0 strongly in H1(Ω) and sn is bounded: otherwise we would have (up to subsequences) 1 = ∫ Ω a(x)u2 n = ∫ Ω a(x)(s2 n + 2wnsn + w2 n)dx = s2 n (∫ Ω a(x) dx+ o(1) ) → −∞. Since sn is bounded, up to subsequences, sn → s and un → s strongly, from which we obtain 1 = ∫ Ω a(x)u2 n → s2 ∫ Ω a(x) ≤ 0, which is a contradiction. keywords: a(x; existence; problem; solution; λ−1 cache: ejde-209.pdf plain text: ejde-209.txt item: #192 of 601 id: ejde-21 author: Zhao, Zhihong; Hu, Huanqin title: Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis date: 2023 words: 8591 flesch: 71 summary: (3.13) Multiplying the equations of system (3.9) by cos 2iπx l and then integrating them over 0 to l, once again combining K1 = 0 yields∫ 1 0 Φ1 cos 2iπx l dx = E1 E0 , ∫ 1 0 Ψ1 cos 2iπx l dx = E2 E0 , (3.14) where E0 = f1g2 − f2g1 − 4i2π2(ξiχ(v∗)f2 + g2d1 + f1d2) l2 + 16i4π4d1d2 l4 , E1 = π2i2(ξibiχ(v∗)f2 + 2M1d2) 2l + (M2f2 −M1g2)l 4 , E2 = π2i2(2ξiχ(v∗)M1 − ξibiχ(v∗)f1 + 2M2d1) 2l + 2π4i4ξibiχ(v∗)d1 l3 + (M1g1 −M2f1)l 4 , obviously, E0 is always nonzero by ξi 6= = (u∗ + 0.05 cosx, v∗ + 0.05 cosx) and fix ξ = −30, which is obviously far away from the critical bifurcation value. keywords: bifurcation; cos; model; predator; prey; state; system; taxis cache: ejde-21.pdf plain text: ejde-21.txt item: #193 of 601 id: ejde-211 author: Chhetri, Maya; Mavinga, Nsoki; Pardo, Rosa title: Bifurcation from infinity with oscillatory nonlinearity for Neumann problems date: 2022 words: 5996 flesch: 75 summary: For each compact set K ⊂ (−∞, λ2) ⊂ R, there exists a constant C = C(K), independent of λ ∈ K, such that ‖w(λ)‖C(Ω) ≤ C‖g1(λ, ·)‖Lr(Ω) , where w satisfies ∫ Ω w = 0 and (2.6), and g1 satisfies (2.4). Therefore, by the Uniform Boundedness Principle, there exists a constant C = C(K) such that ‖w(λ)‖C(Ω) ≤ C(K)‖g1‖Lr(Ω) for any λ ∈ K, as desired. keywords: solutions cache: ejde-211.pdf plain text: ejde-211.txt item: #194 of 601 id: ejde-212 author: Korman, Philip; Schmidt, Dieter S. title: Infinitely many solutions and asymptotics for resonant oscillatory problems date: 2022 words: 5712 flesch: 78 summary: We derive a rather precise asymptotic formula for µ1 = µ1(ξ1) in case |ξ1| is large, and this formula tends to be accurate for small |ξ1| as well. Solution curve µ1 = µ1(ξ1) of (1.1), oscillating to ±∞. Values with |µ1| < 1 are not shown. axes), and to make the resulting picture manageable a logarithmic scale is used for both ξ1 and µ1. keywords: solutions cache: ejde-212.pdf plain text: ejde-212.txt item: #195 of 601 id: ejde-214 author: Ma, Ruyun; Zhao, Zhongzi; Yan, Dongliang title: Connected components of positive solutions of biharmonic equations with the clamped plate conditions in two dimensions date: 2021 words: 5459 flesch: 82 summary: URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu CONNECTED COMPONENTS OF POSITIVE SOLUTIONS OF BIHARMONIC EQUATIONS WITH THE CLAMPED PLATE CONDITIONS IN TWO DIMENSIONS RUYUN MA, ZHONGZI ZHAO, DONGLIANG YAN In memory of Professor Alan C. Lazer Abstract. We show the existence of S-shaped connected com- ponent of positive solutions under suitable conditions on the nonlinearity. keywords: lemma; solutions; theorem; λ1(a(·))/f0 cache: ejde-214.pdf plain text: ejde-214.txt item: #196 of 601 id: ejde-215 author: Mawhin, Jean title: the mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane) date: 2021 words: 4376 flesch: 78 summary: Reit (Reit − z)2 dt and, for z ∈ DR \ {0}, g(z)− g(0) Furthermore, Theorem 3.1 provides a localization z ∈ DR for the obtained zeros. keywords: function; holomorphic; reit; theorem cache: ejde-215.pdf plain text: ejde-215.txt item: #197 of 601 id: ejde-216 author: Maia, Liliane de A.; Oliveira Junior, Jose Carlos; Ruviaro, Ricardo title: Generalized quasilinear equations with critical growth and nonlinear boundary conditions date: 2022 words: 7153 flesch: 81 summary: We study the quasilinear problem − div(h2(u)∇u) + h(u)h′(u)|∇u|2 + u = −λ|u|q−2u+ |u|2·2 ∗−2u in Ω, ∂u ∂η = µg(x, u) on ∂Ω, where Ω ⊂ R3 is a bounded domain with regular boundary ∂Ω, λ, µ > 0, 1 < q < 4, 2 ·2∗ = 12, ∂ ∂η is the outer normal derivative and g has a subcritical growth in the sense of the trace Sobolev embedding. 327 328 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 In this article, we are interested in the quasilinear problem −div(h2(u)∇u) + h(u)h′(u)|∇u|2 + u = −λ|u|q−2u+ |u|2·2 ∗−2u in Ω, ∂u ∂η = µg(x, u) on ∂Ω, (1.3) where Ω ⊂ R3 is a bounded domain with regular boundary ∂Ω, λ, µ > 0, 1 < keywords: g(x; h1(ω; lemma cache: ejde-216.pdf plain text: ejde-216.txt item: #198 of 601 id: ejde-217 author: Ozturk, Eylem; Rossi, Julio D. title: Limit for the p-laplacian equation with dynamical boundary conditions date: 2021 words: 5037 flesch: 78 summary: ∀w ∈ K. When the convex functional Ψ : H → (−∞,+∞] is proper, lower-semicontinuous, and such that min Ψ = 0, it is well known (see [8]) that the abstract Cauchy problem ut + ∂Ψ(u) 3 f, a.e. t ∈ (0, T ), u(0) = u0, has a unique solution for any f ∈ L1(0, T ;H) and u0 ∈ D(∂Ψ). Hence, ∫ ∂Ω B(up)(t)− ∫ ∂Ω B(u0) = ∫ t 0 ∫ ∂Ω ∂B(up) ∂t ≤ ∫ t 0 ∫ ∂Ω fβ(up), here B satisfies B′(s) = β(s). keywords: limit; problem cache: ejde-217.pdf plain text: ejde-217.txt item: #199 of 601 id: ejde-218 author: Pacella, Filomena; Stolnicki, David title: Oscillatory solutions and critical exponents for fully nonlinear equations date: 2021 words: 7268 flesch: 78 summary: = F (X,Z) = ( f(X,Z), g(X,Z) ) , (2.5) where the dot ˙ stands for derivation with respect to t, and f, g are given by for M+ λ,Λ: f(X,Z) = { X(X − (N − 2) + Z λ ) if (X,Z) ∈ R+ λ X(X − (Ñ+ − 2) + Z Λ ) if (X,Z) ∈ R−λ , (2.6a) g(X,Z) = { Z(N − pX − Z λ ) if (X,Z) ∈ R+ λ Z(Ñ+ − pX − Z Λ ) if (X,Z) ∈ R−λ , (2.6b) (2.6c) and for M−λ,Λ: f(X,Z) 154 F. PACELLA, D. STOLNICKI EJDE/SI/01 M0 A0 `+ `+2 `+2 `+1 →→→→→→→→→→→→→→→→→→→→→→ → → → → → → → → → → → → ←← ←← ←← ←← ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↗↗↗↗↗↗↗↗↗↗↗ ↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘↘ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↓↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ ↘ N0 X Z O X = Ñ+ keywords: ←← ↘; ↑ ↑; ↓ ↘; ↗ ↘; ↘ ↑; ↘ ↗; ↘ ↘ cache: ejde-218.pdf plain text: ejde-218.txt item: #200 of 601 id: ejde-219 author: Onete, Florin I.; Papageorgiou, Nikolaos S.; Radulescu, Vicentiu D. title: Twin positive solutions for resonant singular (p,q)-equations date: 2021 words: 5843 flesch: 80 summary: (4.32) Now we return to (4.19), choose h = un − u ∈ W 1,p 0 (Ω), pass to the limit as n→∞ and use (4.32). So, if in (4.23) we choose h = yn − y ∈ W 1,p 0 (Ω), pass to the limit as n → ∞ and use (4.22), (4.21), (4.24), then we obtain lim n→∞ 〈Ap(yn), yn − y〉 = 0, ⇒ yn → y in W 1,p 0 (Ω), hence ‖y‖ = 1, y ≥ 0 (see Proposition 2.1). keywords: 1,p; papageorgiou cache: ejde-219.pdf plain text: ejde-219.txt item: #201 of 601 id: ejde-22 author: Daoues, Adel; Hammami, Amani; Saoudi, Kamel title: Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent date: 2023 words: 7461 flesch: 84 summary: ‖vλ‖p − λ ( 1 1− α + 1 p∗s(t) )∫ Ω |uk|1−α dx+ o(1) ≥ (sp− t) p(N − t) S N−t sp−t + (sp− t) p(N − t) ‖vλ‖p − λ ( 1 1− α + 1 p∗s(t) ) |Ω| p∗s (t)−1+α p∗s (t) × S− 1−α p ‖vλ‖1−α + o(1) ≥ (sp− t) p(N − t) S N−t sp−t + (sp− t) p(N − t) keywords: p∗s(t; |x|t; − t cache: ejde-22.pdf plain text: ejde-22.txt item: #202 of 601 id: ejde-220 author: Recova, Leandro L.; Rumbos, Adolfo J. title: An asymmetric problem at resonance with a one-sided Ahmad-Lazer-Paul condition date: 2021 words: 8417 flesch: 72 summary: Let Ω be a bounded, connected, open subset of RN , for N ≥ 2, with smooth boundary ∂Ω. Consider the Dirichlet problem −∆u = λku+ g(x, u), x ∈ Ω; u = 0, x ∈ ∂Ω, (1.1) where λk is an eigenvalue of the N -dimensional Laplacian −∆ in Ω with Dirichlet boundary conditions, and g : Ω × R → R is continuous and uniformly bounded; that is, |g(x, s)| 6M, for all x ∈ Ω, and s ∈ R, (1.2) 2010 Mathematics Subject Classification. − 2G(x, um)] dx ∣∣∣ 6 C + εm(‖u+ m‖+ ‖u−m‖), for all m. (3.4) Put T (x, s) = g(x, s)s − 2G(x, s), for x ∈ Ω and s ∈ R. keywords: g(x; problem cache: ejde-220.pdf plain text: ejde-220.txt item: #203 of 601 id: ejde-223 author: Yang, Jiaxuan; Li, Yongqing; Wang, Zhi-Qiang title: Radial and non-radial solutions for a nonlinear Schrodinger equation with a constraint date: 2021 words: 5165 flesch: 85 summary: Then c(εn) = inf ‖u‖2=1 (1 2 ∫ RN |∇u|2dx− 1 p ∫ RN Q(εnx)|u|pdx ) ≤ 1 2 ∫ RN |∇vn|2dx− 1 p ∫ RN Q(εnx)|vn|pdx = 1 2 ∫ RN |∇vn|2dx− 1 p qM ∫ RN |vn|pdx+ 1 p ∫ RN (qM −Q(εnx))|vn|pdx 230 J. YANG, Y. LI, Z.-Q. WANG EJDE/SI/01 = cqM + 1 p ∫ RN (qM −Q(εnx))|vn|pdx. lim sup ε→0 c(ε, k) ≤ c k 2−p 2 qM (4.4) where as in (1.5), c k 2−p 2 qM = inf u∈H1,‖u‖2=1 (1 2 ∫ R2 |∇u|2dx− k 2−p 2 qM p ∫ R2 |u|pdx ) . keywords: c(ε; solutions cache: ejde-223.pdf plain text: ejde-223.txt item: #204 of 601 id: ejde-224 author: Simsen, Jacson title: Evolution equations on time-dependent Lebesgue spaces with variable exponents date: 2023 words: 5314 flesch: 82 summary: The family of pullback attractors {Aλ(t) : t ∈ R}, λ ∈ [0,∞) is upper semicontinuous at λ1 in the topology of H. Proof. For t ∈ R and ε > 0, let τ ∈ R be such that distYt ( Uλ1(t, τ)B(τ),Aλ1(t) ) < ε 3 , where ∪λ∈[0,∞)Aλ(τ) ⊂ B(τ) and B(τ) is a nonempty set in Xτ ⊂ keywords: dλ(t; p(x; p(·,t cache: ejde-224.pdf plain text: ejde-224.txt item: #205 of 601 id: ejde-225 author: Ahrami, Mohammed; El Allali, Zakaria title: Lower bounds on the fundamental spectral gap with Robin boundary conditions date: 2022 words: 4278 flesch: 74 summary: Our main results include improvements of the lower bound on the fundamental gap of Robin Schrödinger operators with a convex potential. [6] M. Ashbaugh, D. Kielty; spectral gaps of 1-D Robin Schrödinger operators with single-well potentials, Journal of Mathematical Physics, 61, 091507 (2020). keywords: boundary; gap; robin; schrödinger cache: ejde-225.pdf plain text: ejde-225.txt item: #206 of 601 id: ejde-226 author: Awanou, Gerard title: Discrete Aleksandrov solutions of the Monge-Ampere equation date: 2022 words: 8717 flesch: 82 summary: Let x0 ∈ Ω and φ be a strictly convex quadratic polynomial such that u∗ − φ has a local minimum at x0 with (u∗ − φ)(x0) = 0. For x ∈ Ω we denote by d(x, ∂Ω) the distance of x to ∂Ω. For a subset S of Ω, diam(S) denotes its diameter. Lemma 2.11. keywords: aleksandrov; convergence; convex; function; monge; proof; solution; viscosity cache: ejde-226.pdf plain text: ejde-226.txt item: #207 of 601 id: ejde-228 author: Cho, Manki; Rivas, Mauricio A. title: On the L^2-orthogonality of Steklov eigenfunctions date: 2022 words: 4872 flesch: 70 summary: A consequence of the calculations is a tabulation of the mean value of Steklov eigenfunctions over Ω1α. Introduction This article describes the exact, or near, orthogonality in L2(Ω1α) of the sequence of Steklov eigenfunctions in the case Ω1α is a rectangle in R2. keywords: eigenfunctions; orthogonality; steklov; ω1α cache: ejde-228.pdf plain text: ejde-228.txt item: #208 of 601 id: ejde-229 author: Feng, Xiaobing; Lewis, Thomas; Ward, Kellie title: A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs date: 2022 words: 17555 flesch: 69 summary: Then F̂0, F̃ij , and −F̂ij are all nonnegative definite, and we have (5.10) becomes Ŵ = (I − ρF̂0)W − ρ d∑ i=1 d∑ j=1 ( F̃ij + F̂ij ) − F̂ ( D̂2 hk uhk (xk), D̂2 hk uhk (xk), uhk (xk),xk ) . (4.5) Then, by the mean value theorem, the Lipschitz continuity of F , and the uniform and proper ellipticity of F , there exists a constant K ≥ 0 keywords: boundary; i=1; numerical; operators; order cache: ejde-229.pdf plain text: ejde-229.txt item: #209 of 601 id: ejde-230 author: Guo, Daniel X. title: Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations date: 2022 words: 6752 flesch: 72 summary: Semi-Lagrangian methods have been introduced at the beginning of the eighties Recently, more applications of semi-Lagrangian method are reported. keywords: k d; k d2; method; x k cache: ejde-230.pdf plain text: ejde-230.txt item: #210 of 601 id: ejde-232 author: Lewis, Thomas; Rapp, Aaron; Zhang, Yi title: Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs date: 2022 words: 7283 flesch: 73 summary: By the Cauchy-Schwarz inequality and the fact that [[uch − u]]e = 0 for all e ∈ Eh, we have Bh,γ(uch − u, uch − u γ h) ≤ 1 2 ‖∇+ h,0(uch − u)‖L2(Th)‖∇+ h,0(uch − u γ h)‖L2(Th) + 1 2 ‖∇−h,0(uch − u)‖L2(Th)‖∇−h,0(uch − u γ h)‖L2(Th) + γ ∑ e∈Eh 〈h−1 e [[uch − u γ h]]‖2L2(e) ≤ Bh,γ(uch − u γ h, u c h − u γ h) = Bh,γ(uch − u, uch − u γ h) +Bh,γ(u− uγh, u c h − u γ h). keywords: dwdg; error; solution; u γ; − u cache: ejde-232.pdf plain text: ejde-232.txt item: #211 of 601 id: ejde-234 author: Valdebenito, Dario A. title: On solutions arising from radial spatial dynamics of some semilinear elliptic equations date: 2022 words: 9910 flesch: 70 summary: Previ- ously, related ideas for finding quasiperiodic solutions of elliptic equations on an unbounded strip have been used by Scheurle [38] P. Poláčik, D. Valdebenito; Existence of quasiperiodic solutions of elliptic equations on RN+1 via center manifold and KAM theorems, Journal of Differential Equations 262 (2017), 6109– 6164. keywords: equation; form; hamiltonian; sn−1; solutions; terms; theorem cache: ejde-234.pdf plain text: ejde-234.txt item: #212 of 601 id: ejde-235 author: Huang, Lan-Xin; Wu, Xing-Ping; Tang, Chun-Lei title: Multiple positive solutions for nonhomogeneous Schrodinger-Poisson systems with Berestycki-Lions type conditions date: 2021 words: 6041 flesch: 84 summary: Repeating the proof of Lemma 2.2, we easily obtain on(1) = 〈I ′λ,T (un)− I ′λ,T (u), un − u〉 ≥ min{1,m}〈un, un − u〉 −max{1,m}〈u, un − u〉 + λhT (un) ∫ R3 φunun(un − u) dx− λhT (u) ∫ R3 φuu(un − u) dx + aλ,T (un) 2 〈un, un − u〉 − aλ,T (u) 2 〈u, un − u〉 − ∫ R3 (g1(un)− g1(u))(un 〈un, un − u〉, 8 L.-X. HUANG, X.-P. WU, C.-L. TANG EJDE-2021/01 this shows that ( min{1,m} + aλ,T (un) 2 ) 〈un, un − u〉 → 0 as n → ∞. By (2.2) and (3.2), we have |aλ,T (un)| ≤ λT−2|χ′(T−2‖un‖2)| ∣∣ ∫ R3 φunu 2 n dx ∣∣ < 8λT̃ . keywords: system cache: ejde-235.pdf plain text: ejde-235.txt item: #213 of 601 id: ejde-236 author: Engu, Satyanarayana; Sahoo, Manas R.; Berke, Venkatramana P. title: Solutions to viscous Burgers equations with time dependent source term date: 2021 words: 7006 flesch: 80 summary: ∫ 0 −∞ −wwt dx dt+ ∫ 0 −∞ w2(x, t) dx − 1 2 [ ∫ T 0 ∫ 0 −∞ (u+ v)wwx dx dt− ∫ T 0 (w (u+ v)w)(0, t) dt ] + ∫ T 0 ∫ 0 −∞ w2 x dx dt− ∫ T 0 (wxw)(0, t) dt = 0. (3.8) Similarly for φ = w(x, t)H(T − t)H(x), integral equation (3.7) yields∫ T 0 ∫ ∞ 0 −wwt dx dt+ ∫ ∞ 0 w2(x, t) dx + ∫ T 0 ∫ R w2 x dx dt+ ∫ T 0 [ (wxw)(0+, t)− (wxw)(0−, t) ] dt = 1 2 [ ∫ T 0 ∫ R (u+ v)wwx dx dt+ ∫ T 0 [ ((u+ v)w2)(0+, t)− ((u+ v)w2)(0−, t) ] dt ] , which implies ‖w(·, T )‖22 + 2 ∫ T 0 ‖wx(· , t)‖22 dt ≤ 1 2 ∫ T 0 ∫ R ‖(u+ v)(t)‖∞|w(x, t)‖wx(t)| dx dt ≤ 1 2 ∫ T 0 ‖(u+ keywords: burgers; equation; solutions cache: ejde-236.pdf plain text: ejde-236.txt item: #214 of 601 id: ejde-237 author: Ali, Mageed; Iaia, Joseph A. Iaia title: Existence and nonexistence for singular sublinear problems on exterior domains date: 2021 words: 7844 flesch: 91 summary: ds ≥ ∫ t 0 h(s) ds. (2.15) Integrating (2.15) again and using (2.7) gives vq+1 a (t) q + 1 + ∫ t 0 ∫ ds = ∫ t 0 h(s) ds. keywords: 1+q; 2−α̃; r2−n cache: ejde-237.pdf plain text: ejde-237.txt item: #215 of 601 id: ejde-238 author: Sourdis, Christos title: An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property date: 2021 words: 4623 flesch: 73 summary: In light of the recent density estimates of [23], we expect that the assertions of Theorems 1.1 and 1.2 should also remain valid under the complementary set of assumptions that W ∈ C1 satisfies c|u− a|p ≤W (u) ≤ C|u− a|p, u ∈ Rm, m ≥ 1, for some constants c, C > 0, where p ∈ { (2,∞), n = 2,( 2, 2n n−2 ) , n ≥ 3. (2.8) From (2.2), using again that W ∈ C1, there exists a C4 > 0 such that ‖e‖C0,α(Rn;R) ≤ C4. keywords: solutions; theorem cache: ejde-238.pdf plain text: ejde-238.txt item: #216 of 601 id: ejde-239 author: He, Rui; Liu, Xiangqing title: Localized nodal solutions for parameter-dependent quasilinear Schrodinger equations date: 2021 words: 7081 flesch: 89 summary: It holds that (1) p ·A(z, p) ≥ φ(K)gµ(|p|)|p|, (2) |A(z, p)| ≤ Φ(K)gµ(|p|), (3) |B(x, z, p)| ≤ Φ(K)(1 + gµ(|p|)|p|) for x ∈ RN , z ∈ R, |z| ≤ K, p ∈ RN , where φ, Φ are two functions from R+ to R+ such that φ is decreasing and Φ is increasing. − V (x)v + λ|v|q−2v = 0, v(x)→ 0 as |x| → ∞, (1.1) where x ∈ RN , ε > 0 is a small parameter, Div = ∂v ∂xi , Dzbij(z) keywords: 1,m; χε(x)u2; ∫ rn cache: ejde-239.pdf plain text: ejde-239.txt item: #217 of 601 id: ejde-240 author: Antontsev, Antontsev; Ferreira, Jorge; Piskin, Erhan title: Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities date: 2021 words: 6658 flesch: 82 summary: = ∫ Ω |u|q(·)dx (5.6) for any u ∈ H2 0 (Ω) and 2 ≤ s ≤ q−. Where C > 1 a positive constant and H(t) = −E(t). = (1− σ)H−σ(t)H ′(t) + ε ∫ Ω (u2 t + uutt)dx+ ε ∫ Ω ∇u∇ut dx = (1− σ)H−σ(t)H ′(t) + ε‖ut‖2 − ε‖∆u‖2 + ε ∫ Ω |u|q(·)dx− ε ∫ Ω uut|ut|p(·)−2dx. (5.13) By using the definition of the H(t), it follows that −εq−(1− ξ)H(t) = εq−(1− ξ) 2 ‖ut‖2 + εq−(1− ξ) 2 ‖∆u‖2 − εq−(1− ξ) ∫ Ω 1 q(x) |u|q(·)dx, (5.14) where 0 < ξ < 1. keywords: equation; solutions; ‖ut‖2 cache: ejde-240.pdf plain text: ejde-240.txt item: #218 of 601 id: ejde-241 author: Liu, Lintao; Teng, Kaimin title: Ground state and multiple solutions for critical fractional Schrodinger-Poisson equations with perturbation terms date: 2021 words: 8588 flesch: 86 summary: (3.3) Let R > 0 and γ ∈ R3 with |γ| = 1. = I(tu∞(x − Rγ)), t ∈ (0,∞), γ ∈ R3 with |γ| = 1. keywords: fractional; lemma; sdx; solutions; ∫ r3 cache: ejde-241.pdf plain text: ejde-241.txt item: #219 of 601 id: ejde-242 author: Diaz, Jesus Ildefonso; Hilhorst, Danielle; Kyriazopoulos, Paris title: A parabolic system with strong absorption modeling dry-land vegetation date: 2021 words: 8282 flesch: 76 summary: Now, let x0 ∈ Ω − supp(h0), R : (2.6) In any case, we are specifically interested in the case in which the initial data satisfy 0 ≤ b0 ≤ 1, w0 ≥ 0, h0 ≥ 0, on Ω. (2.7) Concerning the precipitation term p, we assume that p ∈ L∞(QT ) is nonnegative. keywords: problem; solution; system cache: ejde-242.pdf plain text: ejde-242.txt item: #220 of 601 id: ejde-243 author: Giacomoni, Jacques; Gouasmia, Abdelhamid; Mokrane, Abdelhafid title: Existence and global behavior of weak solutions to a doubly nonlinear evolution date: 2021 words: 15385 flesch: 82 summary: = ( ‖u‖p Lp(RN ) + ∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+sp dx dy )1/p . • The space W s,p 0 (Ω) is the set of functions W s,p 0 (Ω) := {u ∈W s,p(RN ) : u = 0 a.e. in RN \ Ω}, and the norm is given by the Gagliardo semi-norm ‖u‖W s,p 0 (Ω) := (∫ RN ∫ RN |u(x)− u(y)|p |x− y|N+sp dx dy )1/p . Then, for any r ≥ 1, ‖vq(t, ·)− vq∞‖Lr(Ω) → 0 as t→∞, where v∞ is the unique stationary solution to (1.12) associated to the potential h∞. This article is organized as follows: In Section 2, we study the stationary non- linear problem v2q−1 + λ(−∆)spv = h0(x)vq−1 + λf(x, v) in Ω, v > in Ω, v = 0 in RN \ Ω, related to the parabolic problem (1.12) and establish the existence and the unique- ness results in case h0 ∈ L∞(Ω) keywords: f(x; fractional; problem; solution; theorem; vq−1; w s; y|n+sp; |x−; ε)q; ε)q−1; ∫ rn; ∫ ω cache: ejde-243.pdf plain text: ejde-243.txt item: #221 of 601 id: ejde-244 author: Ciou, Jyun-Yuan; Tzung-Shin, Tzung-Shin title: Complete classification of bifurcation curves for a multiparameter diffusive logistic problem with generalized Holling type-IV functional response date: 2021 words: 4390 flesch: 84 summary: In this article we study exact multiplicity of positive solutions and shapes of bifurcation curves of (1.1) for parameters m ≥ 1 and q, r > 0. We divide the first quadrant of (q, r)- parameter plane into the disjoint union of three curves Γ1, Γ2, Γ3 and five regions EJDE-2021/10 BIFURCATION CURVES FOR A DIFFUSIVE LOGISTIC PROBLEM 5 R1, R2, R3, R4, R5 defined as follows: Γ1 = { (q, r) : q(a) keywords: bifurcation; curve cache: ejde-244.pdf plain text: ejde-244.txt item: #222 of 601 id: ejde-245 author: Cui, Na; Sun, Hong-Rui title: Existence of solutions for critical fractional p-Laplacian equations with indefinite weights date: 2021 words: 6726 flesch: 80 summary: Then it follows that ξn(x, y)→ |u(x)−u(y)|p−2(u(x)−u(y)) |x−y| N+sp p′ a.e. in RN × RN . ε ( [vε,ρ] p s,p − λ ∫ RN g|vε,ρ|p dx ) − tp ∗ s−1 ε ∫ RN h|vε,ρ|p ∗ s dx, moreover, combining (A2), (A3) and (A4), we deduce that t p∗s−p ε = [vε,ρ] p s,p − λ ∫ RN g|vε,ρ| p dx∫ RN h|vε,ρ|p ∗ s dx ≤ keywords: fractional cache: ejde-245.pdf plain text: ejde-245.txt item: #223 of 601 id: ejde-246 author: Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca title: Multiple solutions for semilinear Robin problems with superlinear reaction and no symmetries date: 2021 words: 10036 flesch: 84 summary: So, if there exists M > 0 such that for a.a. z ∈ Ω x 7→ fλ(z, x) x is nondecreasing on [M,+∞), 6 N. S. PAPAGEORGIOU, C. VETRO, F. VETRO EJDE-2021/12 x 7→ fλ(z, x) x is nonincreasing on (−∞,−M ], then the quasimonotonicity condition (H2)(iii) is satisfied. Let Ω+ = {z ∈ Ω : y(z) > 0}. keywords: fλ(z; h1(ω; intc+ cache: ejde-246.pdf plain text: ejde-246.txt item: #224 of 601 id: ejde-248 author: Maione, Alberto title: H-convergence for equations depending on monotone operators in Carnot groups date: 2021 words: 5179 flesch: 79 summary: Let f ∈ V ∗ and let B : V → V ∗ be defined by 〈B(u), v〉V ∗×V := ∫ Ω ( 〈A(x,∇Gu),∇Gv〉 − f v ) dx ∀u, v ∈ V. Let us show that B is strictly-monotone, coercive and continuous on any finite dimensional subspace of V . The class M(α, β; Ω) is defined as follows. keywords: operators cache: ejde-248.pdf plain text: ejde-248.txt item: #225 of 601 id: ejde-249 author: Yang, Zhipeng; Zhang, Wei; Zhao, Fukun title: Existence and concentration results for fractional Schrodinger-Poisson system via penalization method date: 2021 words: 12530 flesch: 86 summary: Note that un satisfies (−∆)sun + un = Υn, x ∈ R3, where Υn(x) = un(x)− V (εn(x+ ỹn))un(x)− φtunun(x) + g(εn(x+ ỹn), un), x ∈ R3. The fractional Laplacian, (−∆)αu, of a smooth function u : R3 → R, is defined by F((−∆)αu)(ξ) = |ξ|2αF(u)(ξ), ξ ∈ R3. keywords: fractional; lemma; poisson; proof; schrödinger; solutions cache: ejde-249.pdf plain text: ejde-249.txt item: #226 of 601 id: ejde-25 author: Jiang, Shuai; Yin, Li-Feng title: Existence of nontrivial solutions for Schrodinger-Kirchhoff equations with indefinite potentials date: 2023 words: 5782 flesch: 82 summary: = ∫ t 0 g(τ)dτ ≥ c|t|ν for some ν < 4, then problem (1.6) has at least one nontrivial solution. (2.16) We write the integral over R3 \BR as the sum of the integrals over the intersections of R3 \ BR with {V ≥ 0} and {V < 0}. keywords: |∇u|2 cache: ejde-25.pdf plain text: ejde-25.txt item: #227 of 601 id: ejde-252 author: Fan, Jishan; Zhou, Yong title: Uniform regularity of fully compressible Hall-MHD systems date: 2021 words: 3177 flesch: 86 summary: [1 2 ∇|D3b|2 − (D3b · ∇)D3b ] udx + ξ ∫ ( D3 ( b ρ × rot b ) − b ρ ×D3 rot b ) D3 rot bdx = − ∫ rot(D3(b× u)−D3b× u− b×D3u)D3bdx EJDE-2021/17 FULLY COMPRESSIBLE HALL-MHD SYSTEMS 7 − 1 2 ∫ |D3b|2 div udx+ ∫ D3b⊗D3b : ∇udx + ξ ∫ ( D3 ( b ρ × rot b ) ‖1 ρ ‖Lq‖ div u‖L∞ , which gives ‖1 ρ ‖Lq ≤ ‖ 1 ρ0 ‖Lq exp (( 1 + 1 q ) ∫ t 0 ‖ div u‖L∞dτ ) keywords: div; rot cache: ejde-252.pdf plain text: ejde-252.txt item: #228 of 601 id: ejde-253 author: Chen, Yu Xian; Xu, Hong Yan title: Exact forms of entire solutions for Fermat type partial differential equations in C^2 date: 2021 words: 4737 flesch: 75 summary: = sin(z2 − z1 + η1)− cos(z2 − z1 + η1) + η2e −(z1+z2), where η, η1, η2 ∈ C. Secondly, we study the existence and the form of transcendental entire solutions of several second order partial differential equations of Fermat type,[ a1f(z) + a2 ∂f ∂z1 ]2 + URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXACT FORMS OF ENTIRE SOLUTIONS FOR FERMAT TYPE PARTIAL DIFFERENTIAL EQUATIONS IN C2 YU XIAN CHEN, HONG YAN XU Abstract. keywords: differential; equations; solutions; ∂z1 cache: ejde-253.pdf plain text: ejde-253.txt item: #229 of 601 id: ejde-254 author: Pu, Hongling; Li, Shiqi; Liang, Sihua; Repovs, Dusan D. title: Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N date: 2021 words: 8843 flesch: 84 summary: Then for any u ∈ E with u± 6= 0, there is the unique maximum point pair of positive numbers (αu, βu) such that αuu + + βuu − ∈ N λ b . Proof. [22], together with (2.6) and (2.8), we can conclude that there exists (αu, βu) ∈ R+×R+ such that W (αu, βu) = (0, 0), i.e., αuu + + βuu − ∈ N λ b . keywords: iλb; kirchhoff; problem; solutions cache: ejde-254.pdf plain text: ejde-254.txt item: #230 of 601 id: ejde-255 author: Allahverdiev, Bilender P.; Tuna, Huseyin; Isayev, Hamlet A title: Impulsive regular q-Dirac systems date: 2023 words: 3354 flesch: 83 summary: χ21(qt, λ)h2(qt))dqt + q ω(λ)ψ11(ξ, λ)α ∫ a d (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I1, q ω(λ)χ12(ξ, λ) ∫ χ21(qt, λ)h2(qt))dqt + q ω(λ)ψ21(ξ, λ)α ∫ a d (χ12(qt, λ)h1(qt) + χ22(qt, λ)h2(qt))dqt, ξ ∈ I1, q ω(λ)χ22(ξ, λ) ∫ keywords: λ)h1(qt cache: ejde-255.pdf plain text: ejde-255.txt item: #231 of 601 id: ejde-256 author: Barboza, Eudes M.; Miyagaki, Olimpio H.; Pereira, Fabio R.; Santana, Claudia R. title: Henon equation with nolinearities involving Sobolev critical growth in H^1 date: 2021 words: 7121 flesch: 82 summary: Here H1 0,rad(B1) = {u ∈ H1 0 (B1) : u is radial, that is, u(x) = u(|x|),∀x ∈ B1}. First of all, we define W (ε, r) = {u ∈ H1 0,rad(B1);u = u− + turε , u − ∈ H2, t ∈ R}. keywords: 0,rad(b1; 2∗α cache: ejde-256.pdf plain text: ejde-256.txt item: #232 of 601 id: ejde-257 author: Pereira, Ducival; Cordeiro, Sebastiao; Raposo, Carlos; Maranhao, Celsa title: Solutions of Kirchhoff plate equations with internal damping and logarithmic nonlinearity date: 2021 words: 5430 flesch: 83 summary: Integrating (3.7) from 0 to t, 0 ≤ t ≤ tm, we obtain 1 2 ‖umt (t)‖2 + 1 2 ‖∆um(t)‖2 + 1 2 M̂(‖∇um(t)‖2) + 1 2 ‖um(t)‖2 + ∫ t 0 ‖umt (s)‖2ds = 1 2 ‖u1m‖2 + 1 2 ‖∆u0m‖2 + 1 2 M̂(‖∇u0m‖2)− 1 2 ∫ Ω (u0m)2 ln |u0m|2 dx (3.8) + 1 2 ∫ Ω (um(t))2 ln |um(t)|2 dx. Introduction In this article we study the existence and decay properties of global solutions for the nonlinear initial boundary value problem utt + ∆2u+M(‖∇u‖2)(−∆u) keywords: equation; |u|2 cache: ejde-257.pdf plain text: ejde-257.txt item: #233 of 601 id: ejde-258 author: Han, Bang-Sheng; Kong , De-Yu Kong; Shi, Qihong; Wang, Fan title: Periodic traveling waves and asymptotic spreading of a monostable reaction-diffusion equations with nonlocal effects date: 2021 words: 10054 flesch: 78 summary: The map h has the form h(|B|2, ε, δ) =− φ̂(σc) 2 ε2 − ( 1 + 1 + αc − βc 2 φ̂′′(σc) ) δ2 + ς|B|2 +O(|δ|3 + |ε|2|δ|+ |B|4), (2.9) where ς :=− 2(1 + αc − βc)φ̂(σc)− 2(αc − 3βc) 1 + βc ( αc − 1− 5βc + φ̂(σc) ) + αc − 3βc − (1 + αc − βc)φ̂(σc) 4σ2 c − (αc − 2βc) + (1 + αc − βc)φ̂(2σc) (5.1) 20 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22 Substituting (5.1) into (2.7) and comparing the coefficient of B2ei2x and BB, we have e2,0 = αc − 3βc − (1 + αc − βc)φ̂(σc) 4σ2 c − (αc − 2βc) + (1 + αc − βc)φ̂(2σc) ei2x + span(e, ē), and e1,1 = 2(1 + αc − βc)φ̂(σc)− keywords: equation; solutions; state; steady; wave cache: ejde-258.pdf plain text: ejde-258.txt item: #234 of 601 id: ejde-259 author: Zhang, Yajing; Li, Qiaoqin; Pang, Lu title: Existence of multiple positive solutions for fractional Laplace problems with critical growth date: 2021 words: 10426 flesch: 90 summary: − u0 − ∑̀ j=1 (rjn) 2s−N 2 uj (x− xjn rjn )∥∥ Ḣ(RN ) Moreover, either the convergence is strong, or there exist ` ∈ N, nontrivial solutions u1, . . . keywords: lemma cache: ejde-259.pdf plain text: ejde-259.txt item: #235 of 601 id: ejde-260 author: Ren, Yuanyuan; Li, Yongsheng title: Small data blow-up of solutions to nonlinear Schrodinger equations without gauge invariance in L^2 date: 2021 words: 5024 flesch: 89 summary: Then there exist a positive time T = T (ε, ‖f‖L2 , ‖g‖L2) and a unique solution (u, v) ∈ XT×XT of (2.1). ∣∣ ∫ [0,T )×Rn (|u|p1 − |uk|p1)ψ dx dt ∣∣+ ∣∣ ∫ [0,T )×Rn (|v|p2 − |vk|p2)ψ dx dt ∣∣ . keywords: solution cache: ejde-260.pdf plain text: ejde-260.txt item: #236 of 601 id: ejde-263 author: Youssfi, Ahmed; Khatri, Mohamed Mahmoud Ould title: Continuous imbedding in Musielak spaces with an application to anisotropic nonlinear Neumann problems date: 2021 words: 11725 flesch: 86 summary: , N , we denote by νi the ith component of the outer normal unit vector and ai : Ω × R → R is a Carathéodory function such that there exist a locally integrable Musielak-Orlicz function (see definition 1.1 below) Pi : Ω× R+ → R+ with Pi � φi, a positive constant ci and a nonnegative function di ∈ Eφ∗i (Ω) satisfying for all s, t ∈ R and for almost every x ∈ Ω the following assumptions |ai(x, s)| ≤ ci ( di(x) + (φ∗i ) −1(x, Pi(x, s)) ) , (1.2) φi(x, |s|) ≤ ai(x, s)s ≤ Ai(x, s), (1.3) 2010 Mathematics Subject Classification. × R+ → R+ with R � φmax and a nonnegative function D ∈ Eφ∗max (Ω), such that for all s, t ∈ R and for almost every x ∈ Ω, |ϕmax(x, s)| ≤ D(x) + (φ∗max)−1(x,R(x, s)), (1.5) where φ∗max stands for the complementary function of φmax defined below in (2.1). keywords: function; musielak; u(x; φ(ω cache: ejde-263.pdf plain text: ejde-263.txt item: #237 of 601 id: ejde-264 author: Shan, Maria A.; Skrypnik, Igor I.; Voitovych, Mykhailo V. title: Harnack's inequality for quasilinear elliptic equations with generalized Orlicz growth date: 2021 words: 6555 flesch: 82 summary: Now, let δ1 ≤ s < n/(n− 1), and let j be a non-negative integer such that s (n− 1 n )j+1 ≤ δ1 ≤ s (n− 1 n )j . G(x0, u/ρ) ζq dx ≤ keywords: g(x; inequality cache: ejde-264.pdf plain text: ejde-264.txt item: #238 of 601 id: ejde-265 author: Chernysh, Edward title: Weakly monotone decreasing solutions to elliptic Schrodinger integral system date: 2021 words: 3986 flesch: 83 summary: g(x) to state that there exists C,R > 0 such that f(x) ≤ Cg(x) for all x satisfying |x| ≥ R. Theorem 1.3. By (1.3), we may choose R > 0 such that min{φ(x), ψ(x)} ≥ γ0 > 0 whenever |x| ≥ R− 1. keywords: u(x; |x| cache: ejde-265.pdf plain text: ejde-265.txt item: #239 of 601 id: ejde-268 author: Li, Xiaoyan; Yang, Bian-Xia title: Existence and multiplicity for radially symmetric solutions to Hamilton-Jacobi-Bellman equations date: 2021 words: 8316 flesch: 79 summary: Assuming (A5) we exclude the case when the projection of C onto the y-axis is a singleton, which is equivalent to C = {(a1, a2)}. Suppose that f satisfies (A1) and (A2). (a) If f0, f∞ ∈ (0,+∞) with f0 6= f∞, then for k ∈ N, µ ∈ (min{µ ν k f0 , µνk f∞ },max{µ ν k f0 , µνk f∞ }), problem (1.7) has at least one nodal solution uνk, such that νuνk has exactly k − 1 simple zeros in (0, 1) and is positive near 0, where ν ∈ {+,−}. keywords: problem; theorem cache: ejde-268.pdf plain text: ejde-268.txt item: #240 of 601 id: ejde-269 author: Dix, Julio G. title: Improved oscillation criteria for first-order delay differential equations with variable delay date: 2021 words: 5110 flesch: 84 summary: = ∫ t t∗ p(s1) ∫ τ(t) τ(s1) p(s2) ∫ τ2(t) τ(s2) p(s3)· · · ∫ τn−1(t) τ(sn−1) p(sn) dsn . . . For the basic step n = 2, we have∫ t t∗ p(s1) ∫ τ(t) τ(s1) p(s2) ds2 ds1 ≥ ω ∫ t t∗ p(s1) ∫ t s1 p(s2) ds2 ds1 = ω 2! (∫ t t∗ p(s) ds )2 , where the equality follows from Lemma 2.1. keywords: τ(t cache: ejde-269.pdf plain text: ejde-269.txt item: #241 of 601 id: ejde-27 author: White, Luther W.; Malysheva, Tetyana; Karlstrom, Leif title: Estimation of plate parameters from vertical displacement data using a family of plate models date: 2023 words: 9006 flesch: 67 summary: The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. In fact, the problems of estimation of external forces and parameters for plate models have been of great practical interest in all fields of science and engineering where elastic plate models are employed. keywords: displacement; estimation; force; foundation; f̄v; mindlin; mindlin plate; models; parameters; plate; plate model; r3d cache: ejde-27.pdf plain text: ejde-27.txt item: #242 of 601 id: ejde-270 author: Ma, Li Ma; Yang, Guangzhengao title: Hadamard type inequalities via fractional calculus in the space of exp-convex functions and applications date: 2021 words: 6820 flesch: 78 summary: As a matter of fact, the development of mathematical inequalities is very closely related to the advances in the theory of convex function. As we know, the origin of the theory of convex function could be traced back to the literatures from many famous mathematicians, such as Jensen, Hardy, Hadamard. keywords: convex; exp; function; hadamard; inequalities; type cache: ejde-270.pdf plain text: ejde-270.txt item: #243 of 601 id: ejde-271 author: Liu, Zhenhai; Papageorgiou, Nikolaos S. title: Dirichlet (p,q)-equations with gradient dependent and locally defined reaction date: 2021 words: 3587 flesch: 78 summary: Then we can find z0 ∈ Ω such that u(z0) = max Ω̄ u > M . For u ∈ W 1,p 0 (Ω) we define u±(z) = u(z)± for all z ∈ Ω. keywords: 1,p; papageorgiou cache: ejde-271.pdf plain text: ejde-271.txt item: #244 of 601 id: ejde-272 author: Diz-Pita, Erika; Libre, Jaume; Otero-Espinar, M. Victoria title: Phase portraits of a family of Kolmogorov systems depending on six parameters date: 2021 words: 18300 flesch: 82 summary: We shall consider three cases: c0 < 0, c0 > 0, and c0 = 0. Conditions Classification 1.1 a0 > 0, c0 = 0, µ > 0, c2 < 0. keywords: c0µ; case; figure; node; phase; portrait; saddle; singular; stable cache: ejde-272.pdf plain text: ejde-272.txt item: #245 of 601 id: ejde-273 author: Huu-Tai, Pierre Chau; Ducomet, Bernard title: Energy-dependent Hamiltonian in a nuclear optical model date: 2021 words: 7965 flesch: 80 summary: Choosing R = Rδ so large that for r > R, =m(ρ) ≥ 0 and |ρ| ≥ δ one has |e(r, ρ)| > 1 2 e−τr, τ = =m(ρ), we obtain ∫ ∞ R |e(r, ρ)|2dr ≥ e−τR 8τ , ‖R(λ)ΦR‖2 ≥ ‖ΦR‖2e−τR |2e(ρ)| √ 2τ , which completes the proof. ρ U(r′, ρ2)e(r′, ρ) dr′, (2.8) for ρ 6= 0 and =m(ρ) ≥ 0. (2) For any δ > 0 and for r →∞ e(r, ρ) = eiρr(1 + o(1)), ∂re(r, ρ) = eiρr(iρ+ o(1)), (2.9) uniformly with respect to ρ in the domain {=m(ρ) ≥ 0, |ρ| > δ}. keywords: dr′ cache: ejde-273.pdf plain text: ejde-273.txt item: #246 of 601 id: ejde-274 author: Zeng, Shengda; Bai, Yunru; Gasinski, Leszek; Krech, Ireneusz title: Existence of solutions for implicit obstacle problems involving nonhomogeneous partial differential operators and multivalued terms date: 2021 words: 8223 flesch: 79 summary: η − f, u〉 ≥ a3 p− 1 ‖∇u‖pp − αj‖u‖pp − ‖βj‖1 − aK(w)‖u‖ − bK(w) − ‖f‖W 1,p 0 (Ω)∗‖u‖ ≥ ( a3 p− 1 − αjλ−1 1,p)‖∇u‖pp − ‖βj‖1 − aK(w)‖u‖ − bK(w) − ‖f‖W 1,p 0 (Ω)∗‖u‖ ≥ a3 p− 1 ‖u‖p − αjc(θ)θ‖u‖θ − ‖βj‖1 − aK(w)‖u‖ − bK(w) keywords: 1,p; w 1,p cache: ejde-274.pdf plain text: ejde-274.txt item: #247 of 601 id: ejde-275 author: Donyari, Zahra; Zivari-Rezapour, Mohsen; Emamizadeh, Behrouz title: Optimization problems and mathematical analysis of optimal values in Orlicz spaces date: 2021 words: 6068 flesch: 81 summary: We note that for f ∈ Aα, uf is positive, see [8, Lemma 3.4], and that uf ∈W 2,Φ(Ω), [3]. Let f ∈ Aα and h ∈ L∞(Ω) be such that (i) ∫ Ω0 n h− dx = ∫ Ω1 n h+ dx for all n ∈ N. (ii) limn→∞ ‖χΩ0 n h−‖∞ keywords: lemma; uf̂ cache: ejde-275.pdf plain text: ejde-275.txt item: #248 of 601 id: ejde-276 author: Feng, Binhua; He, Zhiqian; Liu, Jiayin title: Blow-up criteria and instability of standing waves for the inhomogeneous fractional Schrodinger equation date: 2021 words: 6829 flesch: 84 summary: EJDE-2021/39 BLOW-UP CRITERIA FOR SCHRÖDINGER EQUATIONS 17 References [1] T. Boulenger, D. Himmelsbach, E. Lenzmann; Blowup for fractional Schrödinger equation, J. Funct. [21] Y. Hong, Y. Sire; On fractional Schrödinger equations in Sobolev spaces, Comm. keywords: blow; equation; schrödinger; u(t cache: ejde-276.pdf plain text: ejde-276.txt item: #249 of 601 id: ejde-277 author: Urus, Nazia; Verma, Amit K. title: Existence and uniqueness results for fourth-order four-point BVP arising in bridge design in the presence of reverse ordered upper and lower solutions date: 2023 words: 7645 flesch: 81 summary: Step 5: Similarly, we deduce that u0 ≤ · · · ≤ ln+1 ≤ ln ≤ · · · ≤ l1 ≤ l0 = l(s). Thus we arrive at, the sequences ln and un such that u0 ≤ u1 ≤ · · · ≤ un ≤ un+1 ≤ · · · ≤ ln+1 ≤ ln ≤ · · · ≤ l1 ≤ l0. keywords: bvp; linear; order; solution cache: ejde-277.pdf plain text: ejde-277.txt item: #250 of 601 id: ejde-278 author: Wang, Wei-Chuan title: Existence of sign-changing solutions for radially symmetric p-Laplacian equations with various potentials date: 2021 words: 5658 flesch: 77 summary: [23] B. Liu; Positive solutions of singular three-point boundary value problems for the one- dimensional p-Laplacian, Comput. [24] R. Ma; Positive solutions for multipoint boundary value problem with a one-dimensional p-Laplacian, Comput. keywords: laplacian; p−1; q−p; solutions cache: ejde-278.pdf plain text: ejde-278.txt item: #251 of 601 id: ejde-279 author: Li, Xinyue; Zhang, Yongli; Zhang, Huiqun; Zhao, Qiulan title: Lie symmetry analysis and conservation laws for the (2+1)-dimensional Mikhalev equation date: 2021 words: 5616 flesch: 76 summary: (3.31) Substituting (3.31) into (1.1), it is easily to obtain the reduced nonlinear PDE with variable coefficients through a straight calculation 2v3fv + v4fvv + c4vfvw − c4wfww − 2v2ffv + wv2ffwv (3.27) 8 X. Y. LI, H. Q. ZHANG, Y. L. ZHANG, Q. L. ZHAO EJDE-2021/41 Solving this equation, we obtain v = (c3 − d3)y − d2x, w = t, u = d4 d2 y + f(w, v). keywords: equation; mikhalëv cache: ejde-279.pdf plain text: ejde-279.txt item: #252 of 601 id: ejde-280 author: Cai, Yuting; Wang, Chuncheng; Fan, Dejun title: Stability and bifurcation in a delayed predator-prey model with Holling-type IV response function and age structure date: 2021 words: 6146 flesch: 83 summary: Let φ1(ω) = arg{(Y − PN)ω2 +RH − Y Q+ (−Nω2 + SH − PY +QN)iω}. = √ ((Y − PN)ω2 + (RH − Y Q))2 + ((SH − PY +QN)ω −Nω3)2 × sin(φ1(ω)). keywords: bifurcation; equation; predator; prey cache: ejde-280.pdf plain text: ejde-280.txt item: #253 of 601 id: ejde-282 author: Papanicolaou, Vassilis G.; Kallitsi, Eva; Smyrlis, George title: Entire solutions for the heat equation date: 2021 words: 11196 flesch: 83 summary: (3.64) Then, from our assumption for the order and type of f(z), the integral in the right- hand side of (3.64) is entire in (t, z), satisfies the heat equation for every t, z ∈ C (e.g., by analytic continuation) and it is clear from (3.64) that F (0, z) = f(z). Using (1.5) in (1.1) yields F (t0 + t, z0 + z) = ∑ j,k≥0 ∂2j+kz F (t0, z0) j!k! tjzk, t, z ∈ C. (1.6) keywords: equation; function; heat; order; sup cache: ejde-282.pdf plain text: ejde-282.txt item: #254 of 601 id: ejde-283 author: Long, Yuhua; Chen, Yining title: Modeling porcine pseudorabies with age structure date: 2021 words: 7110 flesch: 76 summary: Next, define a continuously differentiable function V : R5 + → R as V = α β1 (I1 − I∗1 − I∗1 ln I1 I∗1 ) Owing to (2.1), there holds ε ≤ min{α+ γ, d2 − d1}, then detA ≥d1(d1 + d2 + ξ) (2αd2 + γ(d1 + d2)− ε(α+ d1 + d2)) keywords: disease; pseudorabies; system; −n01 cache: ejde-283.pdf plain text: ejde-283.txt item: #255 of 601 id: ejde-284 author: Wu, Yakui; Sun, Jiawei title: Asymptotic behavior of linearized Boltzmann equations for soft potentials with cut-off date: 2021 words: 8324 flesch: 87 summary: We decompose (λI − B̂(ξ))−1 as follows (λI − B̂(ξ))−1 = (λI − B̂0(ξ))−1 + (λI − B̂0(ξ))−1(I − P (λI − B̂0(ξ))−1)−1P (λI − B̂0(ξ))−1. ∩ %(Âs(ξ)), we have (λI − B̂0(ξ))−1 = (I − (λI − Âs(ξ))−1K0)−1(λI − Âs(ξ))−1. keywords: lemma; operator cache: ejde-284.pdf plain text: ejde-284.txt item: #256 of 601 id: ejde-285 author: You, Song; Zhao, Peihao; Wang, Qingxuan title: Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations date: 2021 words: 8991 flesch: 86 summary: Coupled Choquard equations; positive least energy solution; asymptotic behavior; variational method. This allows to consider positive least energy solution, which is defined as solution (u, v) of (1.4) with positive components and achieving the level inf{E(u, v) : E′(u, v) = 0, (u, v) ∈ H, u > 0 and v > 0}. keywords: energy; solution; |x|α ∗ cache: ejde-285.pdf plain text: ejde-285.txt item: #257 of 601 id: ejde-287 author: Nunes, Ruikson S. O. title: Exact boundary controllability for the wave equation with moving boundary domains in a star-shaped hole date: 2021 words: 5440 flesch: 77 summary: Another interesting point it is to study on exact boundary control problems, in holed domains, for systems of coupled waves equations as proposed in [5, 16]. [4] W. D. Bastos, J. Ferreira; Exact boundary control for the wave equation in a polyhedral time-dependent domain, Appl Math Lett., 12 (1999), 1–5. keywords: boundary; control; equation; wave cache: ejde-287.pdf plain text: ejde-287.txt item: #258 of 601 id: ejde-288 author: Katarina S. Djordjevic, Katarina S. title: Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations date: 2021 words: 10368 flesch: 75 summary: This article studies the asymptotic behavior of positive solutions of the q-difference half-linear equation Dq(p(t)Φ(Dq(x(t)))) + r(t)Φ(x(qt)) (1.2) For recent papers investigating asymptotic behavior of positive solutions of (1.2) see [13, 14, 15]. keywords: solution; varying cache: ejde-288.pdf plain text: ejde-288.txt item: #259 of 601 id: ejde-289 author: Bychkov, Evgeniy; Sviridyuk, Georgy; Bogomolov, Alexey title: Optimal control for solutions to Sobolev stochastic equations date: 2021 words: 5390 flesch: 68 summary: Optimal control problems for models of mathematical physics represent a promising direction We construct the noise space ClL2, l ∈ N, as the space of random processes from CL2, whose trajectories are almost sure (a.s.) keywords: control; equations; operator; problem; process; sobolev; space cache: ejde-289.pdf plain text: ejde-289.txt item: #260 of 601 id: ejde-290 author: Alvarez-Caudevilla, Pablo; Colorado, Eduardo; Ortega, Alejandro title: Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator date: 2021 words: 10098 flesch: 76 summary: Introduction In this work, we analyze the existence of positive solutions of the second order elliptic equation under homogeneous Dirichlet boundary conditions and involving a non-local term, −∆u = γ(−∆)−1u+ |u|p−1u in Ω, u = 0 on ∂Ω, (1.1) where γ is a positive real parameter and Ω is a smooth bounded domain of RN , with N ≥ 3, 1 < p ≤ 2∗ − 1, where 2∗ = 2N N−2 is the critical Sobolev exponent. Here, as customary (−∆)−1u = v, if −∆v = u in Ω, v = 0 on ∂Ω. Thus, (−∆)−1 is a positive linear integral compact operator from L2(Ω) into itself, which is well-defined thanks to the Spectral Theorem. keywords: lemma; problem cache: ejde-290.pdf plain text: ejde-290.txt item: #261 of 601 id: ejde-291 author: Carriao, Paulo Cesar; Costa, Augusto Cesar dos Reis; Miyagaki, Olimpio Hiroshi; Vicente, Andre title: Kirchhoff-type problems with critical Sobolev exponent in a hyperbolic space date: 2022 words: 5182 flesch: 81 summary: We consider the functional J : H1 0,r(Ω)→ R associated with problem (2.1), J(v) = a 2 ‖v‖2 + b 4 ‖v‖4 − λ q ∫ Ω pα|v|q − 1 6 ∫ Ω |v|6, (2.2) whose Gateaux derivative is J ′(v)w = (a+ b‖v‖2) ∫ Ω ( ∇v ·∇w+ 3 4 p2vw ) −λ ∫ Ω pα|v|q−2vw− ∫ Ω |v|4vw. From (2.13), we have d dt J(tvε)|t=tε = 0, thus, atε‖vε‖2 + bt3ε‖vε‖4 − λtq−1 ε ∫ Ω pα|vε|q − t5ε ∫ Ω |vε|6 = 0, EJDE-2021/53 KIRCHHOFF-TYPE PROBLEMS IN A HYPERBOLIC SPACE 7 which implies a‖vε‖2 + bt2ε‖vε‖4 − λtq−2 ε ∫ Ω pα|vε|q − t4ε ∫ Ω |vε|6 = 0. keywords: 0,r(ω; kirchhoff cache: ejde-291.pdf plain text: ejde-291.txt item: #262 of 601 id: ejde-292 author: Wang, Haoyu; Tian, Ge title: Propagating interface in reaction-diffusion equations with distributed delay date: 2021 words: 9017 flesch: 85 summary: ln ε| ⇒ 1− ερ0 ≤ uε(α0ε| ln ε|+ ετ + εt, x) ≤ 1, t ∈ There exists P > 1 such that, for sufficiently small ε > 0, it holds u−η (t, x) ≤ uε(t+ α0ε| ln ε|+ ετ, x), ∀ − ετ ≤ t ≤ 0, x ∈ RN , where α0ε| ln ε| denotes the “generation of interface from below time” appearing in Proposition 2.11. keywords: diffusion; equations; interface; proof cache: ejde-292.pdf plain text: ejde-292.txt item: #263 of 601 id: ejde-293 author: Galiano, Gonzalo; Gonzalez-Tabernero, Victor title: Turing instability analysis of a singular cross-diffusion problem date: 2021 words: 7487 flesch: 72 summary: Then, for k ≥ 1 the linear problem to solve is: Find (un,k1 , un,k2 ) such that for for all χ ∈ Sh 1 τ ( un,k1 − un−1 1 , χ)h + ( dδ11(un,k−1)∂xu n,k 1 + dδ12(un,k−1)∂xu n,k 2 , ∂xχ )h = ( un,k1 (αb1 − βb11u n,k−1 1 − βb12u n,k−1 2 ), χ)h, 1 τ ( un,k2 − un−1 2 , χ)h + ( dδ21(un,k−1)∂xu n,k 1 + dδ22(un,k−1)∂xu n,k 2 , ∂xχ )h = ( un,k2 (αb2 − βb21u n,k−1 1 − βb22u n,k−1 2 )χ)h. − k2 tr(Dδ(u∗)) keywords: b→0; cross; diffusion; instability; problem cache: ejde-293.pdf plain text: ejde-293.txt item: #264 of 601 id: ejde-294 author: Montgomery-Smith, Stephen J.; Oveys, Hesam title: Age-dependent branching processes and applications to the Luria-Delbruck experiment date: 2021 words: 8690 flesch: 75 summary: Cells have the following properties: (1) there are exactly two types of cells: mother cells and daughter cells; (2) all cells are independent of each other, mother cells are identical to other mother cells, and daughter cells are identical to other daughter cells; (3) cell life-span for mother cells and daughter cells are strictly positive, real- valued random variables Tx and Ty, respectively, with distributions P (t) keywords: cell; daughter; function; generating; mother cache: ejde-294.pdf plain text: ejde-294.txt item: #265 of 601 id: ejde-295 author: Ochoa, Pablo; Ruiz, Julio Alejo title: Solving singular evolution problems in sub-Riemannian groups via deterministic games date: 2021 words: 9407 flesch: 83 summary: ε−1sj0〈η̂ − η, ξj0〉+ s20 2 E ( X̂ − X ) + s2j0 2 〈( X̂ − X ) ξj0 , ξj0 〉 + [ F ( t, p, η, X̂ ) −F ( t, p, η,X )] + [ F ( t, p, η̂, X̂ ) −F ( t, p, η, X̂ )] . (5.10) 20 P. OCHOA, J. A. RUIZ EJDE-2021/57 If E(X̂ − X ) > 0, we chose |s0| = λ1 with s0〈η̂ − η, ξ0〉 ≥ 0. lim n→∞ sup q { ((uε)∗ − ϕ)(tn − ε2, pn · δε(q)) } keywords: carnot; equations; groups; p̂−1|g; µε2 cache: ejde-295.pdf plain text: ejde-295.txt item: #266 of 601 id: ejde-297 author: Bunoiu, Renata; Karim, Karim; Timofte, Claudia title: T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains date: 2021 words: 9545 flesch: 73 summary: For u ∈ H1(Ωε), set Tεu = { u1 in Ωε1, −u2 + 2Pεu1 in Ωε2. For u ∈ H1(Ωε), let Tεu = { u1 − 2Qε(u2 −Mε 2(u2)) in Ωε1, −u2 + 2Mε 2(u2) in Ωε2. keywords: periodic; problem cache: ejde-297.pdf plain text: ejde-297.txt item: #267 of 601 id: ejde-298 author: Jesus, Isaias P. de; Cabanillas Lapa, Eugenio; Limaco, Juan title: Controllability for the wave equation with moving boundary date: 2021 words: 4396 flesch: 77 summary: Wave equation; Stackelberg-Nash strategies; controllability; inverse inequality. [2] A. Shao; On Carleman and observability estimates for wave equations on time-dependent domains, Proc. keywords: controllability; equation; wave; w̃1; w̃2 cache: ejde-298.pdf plain text: ejde-298.txt item: #268 of 601 id: ejde-299 author: Chlebowicz, Agnieszka title: Existence of solutions to infinite systems of nonlinear integral equations on the real half-axis date: 2021 words: 7925 flesch: 82 summary: → R+ which is nondecreasing on R+, con- tinuous at 0 and there exist a natural number p and a nonnegative integer q such that for any r > 0 and for x = (xi), y = (yi) ∈ l1 with ‖x‖l1 ≤ r, ‖y‖l1 ≤ r and for t ∈ R+, n ∈ N, n ≥ p+ 1 the following inequality |fn(t, x1, x2, . . .)| ≤ l(r) n+q∑ i=n |xi − yi| holds for x = (xi), y = (yi) ∈ l1 with ‖x‖l1 ≤ r, ‖y‖l1 ≤ r and for t ∈ R+, 1 ≤ n ≤ p. (vii) keywords: equations; n=1; space cache: ejde-299.pdf plain text: ejde-299.txt item: #269 of 601 id: ejde-30 author: Showalter, Ralph E. title: Hilbert Space Methods for Partial Differential Equations date: 1994 words: 11980 flesch: 84 summary: A function T : V →W is called conjugate linear if T (αx+ βy) = ᾱT (x) + β̄T (y) , α, β ∈ K , x, y ∈ V . A set K in the vector space V is convex if for x, y ∈ K and 0 ≤ α ≤ 1, we have αx+(1−α)y ∈ K. keywords: linear; space; theorem cache: ejde-30.pdf plain text: ejde-30.txt item: #270 of 601 id: ejde-301 author: Li, Mengyuan; Liu, Qihuai title: Periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations date: 2021 words: 16403 flesch: 79 summary: × ( (1− p)(1−G √ −2h)−m )(1−G √ −2h G √ −2h )m−1 and D2(G;h, p) = −β2p−1 √ −2h 2p−2 ( G √ −2h+ 1 )−p−1 1−p∑ m=0 Cm1−p (1− p)! if |β| < 1/16, such that ζε ±(0) tends to(1 2 , 1 2 cos g0 sin k0, 1 2 sin g0, √ 2 sin g0 cos k0, √ 2 sin g0 sin k0,− √ 2 cos g0 ) , as ε→ 0 where g0 = ±1 2 arccos (−16β) . keywords: = −; anisotropic; g √; orbits; problem; − √; √ −2h cache: ejde-301.pdf plain text: ejde-301.txt item: #271 of 601 id: ejde-302 author: Idczak, Dariusz title: Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator date: 2021 words: 6736 flesch: 80 summary: ∈ L2. Assume that function f is measurable in t ∈ (0, π), con- tinuously differentiable in (x, u) ∈ Rm × Rr and |f(t, x, u)|, |fx(t, x, u)|, |fu(t, x, u)| ≤ a(t)γ(|x|) + b(t)δ(|u|) (5.1) for (t, x, u) ∈ (0, π)×Rm ×Rr, where a, b ∈ L2 and γ, δ : R+ 0 → R+ 0 are continuous functions. keywords: d((−∆)β; function; j=1; x(t cache: ejde-302.pdf plain text: ejde-302.txt item: #272 of 601 id: ejde-303 author: Pereira, Jardel Morais title: Attractors for dissipative lattice differential equations with local and nonlocal nonlinearities date: 2021 words: 14254 flesch: 81 summary: Since u = (un) and v = (vn) belong to `2, we have∑ n∈Zd (∆dun)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂−i un)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂+i un)vn+ei = − ∑ n∈Zd d∑ i=1 ∂+i un∂ + i vn = − ∑ n∈Zd ∇+un · ∇+vn. This proves Lemma 2.2 if p = 1. ∈ C1(R+; `2), from (3.34), we obtain (−1)2k+1 ∑ n∈Zd ∆2k+1 d un(θnu̇n) = 1 2 d dt ∑ n∈Zd θn|D2k−1vn|2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z (i) 2k−1,n + ∑ n∈Zd d∑ i=1 ∂+i θn [ (∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n) ] , (3.35) where, in view of Lemma 2.1, ∑ n∈Zd d∑ i=1 |z(i)2k−1,n| ≤ C(2k − 1, d)‖(v, v̇)‖2H ≤ 16d2C(2k − 1, d)‖(u, u̇)‖2H , ∑ n∈Zd d∑ i=1 |(∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n)| ≤ (4d)4k+1‖(u, u̇)‖2H . keywords: attractors; i=1; lemma; n∈zd; proof; sup; t≤s≤t+1 cache: ejde-303.pdf plain text: ejde-303.txt item: #273 of 601 id: ejde-304 author: Lin, Xiaolu; Zheng, Shenzhou title: Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent date: 2021 words: 9198 flesch: 82 summary: 〈I ′(un)− I ′(u), un − u〉 = m(‖un‖)〈un, un − u〉s,p −m(‖un‖)〈u, un − u 〉 s,p + ( 〈un, un − u〉s,q − 〈u, un − u〉s,q ) + ∫ Ω ( |un|p ∗ s(α)−2un − |u|p ∗ s(α)−2u )( un − u ) |x|α dx + λ ∫ Ω f(x) ( |un|r−2un − |u|r−2u )( un − u ) |x|c dx. (3.15) (3.21) Let us now put (3.18), (3.19) and (3.21) into (3.15), which yields the inequality o(1) ≥ m(‖un‖) ( 〈un, un − u〉s,p − 〈u, un − u〉s,p ) +m(‖un‖)〈u, un − u〉s,p −m(‖un‖)〈u, un − u〉s,p. (3.22) keywords: lemma; p∗s(α; solutions cache: ejde-304.pdf plain text: ejde-304.txt item: #274 of 601 id: ejde-306 author: Ali, Mageed; Iaia, Joseph title: Infinitely many solutions for a singular semilinear problem on exterior domains date: 2021 words: 7500 flesch: 92 summary: Next integrating (2.15) on (t, R2−N ) and dividing by (R2−N − t) (2.27) Integrating on (t, R2−N ) and using (2.3), (2.4) we obtain |Va(t)| = ∣∣ ∫ R2−N t V ′a(s) ds ∣∣ ≤ ∫ R2−N t |V ′a(s)| ds ≤ ∫ R2−N t (aRN−1 N − 2 + √ 2F0h(R2−N ) ) ds = (R2−N − t) (aRN−1 N − 2 + √ 2F0h(R2−N ) ) ≤ aR N − 2 +R2−N √ 2F0h(R2−N ). keywords: r2−n; va(ma; va(t cache: ejde-306.pdf plain text: ejde-306.txt item: #275 of 601 id: ejde-307 author: Llibre, Jaume; Oliveira, Regilene; Rodrigues, Camila A. B. title: Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq date: 2021 words: 22812 flesch: 77 summary: = y2 − x(x − 1)(x − r) with r > 1 or f(x, y) = (x+ c)(α2x+ γ2y + α2), ẏ = −(γ1/2)(x2 − y2 − 1) + x(γ2x+ α2y + cγ2), where α2(c+ 1) = 0, (H.3) ẋ keywords: case; darboux; eigenvalues; figure; invariant; line; phase; point; portraits; singular; system cache: ejde-307.pdf plain text: ejde-307.txt item: #276 of 601 id: ejde-308 author: Li, Shanbing; Xiao, Yanni; Dong, Yaying title: Diffusive predator-prey models with fear effect in spatially heterogeneous environment date: 2021 words: 14933 flesch: 76 summary: − d− audu,n − b(x)(v∞ − ε) ) udu,n , x ∈ Ω. A standard comparison argument yields udu,n ≤ Udu,n in Ω for all large n, where Udu,n is the unique positive solution of −du,n∆Udu,n = ( r 1 + k(v∞ − ε) It is clear that (1.4) admits a trivial solution (u, v) = (0, 0), two semi-trivial solutions (u, v) = ((r − d)/a, 0) with r > d and (u, v) = (0,m) with m > 0, and positive solutions (u, v) with no component identically zero. keywords: fear; predator; prey; solution cache: ejde-308.pdf plain text: ejde-308.txt item: #277 of 601 id: ejde-309 author: Chiyo, Yutaro; Mizukami, Masaaki title: Existence of bounded global solutions for fully parabolic attraction-repulsion date: 2021 words: 4593 flesch: 76 summary: While finite-time blow-up was proved in the two-dimensional setting when χ > ξ and the initial data satisfy the conditions that ∫ Ω u0 > 8π χ−ξ and that∫ Ω u(x)|x− x0|2 dx (x0 ∈ Ω) is sufficiently small. [18] obtained global existence and boundedness when χ = 0 (or µ > χ − ξ + M with some M > 0 in (1.1)). keywords: system cache: ejde-309.pdf plain text: ejde-309.txt item: #278 of 601 id: ejde-310 author: Cui, Pengxue; Ji, Shuguan title: Existence and nonlinear stability of solitary wave solutions for coupled Schrodinger-KdV systems date: 2021 words: 5474 flesch: 84 summary: ‖q‖2L2(R))φ(φϕ)′dx = 3‖γ1‖2L(R) 1− c− 4 3β(−ω − c2 4 ) ∫ R φ3(x)φ′(x)dx = 0, (4.17) and (p⊥, φ) = ∫ R pφ+ 1 2 (‖p‖2L2(R) + Schrödinger-KdV system; nonlinear stability; solitary wave solution. keywords: solitary; stability; wave cache: ejde-310.pdf plain text: ejde-310.txt item: #279 of 601 id: ejde-312 author: Koroleva, Yulia O.; Yu, Daria title: Estimates of characteristics of a micropolar flow passing through an axially symmetric cell date: 2021 words: 6298 flesch: 72 summary: Micropolar fluid flow; porous medium; weak solution. = 1 N2 ∆v1 + 2 curlω1 − εσ2 N2 v1, L2∆ω1 + 1 2 N2 1−N2 curl v1 − N2 1−N2 ω1 = 0. (2.14) div v2 = 0,( 1 N2 − 1 ) ∇p2 = 1 N2 ∆v2 + 2 curlω2, L2∆ω2 + 1 2 N2 1−N2 curl v2 − N2 1−N2 ω2 = 0. (2.15) keywords: cell; curl; flow cache: ejde-312.pdf plain text: ejde-312.txt item: #280 of 601 id: ejde-313 author: Jleli, Mohamed; Samet, Bessem title: Nonexistence results for hyperbolic type inequalities involving the Grushin operator in exterior domains date: 2021 words: 10494 flesch: 87 summary: −1 p−1 dx ) ≤ C ( R −2θp p−1 R2θ lnR+R −θp p−1Rθ ( p−2 p−1 ) (lnR) −1 p−1 ) ≤ CR −2θ p−1 lnR. 12 M. JLELI, B. SAMET EJDE-2021/75 For N1 ≥ 3 and Rθ < |x| < 2Rθ, proceeding as above, and using Lemma 3.3, we obtain b −1 p−1 |∆xb| p p−1 ≤ C ( R −2θp p−1 ( 1− |x|2−N1 ) Hence, for 1 < p < N1 N1−2 , taking 0 < 2σ < 2p p−1 − N1 and passing to the limit as R → ∞ in the above inequality, we obtain a contradiction with ∫ ∂D1 g(y)dσy > 0. keywords: g(y)dσy; p−1; ∂d1; ∂d2 cache: ejde-313.pdf plain text: ejde-313.txt item: #281 of 601 id: ejde-314 author: Drabek, Pavel; Zahradnikova, Michaela title: Traveling waves for unbalanced bistable equations with density dependent diffusion date: 2021 words: 9340 flesch: 87 summary: Preliminaries Let g : [0, 1]→ R, g ∈ C[0, 1] be such that g(0) = g(s∗) = g(1) = 0 for s∗ ∈ (0, 1) and g(s) ≤ 0, s ∈ (0, s∗), g(s) > 0, s ∈ (s∗, 1). Note that f ∈ L1(0, 1) implies that h = h(t, y, c) satisfies Carathéodory’s conditions, i.e., for a.e. t ∈ keywords: p−1 cache: ejde-314.pdf plain text: ejde-314.txt item: #282 of 601 id: ejde-315 author: Wan, Youyan; Tan, Jinggang title: Standing waves to Chern-Simons-Schrodinger systems with critical exponential growth date: 2021 words: 5311 flesch: 86 summary: 4 Y. WAN, J. TAN EJDE-2021/77 Again we can derive from ∂1A2 − ∂2A1 = − 1 2u 2 that∫ R2 A0|u|2 In fact, since ‖un‖ ≤ c0, J(un)→ c, and J ′(un)→ 0, we have∫ R2 F (un) = 1 2 ‖un‖2 + 1 2 ∫ R2 ( A2 1,n|un|2 +A2 2,n|un|2 ) dx− c+ on(1),∫ R2 f(un)un dx = ‖un‖2 + 3 ∫ R2 ( A2 1,n +A2 2,n ) u2 n dx− εn‖un‖, where εn → 0 as n→∞. By Proposition 2.1 and Sobolev embedding theorem, we obtain ∫ R2 F (un) ≤ 1 2 ‖un‖2 + C‖un‖4 − c+ on(1),∫ R2 f(un)un dx = ‖un‖2 + C‖un‖4 − εn‖un‖. From ‖un‖ ≤ c0, we obtain ∫ R2 f(un)un dx ≤ c0 and ∫ R2 F (un) dx ≤ c0. keywords: chern; schrödinger cache: ejde-315.pdf plain text: ejde-315.txt item: #283 of 601 id: ejde-316 author: Ishibashi, Kazuki title: Hille-Nehari type non-oscillation criteria for half-linear dynamic equations with mixed derivatives on a time scale date: 2021 words: 5499 flesch: 82 summary: Half-linear dynamic equations; nonoscillation; time scale; Riccati dynamic inequality; linear differential equation; linear difference equation. Let T = R and p = 2. keywords: non; theorem cache: ejde-316.pdf plain text: ejde-316.txt item: #284 of 601 id: ejde-317 author: Guo, Ying-Jia; Jiang, Xiao-Meng title: Stochastic Newtonian equations with mean boundary conditions date: 2021 words: 7277 flesch: 82 summary: = ∫ t a Y (s)ds =− t− a b− a ∫ b a ds ∫ s a E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du + ∫ t a ds ∫ s (3.31) Combining (3.29), (3.30) and (3.31), we conclude that for all t ∈ [a, b], y(t) = − t− a b− a ∫ b a ds ∫ s a E[f(u, x̄(u) + y(u))− f(u, x̄(u))]du + ∫ t a ds ∫ s a [f(u, x̄(u) + y(u))− f(u, x̄(u))]du + ∫ t a ds ∫ s a g(u, x̄(u) + y(u))dB(u) keywords: boundary; conditions; equations; f(u; ∫ b; ∫ t cache: ejde-317.pdf plain text: ejde-317.txt item: #285 of 601 id: ejde-318 author: Webb, Jeffrey R. L. title: A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems date: 2021 words: 10145 flesch: 82 summary: Suppose that u ∈ C+[0, T ] satisfies u(t) ≤ a(t) + ∫ t 0 φ(s)u(s) ds for t ∈ If x ∈ L∞+ [0, T ] satisfies the inequality x(t) ≤ a(t) + g(t) ∫ t 0 (t− s)β−1x(s)ds, t ∈ keywords: fractional; u(t cache: ejde-318.pdf plain text: ejde-318.txt item: #286 of 601 id: ejde-32 author: Feng, Zaichun (Z.C.); Li, Y. Charles title: Enrichment paradox and applications date: 2023 words: 3339 flesch: 64 summary: Dynamics of (2.1)-(2.2) when r = 1/2, k = 1, H = 1.69, ν = 2. such as plagues also caused human population to temporarily decrease. But since 1700, human population has been monotonically increasing due to technological advances. keywords: predator; prey cache: ejde-32.pdf plain text: ejde-32.txt item: #287 of 601 id: ejde-320 author: Hafeez, Usman; Lavier, Theo; Williams, Lucas; Korobenko, Lyudmila title: Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators date: 2021 words: 7614 flesch: 76 summary: Let u be a weak solution to (1.1) on Ω = B and define u+ = max{u, 0}, then the following Caccioppoli inequality holds on the ball B∫ {x∈B:u(x)>0} |∇Au+|2 dµ ≤ ∫ {x∈B:u(x)>0} u+‖f‖L∞ dµ, where dµ = dx|B|. Proof. Moreover, if the above estimate holds for q = σ′ then Sobolev inequality (1.2) holds (almost necessity). keywords: inequality; orlicz; sobolev cache: ejde-320.pdf plain text: ejde-320.txt item: #288 of 601 id: ejde-321 author: Bujac, Cristina; Schlomiuk, Dana; Vulpe, Nicolae title: Cubic differential systems with invariant straight lines of total multiplicity seven and four real distinct infinite singularities date: 2021 words: 53541 flesch: 81 summary: = 2−43−9(X − 2Y )(3X − 4mZ)3(3X − 3Y − 2mZ)2(3Y − 2mZ)2 and hence by Lemma 2.6 we have invariant lines of total multiplicity nine, i.e. we are not in the class of systems with invariant lines of total multiplicity exactly seven. 1.2.2. So considering these conditions as well as the conditions (3.145) and (3.146) we calculate H2 = 8r(1 + r)(2 + 2r + u)3 (1 + u)3(r + u)3∆cf V1V2V3, H ′2 = −8r6(1 + r)(1 + r − u)3 (1 + u)3(r + u)3∆cf V1V2V4, where V1 = h(2r − u)(r + u) +m(u− 2)(1 + u), V2 = hr(r + u)(3 + r + u) +m(1 + u)(1 + 3r + u), V3 = hr(r + u)(2r + 4r2 + 2r3 − u+ 2ru+ 3r2u− 8u2 + 2r2u2 − 4u3 + 3ru3 + u4) +m(1 + u)(r3u− 2r − 4r2 − 2r3 − 3ru− 2r2u− 2u2 + 8r2u2 − 3u3 + 4ru3 − u4), V4 = h(r + u)(2r3 + 2r4 − 2r − 2r2 + u− 7ru− 9r2u− r3u+ u2 − 15ru2 − 10r2u2 − u3 − 8ru3 − u4) +m(1 + u)(2 + 2r − 2r2 − 2r3 − u− 9ru− 7r2u+ r3u− 10u2 − 15ru2 + r2u2 − 8u3 − ru3 − u4). keywords: = 0; = −; bujac; case; complex; condition; configurations; direction y; ejde-2021/; family; g −; invariant; lemma; lines; multiplicity; p =; p+ r; parameter; point; possibility; r −; r)(1; real; schlomiuk; singularities; system; total; transformation; u 6=; u =; vulpe cache: ejde-321.pdf plain text: ejde-321.txt item: #289 of 601 id: ejde-322 author: Antontsev, Stanislav; Ferreira, Jorge; Piskin, Erhan; Yuksekkaya, Hazal; Shahrouzi, Mohammad title: Blow up and asymptotic behavior of solutions for a p(x)-Laplacian equation with delay term and variable exponents date: 2021 words: 6660 flesch: 80 summary: = (1− α)H−α(t)H ′(t) + ε ∫ Ω u2 tdx− ε ∫ Ω |∇u|p(x)dx + εb ∫ Ω |u|q(x)dx− εµ1 ∫ Ω uut(x, t)|ut(x, t)| m(x)−2 dx − εµ2 ∫ Ω uz(x, 1, t)|z(x, 1, t)| m(x)−2 dx. dx dρ EJDE-2021/84 BLOW UP AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS 9 + ε ∫ Ω u2 tdx− ε ∫ Ω |∇u|p(x) + εab ∫ Ω |u|q(x)dx − εµ1 ∫ Ω uut(x, t)|ut(x, t)| m(x)−2 dx − εµ2 ∫ Ω uz(x, 1, t)|z(x, 1, t)| m(x)−2 dx. keywords: er(t; m(x; ∫ t; ∫ ω cache: ejde-322.pdf plain text: ejde-322.txt item: #290 of 601 id: ejde-323 author: Feng, Sebert title: Symmetry analysis for a second-order ordinary differential equation date: 2021 words: 4764 flesch: 73 summary: Hence, the general solution of the linearized symmetry condition is ξ = − (q + 2)c0 k2 e k2q q+2x + c1, η = c0e k2q q+2xy. (4.5) Substituting (4.5) into (4.2), we have η = − 2a(x)k1 (q + 1)(q + 2) yq+2 + {a′(x)− a(x)k2}y2 + c(x)y + d(x), (4.6) where q 6= −1 and q 6= −2. keywords: differential; equation; lie; symmetry cache: ejde-323.pdf plain text: ejde-323.txt item: #291 of 601 id: ejde-324 author: Li, Mengni title: Singular Monge-Ampere equations over convex domains date: 2021 words: 7300 flesch: 80 summary: By constructing a family of sub-solutions, we prove the existence and global Hölder estimates of convex solutions to the problem over convex domains. For any point y ∈ Ω, we can find z ∈ ∂Ω be the nearest boundary point to y. Without loss of generality, we assume that the domain Ω satisfies the exterior sphere condition with radius R. By some translations and rotations, we can further assume z = 0, 0 ∈ ∂Ω ∩ ∂BR(y0), Ω ⊂ BR(y0), and the line yz is the xn-axis. keywords: convex; domain; n+α; solution cache: ejde-324.pdf plain text: ejde-324.txt item: #292 of 601 id: ejde-326 author: Fama, Alessio; Restuccia, Liliana title: Coupled porosity-fluid concentration flux-temperature waves in isotropic porous media date: 2021 words: 5166 flesch: 64 summary: j c ∂jci ∂t = −jci + (3ξ3 1 + 2ξ3 2)r,i + ξ5T,i, (6.2) i.e. equation (3.14), when we define βc = ξ5, αc = 3ξ3 1 + 2ξ3 2 . (6.3) 7. Appendix D: Derivation of temperature equation To deduce (3.15), we use (3.3) , (3.11), (3.12)1, and (3.12)5, and the special forms (3.8)2, (3.8)3 , and (4.5) of the tensors Kij , ηij and ν3 ijkl, so that we obtain τ q ∂2T ∂t2 + ∂T ∂t = 3η ∂r ∂t +KT,ii +Dν [ ν3 1δilδjk + ν3 2(δijδkl + δikδjl) ] r,liδjk, (7.1) where ν3 1 , ν3 2 are the 2 significant independent components of the fourth tensor ν3 ijkl and K, η are the only significant components of the second order tensors Kij and ηij . In [25, 26] the constitutive equations and rate equations were obtained (to close the systems of balance equations, see [25]) obeying the objectivity and frame indifference principles keywords: equation; fluid; flux; ijkl; propagation cache: ejde-326.pdf plain text: ejde-326.txt item: #293 of 601 id: ejde-327 author: Dao, Nguyen Anh; Diaz, Jesus Ildefonso title: Logarithmically improved regularity criteria for the Navier-Stokes equations in homogeneous Besov spaces date: 2021 words: 3420 flesch: 82 summary: To be more precise, if∫ T 0 ‖w(τ)‖L∞ dτ <∞ , then the smooth solution u, in C([0, T );W s,p(Rn)), with s > n/p + 1, can be continued beyond t = T . Thanks to the Gagliardo–Nirenberg inequality, we obtain ‖(−∆) s0 2 u(t)‖L2 ≤ ‖u(t)‖1− s0 s L2 ‖(−∆)s/2u(t)‖ s0 s L2 ≤ ‖u‖ 1− s0s L∞(0,T ;L2)‖(−∆)s/2u(t)‖ s0 s L2 for t ∈ (0, T ). keywords: regularity cache: ejde-327.pdf plain text: ejde-327.txt item: #294 of 601 id: ejde-328 author: Aguilera Contreras, Gabriel; Munoz-Rivera, Jaime E. title: Bresse systems with localized Kelvin-Voigt dissipation date: 2021 words: 5540 flesch: 82 summary: More precisely, we consider the system ρ1ϕtt − Sx − lN = 0 in Ĩ × (0,+∞), (1.1) −−lK(zx − lv)x − lK(zxt − lvt)x + lκ(vx + y + lz) + lκ̃(vxt + yt + lzt) = 0. (4.5) EJDE-2021/90 BRESSE SYSTEMS 13 Multiplying (4.3) by vt, and (4.4) by wt, and integrating on [0, `], we obtain∫ ` 0 ( ρ1|vt|2 + ρ2|yt|2 + ρ1|zt|2 + b|yx|2 + κ|vx + y + lz|2 +K|zx − lv|2 ) keywords: beam cache: ejde-328.pdf plain text: ejde-328.txt item: #295 of 601 id: ejde-329 author: Messaoudi, Salim A.; Bouhoufani, Oulia; Hamchi, Ilhem; Alahyane, Mohamed title: Existence and blow up in a system of wave equations with nonstandard nonlinearities date: 2021 words: 11145 flesch: 86 summary: , ωk}, as follows uk(x, t) = Σkj=1aj(t)ωj(x), vk(t) = Σkj=1bj(t)ωj(x), for x ∈ Ω and t ∈ (0, T ), satisfying the approximate problems∫ Ω uktt(x, t)ωjdx+ ∫ Ω A∇uk(x, t) · ∇ωjdx + ∫ Ω |ukt (x, t)|m(x)−2ukt (x, t)ωjdx = ∫ Ω f(x, t)ωjdx,∫ Ω vktt(x, t)ωjdx+ ∫ Ω B∇vk(x, t) · ∇ωjdx + ∫ Ω |vkt (x, t)|r(x)−2vkt (x, t)ωjdx = ∫ Ω g(x, t)ωjdx, (3.5) for j Introduction In this work, we study the following initial-boundary-value problem for the un- knowns u and v: utt − div(A∇u) + |ut|m(x)−2ut = f1(x, u, v) in Ω× (0, T ), vtt − div(B∇v) + |vt|r(x)−2vt = f2(x, u, v) in Ω× (0, T ), u = v = 0 on ∂Ω× (0, T ), u(0) = u0, ut(0) = u1 in Ω, v(0) keywords: blow; existence; p−+1; ρ(v cache: ejde-329.pdf plain text: ejde-329.txt item: #296 of 601 id: ejde-33 author: Allal, Brahim; Fragnelli, Genni; Salhi, Jawad title: Null controllability of coupled systems of degenerate parabolic integro-differential equations date: 2023 words: 8349 flesch: 83 summary: = 0, t ∈ (0, T ),{ y1(t, 0) = y2(t, 0) = 0, if a is strongly degenerate, t ∈ (0, T ), y1(0, x) = y0 1(x), y2(0, x) = y0 2(x), x ∈ (0, 1), (1.1) where Q = (0, T )×(0, 1), ω b (0, 1) is a non-empty open set, 1ω is the corresponding characteristic function, bij := bij(t, x) ∈ L∞(Q) and u = u(t, x) is the distributed control function. keywords: controllability; dt dx cache: ejde-33.pdf plain text: ejde-33.txt item: #297 of 601 id: ejde-330 author: Bellaama, Rachid; Belaidi, Benharrat title: Lower order for meromorphic solutions to linear delay-differential equations date: 2021 words: 7903 flesch: 88 summary: (3.29) We may choose ε sufficiently small satisfying 0 < 3ε < min { µ(Al0)− ρ, µ(Al0)− λ ( 1 Al0 )} , it follows from (3.29) that for r ∈ E6 \ E8, r → +∞, rµ(Al0)−2ε ≤ rρ(f)+ε, this means, µ(Al0) ≤ ρ(f) + (3.30) We choose ε sufficiently small satisfying 0 < 3ε < min { µ(Al0)− ρ, µ(Al0)− λ( 1 Al0 ) } , from (3.30) that for r ∈ E6 \ E8, r → +∞, rµ(Al0)−2ε ≤ rρ(f)−1+ε, this means, µ(Al0) ≤ ρ(f)−1 + 3ε, since ε > 0 is arbitrary, then ρ(f) ≥ µ(Al0) + 1. 12 R. BELLAAMA, B. BELAÏDI EJDE-2021/92 Case 2: β = ρ(S) < µ(Al0) keywords: f(z; µ(al0 cache: ejde-330.pdf plain text: ejde-330.txt item: #298 of 601 id: ejde-331 author: Wang, Lianwen; Mubarak, Abdulrahman title: Monotone solutions of first order nonlinear differential systems date: 2021 words: 5986 flesch: 83 summary: Indeed, if limt→α− x(t) < ∞, it follows from y(t) = y(a) + ∫ t a q(s)g(x(s))ds that lim t→α− y(t) = y(a) + ∫ α a q(t)g(x(t))dt <∞. So (x, y) can be extended to [a, α] and further to a small neighborhood at the right of α. By (H2) both f(x(t)) and g(y(t)) are increasing on [c, α), then y(t) = y(c) + ∫ t c q(s)g(x(s))ds ≤ y(c) + g(x(t)) ∫ t c q(s)ds = g(x(t)) ( y(c) g(x(t)) + ∫ t c q(s)ds ) ≤ g(x(t)) ( y(c) g(x(c)) keywords: solutions; theorem cache: ejde-331.pdf plain text: ejde-331.txt item: #299 of 601 id: ejde-332 author: Ding, Yuanlin; Wang, Jinrong title: Periodic solutions for conformable type non-instantaneous impulsive differential equations date: 2021 words: 7942 flesch: 91 summary: = Qz(ι−k ), ι ∈ (ιk, σk], k = 1, 2, . . = za ∈ Rn, (1.2) and the conformable nonlinear non-instantaneous impulsive differential equation Dσk β z(ι) = Pz(ι) + h(t, z(ι)), ι ∈ (σk, ιk+1], k = 0, 1, 2, . . . keywords: n(a cache: ejde-332.pdf plain text: ejde-332.txt item: #300 of 601 id: ejde-333 author: Liu, Zhiqing; Gao, Cunchen; Fang, Zhong Bo title: Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms date: 2021 words: 8566 flesch: 82 summary: Substituting (3.19)-(3.22) into (3.17), we can derive d dt E (n) 1 (t) + r(1− 3η)‖u(n) tt ‖2Γ2 ≤ 3‖∇u(n)‖Ω1‖∇u (n) t ‖3Ω1 + 3‖∇v(n)‖Ω2‖∇v (n) t ‖3Ω2 − (µ1 − µ2 2 − ζ 2τ )‖u(n) tt ‖2Ω1 − ( ζ 2τ − µ2 2 )‖z(n) t (x, 1, t)‖2Ω1 + r 4η (k′(t))2‖k′′(t− s)‖L1(0,+∞) ∫ t 0 k′′(t− A direct calculation shows that (h ∗ u, ut)Γ2 =− 1 2 d dt [ ∫ Γ2 (h � u)(t)dΓ− (∫ t 0 h(s)ds ) ‖u‖2Γ2 ] − 1 2 h(t)‖u‖2Γ2 + 1 2 ∫ Γ2 (h′ � u)(t)dΓ, (2.1) and ‖(h ◦ u)(t)‖2Γ2 ≤ (∫ t 0 |h(s)|ds )∫ Γ2 (|h| � u)(t)dΓ. (2.2) Differentiating (1.3), we arrive at the following Volterra equation (1 + ‖∇u‖2Ω1 ) keywords: problem; ‖∇u‖2ω1 cache: ejde-333.pdf plain text: ejde-333.txt item: #301 of 601 id: ejde-334 author: hao, Xutong; Zhou, Mingjun; Jing, Xinxin title: Asymptotic behavior of solutions to porous medium equations with boundary degeneracy date: 2021 words: 7208 flesch: 81 summary: = ∫ 1 0 u(x, t)ηδ(x)dx, t ≥ 0. = ∫ 1 0 u(x, t)η(x)dx, t ≥ 0. keywords: problem cache: ejde-334.pdf plain text: ejde-334.txt item: #302 of 601 id: ejde-335 author: Neto, Antonio Francisco title: Extending Putzer's representation to all analytic matrix functions via omega matrix calculus date: 2021 words: 6876 flesch: 76 summary: exp(tA)φ0, where the matrix exponential exp(A) is defined by exp(A) = ∑ k≥0 Ak/k!, (1.2) which is a matrix valued convergent power series for any A ∈ CN×N . = 1 Γ(α) ∫ t 0 (t− s)α−1f(s)ds, t > 0, where Γ(α) is the Euler’s gamma function keywords: calculus; d−1∑; fractional; k=0; lemma; matrix; theorem cache: ejde-335.pdf plain text: ejde-335.txt item: #303 of 601 id: ejde-336 author: aguchi, Dai; Tsuchiya, Takahiro title: Newton-Kantorovitch method for decoupled forward-backward stochastic differential equations date: 2021 words: 5975 flesch: 83 summary: [0, T ]→ Rm×k adapted : ‖Z‖H2 <∞}, where the norms ‖ · ‖L2 T , ‖ · ‖S2m , and ‖ · ‖H2 are defined by ‖Y ‖L2 T = ‖Y ‖S2m = {E[ sup 0≤s≤T |Y (s)|2]}1/2, ‖Z‖H2 = {E[ ∫ T 0 |Z(s)|2 ds]}1/2. For α ∈ R, we introduce the weighted norm ‖(Y,Z)‖2α = E[ sup 0≤s≤T eαs|Y (s)|2] + E[ ∫ T 0 eαs|Z(s)|2 ds]. keywords: t t cache: ejde-336.pdf plain text: ejde-336.txt item: #304 of 601 id: ejde-337 author: Bieske, Thomas; Blackwell, Keller title: Generalizations of the drift Laplace equation in the Heisenberg group and Grushin-type spaces date: 2021 words: 4710 flesch: 84 summary: We begin with R3 using the coordinates (x1, x2, x3) and consider the linearly independent vector fields {X1, X2, X3}, defined by: X1 = ∂ ∂x1 − x2 2 ∂ ∂x3 , X2 = ∂ ∂x2 + x1 2 ∂ ∂x3 , X3 = ∂ ∂x3 which obey the relation [X1, X2] = X3. − a)3n(y2 − b) (α+ β − 1)gα+β−2hα+β−2 (4.7) and 2∑ i=1 Yi‖∇0f‖2(Yif) = 4c3(n+ 1)3(α2 + β2)(y1 − a)3n−1g2α+β−3hα+2β−3 × ( (αh+ βg) ( ngh+ c2(n+ 1)(α+ β − 1)(y1 − a)2n+2 ) + ic(n+ 1)2(y1 − a)n+1(y2 − b)(α+ β − 1)(αh− βg) ) , ‖∇0f‖2(Y1Y1f + Y2Y2f) keywords: equation; group; theorem cache: ejde-337.pdf plain text: ejde-337.txt item: #305 of 601 id: ejde-338 author: Lv, Huilin; Zheng, Shenzhou; Feng, Zhaosheng title: Existence results for nonlinear Schrodinger equations involving the fractional (p,q)-Laplacian and critical nonlinearities date: 2021 words: 10033 flesch: 81 summary: Let vn = un − u and Jε(vn) → d. By Brezis-Lieb Lemma in [11] and [28, Lemma 3.3] we find that |vn| q∗s2 q∗s2 = |un| q∗s2 q∗s2 − |u| q∗s2 q∗s2 + on(1). Indeed, we would like to stress that standard arguments used to investigate the linear case p = q = 2 seem to be inapplicable to the nonlinear case on account of EJDE-2021/100 EXISTENCE RESULTS FOR NONLINEAR SCHRÖDINGER EQUATIONS 3 the lack of Hilbertian structure of W s,p(Rn) for p 6= 2. keywords: lemma; proof; q∗s2 cache: ejde-338.pdf plain text: ejde-338.txt item: #306 of 601 id: ejde-339 author: Bhimani, Divyang G. title: Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaces date: 2021 words: 10091 flesch: 88 summary: = (V ∗ |u|2), u(0) = u0, ut(0) = u1, (1.2) where u(t, x) is a complex valued function of (t, x) ∈ R × Rd, i = √ −1, ut = ∂ ∂t , utt = ∂2 ∂2t , I is the identity operator, ∆ is the Laplace operator, u0 and u1 are complex valued functions of x ∈ Rd, ∗ denotes the convolution in Rd, and V is of the type V (x) = λ |x|γ , λ ∈ R, x ∈ Rd, 0 < γ < d. (1.3) The stationary equation −∆u+(V ∗|u|2)u = ∫ Rd f(w) e2πix·wdw, x ∈ Rd, and keywords: modulation; mp,1; proposition; spaces cache: ejde-339.pdf plain text: ejde-339.txt item: #307 of 601 id: ejde-34 author: Tong, Zhi-Juan; Chen, Jianqing; Wang, Zhi-Qiang title: Non-radial normalized solutions for a nonlinear Schrodinger equation date: 2023 words: 5794 flesch: 83 summary: Choose nm →∞, as m→∞ such that ∣∣bm − ∫ TRm |unm |2dx ∣∣ ≤ 1 m , ∣∣∣bm+1 − ∫ TRm+1 |unm |2dx ∣∣∣ ≤ 1 m . − C0 rn = (‖vn‖22 2b ∫ RN ∣∣√b∇vn ‖ vn‖2 ∣∣2 − ‖vn‖p2 pbp/2 ∫ RN ∣∣√bvn ‖vn‖ 2 ∣∣pdx) + (‖ωn‖22 2b ∫ RN ∣∣√b∇ωn ‖ωn‖2 ∣∣2 − ‖ωn‖p2 pbp/2 ∫ RN ∣∣√bωn ‖ωn‖ 2 ∣∣pdx)− C0 rn ≥ ‖vn‖ 2 2 b S(Rn, b) + (‖vn‖22 pb − ‖vn‖ p 2 pbp/2 )∫ RN ∣∣√bvn ‖vn‖2 ∣∣pdx+ ‖ωn‖22 b S(Rn, b)− C0 rn . keywords: lim; solutions cache: ejde-34.pdf plain text: ejde-34.txt item: #308 of 601 id: ejde-340 author: Xia, Pengcheng; Su, Yu title: p-Laplacian equation with finitely many critical nonlinearities date: 2021 words: 3928 flesch: 90 summary: By Lemma 3.1, there exists tv̄ > 0 such that tv̄ v̄ ∈ N ζ . Introduction We consider the p-Laplacian equation −∆pu− ζ |u|p−2u |x|p = k∑ i=1 ( Iαi ∗ |u| p∗αi ) |u|p ∗ αi −2u+ |u|p ∗−2u, x ∈ RN , (1.1) where N > 3, p ∈ (1, N), ζ ∈ (0,Λ), Λ = (N−pp )p, ∆p := div(|∇u|p−2∇u) is the p-Laplacian, p∗αi = p 2 (N+αi N−p ) are the Hardy-Littlewood-Sobolev critical upper exponents, and the parameters αi satisfy the following assumption: (H1) 0 keywords: d1,p(rn cache: ejde-340.pdf plain text: ejde-340.txt item: #309 of 601 id: ejde-341 author: Zhu, Min; Wang, Ying; Chen, Lei title: Curvature blow-up for the periodic CH-mCH-Novikov equation date: 2021 words: 5914 flesch: 88 summary: Preliminaries To discuss the wave breaking phenomenon of the periodic CH-mCH-Novikov equation (1.1), we rewrite it as ut = −k1G ∗ (2uxm+ umx)− k2G ∗ ((u2 − u2x)m)x − k3G ∗ (u2mx + 3uuxm), t > 0, x ∈ S, u(0, x) = u0(x), x ∈ S, (2.1) where G(x) = cosh(x−[x]− 1 2 ) 2 sinh(1/2) , [x] represents the largest integer part of x, and G(x) is the fundamental solution of (1 − ∂2x)−1 on the unit circle S = R/Z, that is for any x ∈ S. Let G(x) = Λ1(x)+Λ2(x), where Λ1(x) = ex−[x]− 1 2 4 sinh( 1 2 ) and Λ2(x) [Λ1 ∗ (u− ux)3 − Λ2 ∗ (u+ ux)3] − k1[Λ2 ∗ (u2 + 1 2 u2x)− Λ1 ∗ (u2 + 1 2 u2x)], ûx ′ (t) = k1(û2 − 1 2 ûx 2 ) + k2( 1 3 û3 − ûûx2) + k3û 2 (û2 − ûx2) − ( k2 3 + k3 2 ) keywords: equation; û2; ûx cache: ejde-341.pdf plain text: ejde-341.txt item: #310 of 601 id: ejde-342 author: Zhao, Qingjian; Shi, Shaoyun; Li, Wenlei title: Dynamics of flocking models with two species date: 2021 words: 8812 flesch: 77 summary: ∈M. With the assumptions (A1) and (A3), by differentiat- ing H(x, u, y, v) along the solution with the respect of time, we have Ḣ = 1 2 ∑ k 6=l φ(‖xk − xl‖ − d1)〈~e(xk, xl), φij =  φ1(‖qj − qi‖) = ρ1(‖qj − qi‖)φ̂(‖qj − qi‖ − d1), i, j = 1, . . keywords: agents; flocking; groups; interaction; j=1; model; system cache: ejde-342.pdf plain text: ejde-342.txt item: #311 of 601 id: ejde-345 author: Pinelas, Sandra; Tunç, Osman; Korkmaz, Erdal; Tunç, Cemil title: Existence and stabilization for impulsive differential equations of second order with multiple delays date: 2024 words: 6507 flesch: 80 summary: = { ψ(t), t0 ≤ t ≤ t0 − τN y1(t), t0 ≤ t ≤ T is a solution of (1.8), for all t ∈ I. Let I1 = [t1 − τN , T ]. It follows that N∑ i=1 (τi) ≤ t1 − t0 ≤ ¯̀, d̄1 = p̄1 − N∑ i=1 τi(gi + a0σi). keywords: differential; equations; i=1; n∑ i=1; t t0; tunç cache: ejde-345.pdf plain text: ejde-345.txt item: #312 of 601 id: ejde-349 author: Mustafa, Muhammad I. title: Optimal energy decay rates for viscoelastic wave equations with nonlinearity of variable exponent date: 2023 words: 6457 flesch: 83 summary: ≤ − (∫ t 0 g(s)ds− δ )∫ Ω u2 t dx+ δ ∫ Ω |∇u|2 dx+ c[Cα + 1] δ (h ◦ ∇u)(t) + cδ(g ◦ ∇u)(t) + a2 4 ∫ Ω∗ |ut|2m(x)−2 dx+ a ∫ Ω Cδ(x)|ut|m(x) dx. (3.7) Proof. [see (2.4)], then the use of hypothesis (2.2), (3.1), and Jensen’s inequality leads to∫ t 0 g(s) ∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ≤ I(t) I(t) ∫ t 0 H −1 (−g′(s) ξ(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ≤ I(t)H −1 ( 1 I(t) ∫ t 0 (−g′(s) ξ(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ) ≤ I(t)H −1 ( 1 I(t)ξ(t) ∫ t 0 ( − g′(s) )∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ) ≤ I(t)H −1 (−2E′(t) I(t)ξ(t) ) . keywords: 2m1−2; e(t; ∫ t; ∫ ω cache: ejde-349.pdf plain text: ejde-349.txt item: #313 of 601 id: ejde-350 author: Su, Yu; Chen, Haibo; Liu, Senli; Fang, Xianwen title: Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent date: 2020 words: 6949 flesch: 82 summary: = λ 3−2s 2 n un(λnx+ xn) where λn > 0, xn ∈ R3 and xn λn →∞ as n→ +∞, they derived that vn ⇀ v in Ds,2(R3) and∫ R3 vkφ |x+ xk λk |2s → 0 as k → +∞ (1.10) for any φ ∈ Ds,2(R3). In particular the Schrödinger equa- tion for the wave function of an electron interacting with a polar molecule (supposed to be point-like) can be written as H = − ~ 2m ∆ + e x ·D |x|3 − E, where D is the dipole moment of the molecule, e and m denote the charge and the mass of the electron (see [26]). keywords: ds,2; lim; rad(r3; schrödinger; system cache: ejde-350.pdf plain text: ejde-350.txt item: #314 of 601 id: ejde-352 author: Wu, Hong-Jie; Han, Bang-Sheng; Mi, Shao-Yue; Shen, Liang-Bin title: Traveling wave solutions for three-species nonlocal competitive-cooperative systems date: 2023 words: 6722 flesch: 82 summary: + [e−λcx −Ae−(λc+ε)x](Zc1e −λcx + b1e −ζcx + c1e −ηcx) < −Aκce−(λc+ε)x + e−λcx(Zc1e −λcx + b1e −ζcx + c1e −ηcx) = e−(λc+ε)x[−Aκc + Zc1e −(λc−ε)x + b1e −(ζc−ε)x + c1e −(ηc−ε)x] < 0, − d2q′′c − cq ′ c − r2qc + r2qc(φ2 ∗ qc)− b2r2qclc + c2r2qcpc = (−d2ζ2c + cζc − r2)e−ζcx +Be−(ζc+ε)x[d2(ζc + ε)2 − c(ζc + ε) + r2] + r2[e−ζcx −Be−(ζc+ε)x] ( Zc2e −ζcx − b2e−ηcx + b2De −(ηc+ε)x + c2e −λcx ) < e−(ζc+ε)x[−Bιc + r2Z c 2e −(ζc−ε)x + r2b2De −ηcx + r2c2e −(λc−ε)x] < 0, and − d3l′′c − cl ′ c − r3lc + r3lc(φ3 ∗ lc)− b3r3lcqc + c3r3lcpc = (−d3η2c + cηc − r3)e−ηcx +De−(ηc+ε)x[d3(ηc + ε)2 − c(ηc + ε) + r3] + r3[e−ηcx On the other hand, combining with (2.3)-(2.5), it is easy to calculate − p̃′′c − cp̃′c + (φ1 ∗ u0 + b1v0 + c1w0)p̃c ≤ −p̃′′c keywords: lim; species; u(x; v(x; w(x; wave cache: ejde-352.pdf plain text: ejde-352.txt item: #315 of 601 id: ejde-353 author: Li, Yuxin; Chang, Xiaojun; Feng, Zhaosheng title: Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems date: 2023 words: 7999 flesch: 82 summary: For c ∈ (0, c0), I(u) restricted to Λ(c) is coercive on H1(R3). They showed that system (1.3) admits ground state solutions under certain conditions of V and f . keywords: m(c; podolsky; schrödinger; solutions; system cache: ejde-353.pdf plain text: ejde-353.txt item: #316 of 601 id: ejde-354 author: Purushothaman, Ganesh; Suresh, Kannan; Tunc, Ercan; Thandapani, Ethiraju title: Oscillation criteria of fourth-order nonlinear semi-noncanonical neutral differential equations via a canonical transform date: 2023 words: 4489 flesch: 78 summary: [14] N. Kilinc Gecer, P. Temtek; Oscillation criteria for fourth order differential equations, J. Inst. Then (i) if y(t) ∈ S1, then y(t) A1(t) is decreasing for t ≥ t1 for some t1 ≥ t0; (ii) if y(t) ∈ S3, then y(t) Q3(t) is decreasing and L1y(t) ≥ Q2(t)L3y(t) for t ≥ t1 for some t1 ≥ t0. keywords: differential; equations; order; y(t cache: ejde-354.pdf plain text: ejde-354.txt item: #317 of 601 id: ejde-355 author: Musolino, Paolo; Dutko, Martin; Mishuris, Gennady title: Asymptotic analysis of perturbed Robin problems in a planar domain date: 2023 words: 8442 flesch: 75 summary: Boundary value problems with degenerating or perturbed boundary conditions have been analyzed by many authors. Singularly perturbed boundary value problem; Laplace equation; nonlinear Robin condition; perforated planar domain; integral equation. keywords: log; log ε; problem; εδ(ε; ρ(ε; ∂ωi; ∂ωo cache: ejde-355.pdf plain text: ejde-355.txt item: #318 of 601 id: ejde-356 author: Feckan, Michal; Marynets, Kateryna title: Non-local fractional boundary value problems with applications to predator-prey models date: 2023 words: 5830 flesch: 69 summary: Let us fix values of parameters ξ ∈ Dξ and η ∈ Dη and show that {uqm(t, ξ, η) : t ∈ = ∪ξ∈DΞρ(ξ), Ξρ(ξ) := {ξ ∈ Rn : |ξ − z| ≤ ρ}for all ξ. (A3) keywords: t t cache: ejde-356.pdf plain text: ejde-356.txt item: #319 of 601 id: ejde-36 author: Xiao, Qingkun; Gao, Hongjun title: Stochastic attractor bifurcation for the two-dimensional Swift-Hohenberg equation with multiplicative noise date: 2023 words: 9360 flesch: 75 summary: ξ3 − 3 2 ξ33 − 3ξ21ξ3 − 3ξ22ξ3 − 3ξ3ξ 2 4 − 3 2 ξ21ξ4 + 3 2 ξ22ξ4)dt+ σξ3 ◦ dWt, dξ4 = (P ( √ 2π L )ξ4 − 3 2 ξ34 − 3ξ21ξ4 − 3ξ22ξ4 − 3ξ23ξ4 − 3 2 ξ21ξ3 + 3 2 ξ22ξ3)dt+ σξ4 ◦ dWt. ξ1 − 3 2 ξ31 − 3ξ1ξ 2 2 − 3ξ1ξ 2 3 − 3ξ1ξ 2 4)dt+ σξ1 ◦ dWt, (5.8) dξ2 = (P ( √ m2 + n2π L )ξ2 − 3 2 ξ32 − 3ξ21ξ2 − 3ξ2ξ 2 3 − 3ξ2ξ 2 4)dt+ σξ2 ◦ dWt, (5.9) dξ3 = (P ( √ m2 + n2π L )ξ3 − 3 2 ξ33 − 3ξ21ξ3 − 3ξ22ξ3 − 3ξ3ξ 2 4)dt+ σξ3 ◦ dWt, (5.10) dξ4 = (P ( √ m2 + n2π L )ξ4 − 3 2 ξ34 − 3ξ21ξ4 − 3ξ22ξ4 − 3ξ23ξ4)dt+ σξ4 ◦ dWt. keywords: al(θtω; bifurcation; equation; stochastic cache: ejde-36.pdf plain text: ejde-36.txt item: #320 of 601 id: ejde-360 author: Shillor, Meir; Kadhim, Thanaa Ali title: Analysis and simulations of the HANDY model with social mobility, renewables and nonrenewables date: 2023 words: 9277 flesch: 75 summary: − γc, J12 = γe, J13 = 0, J14 = 0, J15 = 0. J21 = γc, J22 = βe − αm − γe, J23 = 0, J24 = 0, J25 = 0. Then, Cc = ηsxc, Ce = ηκ1sxe, and αc = αM − η(αM − αm), αe = αM − κ1η(αM keywords: case; model; states; system; wealth cache: ejde-360.pdf plain text: ejde-360.txt item: #321 of 601 id: ejde-361 author: Liu, Mengqian; Wu, Zhigang title: Space-time behavior for radiative hydrodynamics model with or without heat conduction date: 2023 words: 9338 flesch: 78 summary: − ν2g1 − ν1g3. + · · · , Ĝ12 = −i R(κρ̄ + 4w̄3 C3 v ρ̄ 4 ) c2Cv(Cv +R) ξT eλ1t − 1 2c ξT |ξ| (eλ2t − eλ3t) + · · · , Ĝ13 = − R c2Cv eλ1t + R 2c2Cv (eλ2t + eλ3t) + · · · , Ĝ21 = −i (κρ̄ + 4w̄3 C3 v ρ̄ 4 ) Cv +R ξeλ1t keywords: 4w̄3; cv(2µ+; function; green; pointwise; |ξ|2 cache: ejde-361.pdf plain text: ejde-361.txt item: #322 of 601 id: ejde-362 author: Drissi, Amor; Ghanmi, Abdeljabbar; Repovs, Dusan D. title: Singular p-biharmonic problems involving the Hardy-Sobolev exponent date: 2023 words: 4748 flesch: 81 summary: dx− 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α) ≥ 1 p ‖ϕ‖p − µ r c1‖ϕ‖rr,f − 1 p∗(α) S − p ∗(α) p p∗ ‖ϕ‖p ∗(α) ≥ 1 p ‖ϕ‖p − µ r c1‖f‖ p∗ p∗−r S −r/p p∗ ‖ϕ‖r − 1 p∗(α) dx − p∗(α) p∗(α) ∫ RN |x|−αϕp ∗(α) keywords: p∗(α cache: ejde-362.pdf plain text: ejde-362.txt item: #323 of 601 id: ejde-363 author: Alsaedi, Ahmed; Ahmad, Bashir; Kirane, Mokhtar; Nabti, Aberrazak title: Lifespan of solutions of a fractional evolution equation with higher order diffusion on the Heisenberg group date: 2020 words: 3847 flesch: 81 summary: A function u is called a local weak solution of (1.1)– (1.2), if u ∈ C([0, T );Lploc(R2N+1)) and satisfies λ ∫ T 0 ∫ R2N+1 Iα0|t|u| pφ(η, t) dη [13] considered the equation i∂tu+ ∆u = λ Γ(α) ∫ t 0 (t− s)α−1|u(s)|p ds, x ∈ RN , t > 0, (1.4) with u(x, 0) = f(x), f ∈ L1(RN ) and proved that if 1 < p ≤ 1+2(α+1)/(N−2α)+, λ ∈ C\{0}, λ1 > 0 and ∫ RN f2(x) dx < 0, then equation (1.4) has no global weak solutions. keywords: r2n+1 cache: ejde-363.pdf plain text: ejde-363.txt item: #324 of 601 id: ejde-364 author:  Allahverdiev, Bilender P.; Tuna, Huseyin title: Properties of the resolvent of singular q-Dirac operators date: 2020 words: 4318 flesch: 83 summary: For each non-real number λ, we have χq−n(x, λ)→ χ(x, λ) and∫ q−n 0 ‖χq−n(x, λ)‖2Edqx→ ∫ ∞ 0 ‖χ(x, λ)‖2Edqx, n→∞. 4 B. P. ALLAHVERDIEV, H. TUNA EJDE-2020/03 Putting Gq−n(x, t, λ) = { χq−n(x, λ)ϕT (t, λ), t ≤ x ϕ(x, λ)χTq−n(t, λ), t > x ( [χq−n1(x, λ)ϕ1(t, λ) χq−n1(x, λ)ϕ2(t, λ) χq−n2(x, λ)ϕ1(t, λ) χq−n2(x, λ)ϕ2(t, λ) ) , t ≤ x( ϕ1(x, λ)χq−n1(t, λ) ϕ1(x, λ)χq−n2(t, λ) ϕ2(x, λ)χq−n1(t, λ) ϕ2(x, λ)χq−n2(t, λ) ) , x < t, (3.6) we have (Rq−nf)(x, λ) = y(x, λ) = ∫ q−n 0 Gq−n(x, t, λ)f(t)dqt, λ ∈ C, (3.7) where y(x, λ) = ( y1(x, λ) y2(x, λ) ) and f(·) = ( f1(·) f2(·) ) ∈ H. λ µ Im{m(σ + iτ)}dσ, z = σ + iτ, τ > 0. (5.2) Proof. keywords: function; q−n; resolvent cache: ejde-364.pdf plain text: ejde-364.txt item: #325 of 601 id: ejde-365 author: Liu, Xiang; Jia, Baoguo; Gensler,  Scott; Erbe, Lynn; Peterson, Allan title: Convergence of approximate solutions to nonlinear Caputo nabla fractional difference equations with boundary conditions date: 2020 words: 7482 flesch: 86 summary: = 0 for t ∈ Na+1, where χ[a,∞)(t) = { 1, t ∈ Na, 0, t /∈ Then the Green function for the BVP (Lax)(t) = 0, t ∈ Nb−1 a+1, x(a) = 0, x(b) keywords: a+1; nb−1 cache: ejde-365.pdf plain text: ejde-365.txt item: #326 of 601 id: ejde-368 author: Yao, Shuai; Sun,  Juntao; Wu, Tsung-Fang title: Stationary quantum Zakharov systems involving a higher competing perturbation date: 2020 words: 6442 flesch: 85 summary: SYSTEMS 9 < p− 2 4p ‖u‖2λ for u ∈ N (2) λ,µ, and so h′′u(1) Hence, if u ∈ Xλ is a critical point of Iλ,µ, then (u, φK,u) is a solution of system (1.4). keywords: k(x)φk; λ,µ cache: ejde-368.pdf plain text: ejde-368.txt item: #327 of 601 id: ejde-37 author: Lv, Jiaojiao; Wang, Jinrong; Liu, Rui title: Hyers-Ulam stability of linear quaternion-valued differential equations date: 2023 words: 5794 flesch: 77 summary: [28] X. Zhang; Global structure of quaternion polynomial differential equations, Communications in Mathematical Physics, 303 (2011), 301–316. [4, 5] studied the Hyers-Ulam stability of first-order matrix differential equations by using the norm estimation of exponential functions of quaternion matrices and derived the Hyers-Ulam stability of linear quaternion- valued differential equations by using the Laplace transform. keywords: hyers; quaternion; stability; ulam cache: ejde-37.pdf plain text: ejde-37.txt item: #328 of 601 id: ejde-371 author: Le Trong Thanh, Bui; Ngoc Quoc Thuong, Nguyen title: Passing to the limit on small parameters for generalized viscous Cahn-Hilliard type equations with nonlinear source date: 2020 words: 5768 flesch: 80 summary: + ε 2 |∇uε|2dx+ ∫ t 0 ∫ Ω |∇vε|2dx ≤ C1 ∫ t 0 ∫ Ω f2(uε)dx+ C2. 12 B. L. T. THANH, N. N. Q. THUONG EJDE-2020/07 Now using assumption (H4),∫ Ω Φ(uε) (2.24) Integrate (2.24) over (0, t) with t ≤ T ∗; then thanks to (2.21) we have t‖v‖2−1 + δt‖v‖2 ≤ c ∫ t 0 (s‖v‖2 + δs‖v‖2)ds+ ∫ t 0 ‖v‖2−1 + δ‖v‖2ds (2.25) ≤ c ∫ t 0 (s‖v‖2 + δs‖v‖2)ds+Q(‖u0‖H2). keywords: equation; problem cache: ejde-371.pdf plain text: ejde-371.txt item: #329 of 601 id: ejde-376 author: Kijowski, Antoni title: Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure date: 2020 words: 12152 flesch: 79 summary: Then for each K b Ω∥∥∆hf |h| ∥∥ Lp(K) ≤ C‖∇f‖Lp(Ω), for some constant C > 0 and all h ∈ Rn, 0 < 2|h| < dist(K, ∂Ω). (2) Suppose that 1 < p < ∞, K b Ω, function f ∈ Lp(K) and there exists constant C > 0 Harmonic function; mean value property; metric measure space; Minkowski functional; norm induced metric; Pizzetti formula; weighted Lebesgue measure. keywords: functions; harmonic; mean; theorem; value cache: ejde-376.pdf plain text: ejde-376.txt item: #330 of 601 id: ejde-379 author: Zhao, Junfang; Liu, Xiangqing title: Ground state solutions for quasilinear equations of Kirchhoff type date: 2020 words: 5928 flesch: 83 summary: Given u, ϕ ∈ X with the property that ∫ Ω u2|∇ϕ|2 dx < +∞ and ∫ Ω |∇u|2ϕ2 dx < +∞, for example ϕ ∈ C∞0 (Ω), ϕ = u, u+ or u−, where u+ = max{u, 0}, u− = min{u, 0}, we can define the derivative of I in the direction ϕ at u, denoted by 〈DI(u), ϕ〉 as 〈DI(u), ϕ〉 = lim t→0+ 1 t (I(u+ tϕ)− I(u)) For u ∈ X, define γ+(u) = 〈DI(u), u+〉 = ∫ Ω (a|∇u+|2 + 2bu2 +|∇u+|2) dx + ∫ Ω (c|∇u|2 + du2|∇u|2) dx ∫ Ω (c|∇u+|2 + 2du2 +|∇u+|2) dx − ∫ Ω f(u+)u+ dx , γ−(u) = 〈DI(u), u−〉 = ∫ Ω (a|∇u−|2 + 2bu2 −|∇u−|2) dx + ∫ Ω (c|∇u|2 + du2|∇u|2) dx ∫ Ω (c|∇u−|2 + 2du2 −|∇u−|2) dx − ∫ Ω f(u−)u− dx (1.9) and S∗ = {u : u ∈ X, γ+(u) = 0, u+ 6= 0; γ−(u) = 0, u− 6= 0}, c∗ = inf u∈S∗ I(u) . keywords: kirchhoff; solutions; ∫ ω cache: ejde-379.pdf plain text: ejde-379.txt item: #331 of 601 id: ejde-38 author: Bansil, Mohit; Kitagawa, Jun title: An optimal transport problem with storage fees date: 2023 words: 11780 flesch: 76 summary: Since F ∗∗ ≤ F , we also have mF∗∗ < ∞. Then by strong duality combined with Proposition 5.1 below, we see (since F ∗ = F ∗∗∗) 0 By [9, Corollary 13.3.3] and since F ∗∗∗ = F ∗, we see that F ∗∗ also has Lipschitz constant L, then a calculation similar to (6.2) with F ∗∗ replacing F shows that the EJDE-2023/22 AN OPTIMAL TRANSPORT PROBLEM 23 pair (T, λ∞) minimizes (1.2) with storage fee function F ∗∗. Thus, by the strong duality Theorem 3.3 we have mF −MF = mF −mF∗∗ = F (λ∞)− F ∗∗(λ∞) keywords: function; problem; theorem; x×y; − ∫ cache: ejde-38.pdf plain text: ejde-38.txt item: #332 of 601 id: ejde-380 author: Silva, Kaye; Moreno Sousa, Steffanio title: Multiplicity of positive solutions for a gradient type cooperative/competitive elliptic system date: 2020 words: 6255 flesch: 86 summary: For λ, µ ∈ R and w ∈ X we introduce Hλ,µ(w) Moreover, ψλ,µ,w is decreasing; We start with the study of N+ λ,µ. Observe from Proposition 2.2 that if N+ λ,µ 6= ∅ then there exist (λ, µ) ∈ R2 and w ∈ X such that Hλ(w) < 0 or equivalently∫ |∇u|2 + ∫ |∇v|2 − µ ( ∫ |u|2 + ∫ |v|2 ) 2 ∫ uv < λ, therefore we are led to the study of the function λmin(µ;w) := ‖w‖2 − µ‖w‖22 2 ∫ uv , w ∈ X , ∫ uv > 0. (2.1) keywords: λ,µ cache: ejde-380.pdf plain text: ejde-380.txt item: #333 of 601 id: ejde-381 author: Milla Miranda, Manuel; Medeiros, Luiz A.; Louredo, Aldo T. title: Global solutions to a quasilinear hyperbolic equation date: 2020 words: 4924 flesch: 87 summary: = ∫ t 0 u′m(τ)dτ + u0, we obtain that (um) is bounded in L∞loc(0,∞;H1 Γ0 (Ω)). We obtain 1 2 d dt |u′m|2 + n∑ i=1 ( σ′i (∂um ∂xi )∂u′m ∂xi , ∂u′′m ∂xi ) + ‖u′′m‖2 + 1 2 d dt |u′′m|2L2(Γ1) = 0. (4.4) keywords: i=1; ∂xi cache: ejde-381.pdf plain text: ejde-381.txt item: #334 of 601 id: ejde-382 author: Yang, Jie; Chen, Haibo; Feng, Zhaosheng title: Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities date: 2020 words: 8961 flesch: 88 summary: = t2m − Ātq − B̄t2 ∗ s . Iλ(t+(u1 − u)(u1 − u)) ≤ Iλ(u1 − u), which implies that u1 is a local minimizer of Iλ in E0. keywords: lemma; problem cache: ejde-382.pdf plain text: ejde-382.txt item: #335 of 601 id: ejde-383 author: Chen, Qing; Wu, Guochun; Zhang, Yinghui; Zou,  Lan title: Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential date: 2020 words: 8512 flesch: 83 summary: Furthermore, we have from (2.3) that for k = l − 1, 〈∇l−1N2,∇l−1u〉 = 〈∇l−1 ( − u · ∇u ) ,∇l−1u〉+ 〈 ∇l−1 ( − (P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l−1u 〉 + 〈 ∇l−1 (µ ρ %∇u ) ,∇lu 〉 + 〈 ∇l−1 ( ∇ (µ ρ % ) · ∇u ) ,∇l−1u 〉 + 〈 ∇l−1 (µ+ ν ρ %div u ) ,∇l−1 div u 〉 + 〈 ∇l−1 ( ∇ (µ+ ν ρ % ) div u ) ,∇l−1u 〉 . δ ( ‖∇l%‖2L2 + ‖∇lu‖2L2 ) , (4.26) EJDE-2020/102 TRAVELING WAVES FOR A CHEMOTAXIS MODEL 15 where (6.9) is used, and for k = l, 〈∇lN2,∇lu〉 = 〈∇l(−u · ∇u),∇lu〉+ 〈 ∇l−1 ((P ′(ρ) ρ − P ′(1) ) ∇% ) ,∇l div u 〉 + 〈 ∇l−1 (µ ρ %∆u ) ,∇l div u 〉 + 〈 ∇l−1 (µ+ ν ρ %∇div u ) ,∇l div u 〉 . keywords: decay; proposition; solution; system; |ξ|; ′(1 cache: ejde-383.pdf plain text: ejde-383.txt item: #336 of 601 id: ejde-385 author: Li, Min title: Time periodic solutions for the non-isentropic compressible quantum hydrodynamic equations with viscosity in R^3 date: 2020 words: 12759 flesch: 78 summary: − ‖u‖2L2 − 2 3 ‖ρ‖2L2 − 3 2 ‖s‖2L2 − 5ε 3 ‖∇ρ‖2L2 − µ‖∇u‖2L2 − κ‖∇s‖2L2 − µ 3 ‖div u‖2L2 + δ‖u‖2L2 + Cτ ( ‖(u, ρ, s)‖L∞ + ‖(ρ, s,∇ρ)‖2L∞ ) ‖(u,∇u,∇s,∇ρ)‖2L2 + Cτ‖fR‖2L2 + Cε‖∇s‖2L2 + C~4‖∇ div u‖2L2 ≤ 6∑ i=1 R2,i − 4κ 3 ∫ ΩR ∇s · ∇ρ− 2 ∫ ΩR ρs− 4κ 9 ‖∇ρ‖2L2 − ‖u‖2L2 − 2 3 ‖ρ‖2L2 − 3 2 ‖s‖2L2 − 5ε 3 ‖∇ρ‖2L2 keywords: = −; div; estimates; proof; quantum; solutions; system; time; ~2τ; τρ)2; − ∫; ∫ ωr cache: ejde-385.pdf plain text: ejde-385.txt item: #337 of 601 id: ejde-388 author: Feng, Zaichun; Li, Y. Charles title: Short term unpredictability of high Reynolds number turbulence - rough dependence on initial data date: 2020 words: 7988 flesch: 62 summary: Since the perturbation equations are linear, such perturbation solutions generated from single Fourier modes form a base of superposition. Now we choose more general initial perturbations to the base solution initial condition (5.12)-(5.13) as follows du1(0) keywords: base; equations; perturbations; solutions; turbulence cache: ejde-388.pdf plain text: ejde-388.txt item: #338 of 601 id: ejde-389 author: Kong, Huihui; Lian, Ruxu title: Free boundary value problem for compressible magnetohydrodynamic equations date: 2020 words: 5310 flesch: 84 summary: From (4.15)2 we find that d dτ ∫ 1 0 u(ξ, τ)dξ = 0, (4.17) and without loss of generality, we can renormalize ∫ 1 0 u0(ξ)dξ to be zero, then we denote w = u− 1 1 + τ ∫ ξ 0 1 ρ dζ + 1 1 + τ ∫ 1 0 ∫ ξ 0 1 ρ dζdξ. dx+ ∫ b(t) a(t) (2µ+ ρβ)u2xdx + ν ∫ b(t) a(t) H2 xdx = 0, (3.2) which leads to (3.1) after the integrating with respect to t ∈ keywords: a(t; ∫ b(t; ∫ t cache: ejde-389.pdf plain text: ejde-389.txt item: #339 of 601 id: ejde-39 author: Il'yasov, Yavdat; da Silva, Edcarlos Domingos; da Silva, Maxwell Lizete title: Prescribed energy saddle-point solutions of nonlinear indefinite problems date: 2023 words: 5440 flesch: 78 summary: Then u = (u+ + u−) ∈ W, u± ∈ W±, and c0‖u‖21 ≤ ‖u‖2W ≤ c1‖u‖21, ∀u ∈ W , where 0 < c0, c1 < +∞ do not depend on u ∈ W . This by the Sobolev inequalities implies∫ G(x, u)dx ≤ ε 2 C1‖u‖21 + C2(ε)‖u‖γ1 , u ∈W, (3.1) 6 Y. IL’YASOV, E. D. SILVA, M. L. SILVA EJDE-2023/23 where C1, C2(ε) ∈ (0,+∞) do not depend on u ∈ W and C1 does not depend on ε > 0. keywords: energy; g(x; nonlinear; solutions cache: ejde-39.pdf plain text: ejde-39.txt item: #340 of 601 id: ejde-390 author: Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca title: Positive and nodal solutions for nonlinear nonhomogeneous parametric Neumann problems date: 2020 words: 8954 flesch: 82 summary: If ũλ 6= ûλ, then we can find z0 ∈ Ω such that ûλ(z0) < ũλ(z0) implies ûλ(z0) < ûλn(z0) for all n ≥ n0 (see (3.52)). By Proposition 3.2, we can find uθ ∈ Sθ ⊆ D+, u0 ∈ Sλ ⊆ D+ and uη ∈ Sη ⊆ D+ such that uθ − u0 ∈ int Ĉ+ and u0 − uη ∈ int Ĉ+, ⇒ u0 ∈ intC1(Ω)[uη, uθ]. keywords: 1,p(ω; a.a; f(z cache: ejde-390.pdf plain text: ejde-390.txt item: #341 of 601 id: ejde-392 author: Tian, Hong; Zheng,  Shenzhou title: Orlicz estimates for general parabolic obstacle problems with p(t,x)-growth in Reifenberg domains date: 2020 words: 10906 flesch: 79 summary: Therefore, from (3.20) it follows that − ∫ K5 zi |Dw|2p(t,x)−pi dx dt ≤ − ∫ K5 zi |Dw|p(t,x)(1+ω(Γ(48χρi) α)) By Lemma 3.5 it holds − ∫ Kκ 4r(z) |Dw|p(t,x) dx dt ≤ c2κ with c2 > 1. keywords: p(t; parabolic cache: ejde-392.pdf plain text: ejde-392.txt item: #342 of 601 id: ejde-394 author: Fang, Yue; Li, Kaiqiang; Xu, Xin title: Global classical solutions to equatorial shallow-water equations date: 2023 words: 5895 flesch: 77 summary: + 1 8 ‖ωy‖2. (3.47) Similar derivations show that∫ Ω ωy∇ · U dx ≤ ‖ωy‖‖∇ · U‖ ≤ ‖ωy‖‖U‖1 ≤ C3 ( W (t)3/2 + E(t) + ‖ω‖2 ) + 1 8 ‖ωy‖2. Since ∫ Ω ωttU · ∇ωtt dx = ∫ Ω ∇ · (ω 2 tt 2 U)− ω2 tt 2 ∇ · U dx = − ∫ Ω ω2 tt 2 ∇ · U dx ≤ C3W (t)3/2, (3.56) and∫ Ω yωtt∇ · keywords: t)3/2 cache: ejde-394.pdf plain text: ejde-394.txt item: #343 of 601 id: ejde-395 author: Yan Xu, Hong; Tu, Jin title: Existence of rational solutions for q-difference Painleve equations date: 2020 words: 6064 flesch: 83 summary: [14] R. Korhonen, Z. T. Wen; Existence of zero-order meromorphic solutions in detecting q- difference Painlevé equations, Trans. [19] Z. T. Wen; Meromorphic solutions to difference Painlevé equations I and II, Electronic J. Diff. keywords: difference; equations; f(z cache: ejde-395.pdf plain text: ejde-395.txt item: #344 of 601 id: ejde-396 author: Cao Labora, Daniel; Rodriguez-Lopez, Rosana; Belmekki,  Mohammed title: Existence of solutions to nonlocal boundary value problems for fractional differential equations with impulses date: 2020 words: 6197 flesch: 85 summary: =  B [ − α0(ξ0 − s)δ−1(δ − β(1− t1)) −α1β(1− t1)1−δ(ξ1 − t1)δ−1(t1 − s)δ ] +Bβ(t1 − s)δ, 0 ≤ s ≤ ξ0, B[−α1β(1− t1)1−δ(ξ1 − t1)δ−1(t1 − s)δ + β(t1 − s)δ], 0 ≤ ξ0 ≤ s ≤ t1, −Bα1(ξ1 − s)δ−1(δ − β(1− t1)) − t1)δ−1(t1 − s)δ + β(t1 − s)δ] if 0 ≤ ξ0 ≤ t < s ≤ t1 −Cα1(ξ1 − s)δ−1(δ − β(1− t1)) keywords: β(1− t1; − β(1− cache: ejde-396.pdf plain text: ejde-396.txt item: #345 of 601 id: ejde-397 author: Kostic, Marko title: Abstract degenerate Volterra inclusions in locally convex spaces date: 2023 words: 31503 flesch: 73 summary: Then the mapping t 7→ R(t), t ∈ (0, τ) is infinitely differentiable in L(X) and, for every compact set K ⊆ (0, τ), there exists hK > 0 such that the set {h n K dn dtnR(t) Mn : t ∈ K, n ∈ N0} is equicontinuous. ∈ A, t ∈ keywords: abstract; degenerate; equations; f ∈; families; family; following; function; k ∈; linear; mlo; n ∈; operator; p ∈; resolvent; solution; subgenerator; t ∈; theorem; y ∈; λ ∈; ∈ c; ∈ c([0,∞; ∈ d(a; ∈ l(x; ∈ l1; ∈ ω cache: ejde-397.pdf plain text: ejde-397.txt item: #346 of 601 id: ejde-398 author: Dou, Xuechao; Sun, Juntao title: Local well-posedness and standing waves with prescribed mass for Schrodinger-Poisson systems with a logarithmic potential in R^2 date: 2023 words: 4753 flesch: 84 summary: Inspired by [24], Cingolani and Weth [11] developed a variational framework of ((1.3) with W (x) ≡ 0 in the smaller Hilbert space X := { u ∈ H1(R2) : ∫ R2 ln(1 + |x|)u2dx <∞ } , endowed with the norm ‖u‖2X := ∫ R2 (|∇u|2 + u2(1 + ln(1 + |x|2))) dx dy − ∫ R2 F (u) dx. (1.5) under the constraint S(c) := { u ∈ H : ∫ R2 u2dx = c } , where H := { u ∈ H1(R2) : ∫ R2 ln(1 + |x|2)u2dx <∞ } , endowed with the norm ‖u‖H := ‖u‖H1 + ‖u‖∗, here ‖u‖2∗ = ∫ R2 ln(1 + |x|2)u2dx. keywords: poisson; schrödinger cache: ejde-398.pdf plain text: ejde-398.txt item: #347 of 601 id: ejde-4 author: Zhang, Mengqing; Tian, Jing; Zou, Keyue title: Asymptotic stability of a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps date: 2023 words: 6444 flesch: 83 summary: t 0 E sup s∈[0,t] |X(r)− x(r)|2ds + ((2λ1 + 1)L2 1 + 2ρ2 1 + 1)E ∫ t 0 |Ψ(s)− ψ̄(s)|2ds + E sup s∈[0,t] ∫ t 0 2(X(s)− x(s), (G1x(Ψ(s))−G1x(ψ̄(s)))dw(s)) + 2E sup s∈[0,t] ∫ t 0 (X(s)− x(s), (J1x(Ψ(s))− J1x(ψ̄(s)))dÑ(s)). (4.1) By the BDG inequality, we have E sup s∈[0,t] ∫ = − ∫ t tk ∂x(s) ∂a ds+ ∫ t tk [H1x +H2x]ds+ ∫ t tk G1xdw(s) + ∫ t tk J1xdN(s), thus |x(t)− x̄(t)|2 ≤ 4| ∫ t tk ∂x(s) ∂a ds|2 + 4| ∫ t tk [H1x +H2x]ds|2 + 4| ∫ t tk G1xdw(s)|2 + 4| ∫ t tk J1xdN(s)|2 ≤ 4∆ ∫ t tk |∂x(s) ∂a |2ds+ 4∆ ∫ t tk |H1x +H2x|2ds+ 4| ∫ t tk G1xdw(s)|2 + 8|λ1 ∫ t tk J1xds|2 ≤ 4∆ ∫ t tk |∂x(s) ∂a |2ds+ 8∆[ ∫ t tk (|H2 1x|+ |H2 2x|)ds] + 4| ∫ t tk G1xdw(s)|2 + 8| ∫ t tk J1xdN̄(s)|2 + 8|λ1 ∫ t tk J1xds|2. by Lemma 3.1, E sup t∈[0,T ] |x(t)− x̄(t)|2 ≤ 5E sup t∈[0,T ] max k=0,1,...,N−1 ∣∣ ∫ t tk G1x(ψ)dw(s) ∣∣2 + 8E sup t∈[0,T ] max k=0,1,...,N−1 ∣∣ ∫ t tk J1x(ψ)dÑ(s) ∣∣2 + 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆. 12 M. ZHANG, J. TIAN, K. ZOU EJDE-2023/02 According to the Doob inequality, we obtain E sup t∈[0,T ] |x(t)− x̄(t)|2 ≤ 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆ + 5 max k=0,1,...,N−1 ∫ (k+1)∆ k∆ E|G1x(x, y)|2ds + 8λ1 max k=0,1,...,N−1 ∫ (k+1)∆ k∆ E|J1x(x, y)|2ds ≤ 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆ + 5L2 1C∆ + 8λ1L 2 1C∆. (3.7) keywords: solution; sup; system; t tk; x(t cache: ejde-4.pdf plain text: ejde-4.txt item: #348 of 601 id: ejde-40 author: Chicone, Carmen; Swanson, Richard title: Linearization via the Lie derivative date: 2000 words: 6752 flesch: 65 summary: = 0, • the partial derivatives Fx and Fy are Lipschitz in Ω, and • the partial derivative Fz is Lipschitz in Ωxy uniformly with respect to z ∈ Ωz and Hölder in Ωz uniformly with respect to (x, y) ∈ Ωxy with Hölder exponent µ. System (3.6) satisfies the (1, µ) spectral gap condition if (1 + µ)c < b. We will show that system (3.6) can be linearized by a C1 near-identity trans- formation of the form u = x+ α(x, y, z), v = y + β(x, y, z), w = z. (3.8) The proof of this result is given in three main steps: an invariant manifold theorem for a system with a spectral gap is used to find a preliminary near- identity C1 map, as in display (3.8), that transforms system (3.6) into a system of the same form but with the new function F = (f, g) “flattened” along the coordinate subspace corresponding to the invariant manifold. Equivalently, the identity Dγ(z)Cz −Aγ(z) = F (γ(z), z) (3.12) holds for all z in the domain of γ. keywords: linearization; origin; theorem; vector cache: ejde-40.pdf plain text: ejde-40.txt item: #349 of 601 id: ejde-401 author: Peng, Peng; Wang,  Jinrong; O'Regan, Donal title: Periodicity of non-homogeneous trajectories for non-instantaneous impulsive heat equations date: 2020 words: 2331 flesch: 81 summary: [8] J. Wang, M. Fečkan; Non-instantaneous impulsive differential equations, IOP Publishing, 2018. For any s ∈ I and t ∈ R+, we have ‖G(t, s)‖ ≤ (βγ)r(s,t), where β = supi≥1 supt∈(ti,si] ‖Bi(t)‖ and γ = supi≥1 ‖E + Ii‖. Proof. keywords: g(t cache: ejde-401.pdf plain text: ejde-401.txt item: #350 of 601 id: ejde-402 author: Gasull, Armengol; Torregrosa,  Joan; Zhang, Xiang title: Piecewise linear differential systems with an algebraic line of separation date: 2020 words: 6202 flesch: 79 summary: − y = 0. (3.4) − y = 0, (3.5) that passes trough the origin. keywords: curve; cycles; differential; limit; linear; piecewise; system cache: ejde-402.pdf plain text: ejde-402.txt item: #351 of 601 id: ejde-403 author: Matveeva, Inessa I. title: Exponential stability of solutions to nonlinear time-varying delay systems of neutral type equations with periodic coefficients date: 2020 words: 4484 flesch: 79 summary: Taking into account that y(t) satisfies (1.1), we have d dt V (t, y) = 〈 d dt H(t)y(t), y(t) 〉 + 〈 H(t)z(t), y(t) 〉 + 〈 H(t)F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) ) , y(t) 〉 + 〈 H(t)y(t), z(t) 〉 + 〈 H(t)y(t), F ( t, y(t), y(t− τ(t)), d dt y(t− τ(t)) )〉 + 〈K(0)y(t), y(t)〉 − ( 1− d dt τ(t) ) 〈K(τ(t))y(t− τ(t)), y(t− τ(t))〉 6 I. I. MATVEEVA EJDE-2020/20 + ∫ t t−τ(t) 〈 d dt K(t− s)y(s), y(s) 〉 ds+ 〈 L(0)z(t), z(t) 〉 + 〈 L(0)F ( t, y(t), y(t− τ(t)), EJDE-2020/20 EXPONENTIAL STABILITY OF SOLUTIONS 7 Consider the group of the summands containing F ( t, y(t), y(t − τ(t)), ddty(t − τ(t)) ) and denote them by W (t). keywords: y(t−; y(t− τ(t; τ(t cache: ejde-403.pdf plain text: ejde-403.txt item: #352 of 601 id: ejde-404 author: Cuesta, Mabel; Leadi, Liamidi; Nshimirimana, Pascaline title: Maximum and antimaximum principles for the p-Laplacian with weighted Steklov boundary conditions date: 2020 words: 7037 flesch: 79 summary: = e+ e−1 e− e−1 = β = λ̂1(m). = − inf { ‖u‖p1,p; I(u) = −1 and u ∈ Q } , (4.2) where Q := { u ∈W 1,p(Ω);∃B(x0, r) s.t u|B(x0,r)∩Ω ≡ 0 keywords: λ1(m; λ̂1(m cache: ejde-404.pdf plain text: ejde-404.txt item: #353 of 601 id: ejde-407 author: Jia, Jiwei; Ding,  Jian; Liu, Siyu; Liao, Guidong; Li, Jingzhi; Duan, Ben; Wang, Guoqing; Zhang, Ran title: Modeling the control of COVID-19: impact of policy interventions and meteorological factors date: 2020 words: 9860 flesch: 64 summary: For this reason alone, collecting massive data re- lating to COVID-19 and analyzing the inherent linkage among these data are of great importance for the next step of control strategy. Simulations for most provinces can conduce to understand the effect of control strategy in China. keywords: china; control; covid-19; data; days; disease; hubei; isolation; model; peak; period; province; sars; strategy; transmission cache: ejde-407.pdf plain text: ejde-407.txt item: #354 of 601 id: ejde-408 author: Geba, Dan-Andrei; Lin, Bai title: Almost optimal local well-posedness for modified Boussinesq equations date: 2020 words: 3918 flesch: 81 summary: Fx, (2.1) where t ∈ R is arbitrary, yet fixed. p∏ j=1 ‖(vj0, v j 1)‖Hs×Hs would hold uniformly for t ∈ keywords: equation cache: ejde-408.pdf plain text: ejde-408.txt item: #355 of 601 id: ejde-41 author: Mo, Yichun; Zhu, Min; Feng, Binhua title: Blow-up criteria and instability of standing waves for the fractional Schrodinger Poisson equation date: 2023 words: 9267 flesch: 88 summary: C −D = d(ω), (4s+ 2r − 3)A+ (2s+ 2r − 3)B + (4s+ 2r − 3)C − ((s+ r)(p+ 2)− 3)D = 0, 2(µ(4s+ 2r − 3) + Then∫ R3 (|x|−(3−2r) ∗ |un|2)|un|2dx = ∫ R3 (|x|−(3−2r) ∗ |un − u|2)|un − u|2dx + ∫ R3 (|x|−(3−2r) ∗ |u|2)|u|2dx+ ◦(1). keywords: lemma; standing; waves cache: ejde-41.pdf plain text: ejde-41.txt item: #356 of 601 id: ejde-411 author: dos Santos, Gelson C. G.; Figueiredo, Giovany M.; Tavares,  Leandro S. title: Sub-super solution method for nonlocal systems involving the p(x)-Laplacian operator date: 2020 words: 8639 flesch: 84 summary: Using Hölder’s inequality we have∣∣ ∫ Ω 〈 |∇uin|pi(x)−2∇uin − |∇u|pi(x)−2∇ui,∇(uin − u) 〉∣∣ ≤ |uin Since λn → λ and Hi(T1z 1 n, T2z 2 n)→ Hi(T1z 1, T2z 2) in Lp ′ i(x)(Ω) for i = 1, 2 we have∣∣ ∫ Ω 〈 |∇uin|pi(x)−2∇uin − |∇u|pi(x)−2∇ui,∇(uin − u) 〉∣∣→ 0. keywords: a(x; theorem cache: ejde-411.pdf plain text: ejde-411.txt item: #357 of 601 id: ejde-412 author: Louis-Rose, Carole title: Null controllability from the exterior of fractional parabolic-elliptic coupled systems date: 2020 words: 6645 flesch: 85 summary: Integrating over (0, 1)× (0, T ), we obtain∫ 1 0 ∫ T 0 (∂tu+ (−d2 x)su)ϕdx dt = ∫ 1 0 ∫ T 0 (au+ bv)ϕdx dt,∫ 1 0 ∫ T 0 σ(−d2 x)sv dx dt = ∫ 1 0 ∫ T 0 (cu+ dv)σ dx dt. ∫ 1 0 ∫ T 0 u∂tϕdx dt + c1,s 2 ∫ R2 ∫ T 0 (u(x)− u(y))(ϕ(x)− ϕ(y)) |x− y|1+2s dx dy dt− ∫ R\(0,1) ∫ T 0 keywords: λn−d cache: ejde-412.pdf plain text: ejde-412.txt item: #358 of 601 id: ejde-413 author: Zhang, Zhifei title: Stabilization of the wave equation with variable coefficients and a dynamical boundary control date: 2020 words: 7962 flesch: 84 summary: dt = ∫ T 0 ∫ l 0 η(x, t) ∫ l 0 M(x, ξ, t)u(ξ, t)dξ dx dt + ∫ T 0 ∫ l 0 η(x, t)[S̃ξ(x, 0, t)a(0, t)u(0, t)− S̃ξ(x, l, t)a(l, t)u(l, t)] dt = ∫ T 0 ∫ l 0 η(x, t) ∫ l 0 M(x, ξ, t)un−1(ξ, t)dξ dx dt + ∫ T 0 ∫ l 0 η(x, t)[S̃ξ(x, 0, t)a(0, t)un−1(0, t)− S̃ξ(x, l, t)a(l, t)un−1(l, t)] keywords: conditions; equations; problem; solution; ∫ l; ∫ τ cache: ejde-413.pdf plain text: ejde-413.txt item: #359 of 601 id: ejde-415 author: Pulkina, Ludmila S. title: Nonlocal problems for hyperbolic equations from the viewpoint of strongly regular boundary conditions date: 2020 words: 7962 flesch: 84 summary: dt = ∫ T 0 ∫ l 0 η(x, t) ∫ l 0 M(x, ξ, t)u(ξ, t)dξ dx dt + ∫ T 0 ∫ l 0 η(x, t)[S̃ξ(x, 0, t)a(0, t)u(0, t)− S̃ξ(x, l, t)a(l, t)u(l, t)] dt = ∫ T 0 ∫ l 0 η(x, t) ∫ l 0 M(x, ξ, t)un−1(ξ, t)dξ dx dt + ∫ T 0 ∫ l 0 η(x, t)[S̃ξ(x, 0, t)a(0, t)un−1(0, t)− S̃ξ(x, l, t)a(l, t)un−1(l, t)] keywords: conditions; equations; problem; solution; ∫ l; ∫ τ cache: ejde-415.pdf plain text: ejde-415.txt item: #360 of 601 id: ejde-417 author: Liang, Chen; Yan, Lixu; Fu, Yongqiang title: Existence of solutions to stochastic p(t,x)-Laplace equations and applications date: 2024 words: 5721 flesch: 84 summary: L2dt− ∫ T 0 ∫ Σ η∇φdxdt + ∫ T 0 ⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt+ ∫ T 0 (g(t), φ)L2dt + ∫ T 0 (φ, σdW (t))L2 . (2.2) The series converges strongly to ∫ T 0 σ(s)dB(s) in LF 2 (Ω, C([0, T ], O)). keywords: stochastic cache: ejde-417.pdf plain text: ejde-417.txt item: #361 of 601 id: ejde-418 author: Sremr, Jiri title: Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term date: 2023 words: 10802 flesch: 81 summary: [6] X. Han, Y. He, H. Wei; Existence of positive periodic solutions for a nonlinear system of second-order ordinary differential equations, Electron. [14] J. Šremr; Bifurcation of positive periodic solutions to non-autonomous undamped duffing equations, Math. keywords: problem; solution; t ∈; theorem cache: ejde-418.pdf plain text: ejde-418.txt item: #362 of 601 id: ejde-42 author: Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca title: Positive solutions for singular (p,q)-Laplacian equations with negative perturbation date: 2023 words: 3977 flesch: 77 summary: In (3.11) we use the test function h = ũn ∈ W 1,p 0 (Ω). Introduction Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this paper we study the following singular Dirichlet (p, q)-equation −∆pu(z)−∆qu(z) keywords: 1,p cache: ejde-42.pdf plain text: ejde-42.txt item: #363 of 601 id: ejde-421 author: Brindle, Darin; N'Guerekata, Gaston M. title: S-asymptotically omega-periodic mild solutions to fractional differential equations date: 2020 words: 5117 flesch: 79 summary: = ∫ t 0 (t− s)α−2 Γ(α− 1) Au(s)ds+ f(t, u(t)), 1 < α < 2, t ≥ 0 (2.1) u(0) = u0 + g(u) . Therefore, u′(t) = ∫ t 0 (t− s)α−2 Γ(α− 1) Au(s)ds+ f(t, u(t)), 1 < α < 2, t ≥ 0, (5.1) u(0) = u0 + g(u) keywords: periodic cache: ejde-421.pdf plain text: ejde-421.txt item: #364 of 601 id: ejde-422 author: Fan, Xiaoting; Wang, Shu; Xu, Wen-Qing title: Initial-boundary layer associated with the 3-D Boussinesq system for Rayleigh-Benard convection date: 2020 words: 7970 flesch: 79 summary: = T 0 0 , (4.13) where the remainders are RεIn,u = − ∞∑ i=1 ( √ ε)i(ε[∂tu In,i + i∑ j=0 uIn,j · ∇uIn,i−j ] +∇pIn,i + 1 Ek e3 × uIn,i −∆uIn,i −Rae3T In,i), and RεIn,T = − ∞∑ i=1 ( √ ε)i ( ∂tT In,i + i∑ j=0 uIn,j · ∇T It follows from the divergence formula, divergence theorem, (4.46) and the bound- ary condition (4.48) that J5 = − ∫ X uεa · ∇ ( (T εe )2 2 ) dx dy dz = − ∫ X ∇ · ( uεa (T εe )2 2 ) keywords: b,0; boundary; t ε cache: ejde-422.pdf plain text: ejde-422.txt item: #365 of 601 id: ejde-423 author: Tunc, Ercan; Grace, Said R. title: Oscillatory behavior of solutions to third-order nonlinear differential equations with a superlinear neutral term date: 2020 words: 4215 flesch: 79 summary: Oscillation of solutions; asymptotic behavior; neutral differential equation. [5] P. Das; Oscillation criteria for odd order neutral equations, J. Math. keywords: differential; equations; order cache: ejde-423.pdf plain text: ejde-423.txt item: #366 of 601 id: ejde-424 author: Tuan Duy, Nguyen; Long Phi, Le; Thanh Son, Nguyen title: Hardy and Caffarelli-Kohn-Nirenberg inequalities with nonradial weights date: 2020 words: 4027 flesch: 79 summary: Hardy inequality; Caffarelli-Kohn-Nirenberg inequality; monomial weight; radial derivation; best constant. [29] Lam, N.; A note on Hardy inequalities on homogeneous groups. keywords: inequalities; nirenberg; rn∗ cache: ejde-424.pdf plain text: ejde-424.txt item: #367 of 601 id: ejde-427 author: Sun, Zhongyuan; Wang, Jinfeng title: Dynamics and pattern formation in diffusive predator-prey models with predator-taxis date: 2020 words: 6106 flesch: 73 summary: A reaction diffusion model with stage structure for the predator was proposed in [8], ∂u ∂t − d∆u = bv −mu, x ∈ Ω, t > 0, ∂v ∂t − d∆v = ruw − v, x ∈ Ω, t > 0, ∂w ∂t − d1∆w = (a− w)w − εvw − uw, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = 0, x ∈ ∂Ω, u(x, 0) ≥ 0, v(x, 0) ≥ 0, w(x, 0) ≥ 0, x ∈ Ω, (1.1) where u(x, t), v(x, t) and w(x, t) represent the densities of mature predator, im- mature predator and prey respectively at position x and time t; Ω is a bounded domain in RN , N ≥ 1 with smooth boundary ∂Ω and unit outer normal ν; the ho- mogeneous Neumann boundary condition indicates that the predator-prey system is self-contained with zero population flux across the boundary. Predators are assumed to move randomly in their habitats, and prey mobiles to avoid the mature predators. keywords: predator; prey; taxis cache: ejde-427.pdf plain text: ejde-427.txt item: #368 of 601 id: ejde-428 author: Charro, Fernando title: Explicit solutions of Jensen's auxiliary equations via extremal Lipschitz extensions date: 2020 words: 2551 flesch: 74 summary: In this note we prove that McShane and Whitney’s Lipschitz ex- tensions are viscosity solutions of Jensen’s auxiliary equations which are known to have a key role in Jensen’s celebrated proof of uniqueness of infinity har- monic functions, and therefore of absolutely minimizing Lipschitz extensions. Lipschitz extension; McShane-Whitney extension; infinity Laplacian. keywords: lipschitz cache: ejde-428.pdf plain text: ejde-428.txt item: #369 of 601 id: ejde-429 author: Cao, Feng; Gao, Lu title: Transition fronts of two species competition lattice systems in random media date: 2020 words: 9883 flesch: 86 summary: U(x+ ∫ t 0 c(s;ω, µ)ds, t;ω) = u ( x+ ∫ T 0 c(s; θt−Tω, µ)ds, T ;U(·+ ∫ t−T 0 c(s;ω, µ)ds, t− T ;ω), V (·+ ∫ t−T 0 c(s;ω, µ)ds, t− T ;ω), θt−Tω ) > u∗(t;ω)− 2ε, ∀t ∈ R, x ≤ −N, and hence limx→−∞ U(x+ ∫ t 0 c(s;ω, µ)ds, t;ω) = u∗(t;ω) uniformly in t ∈ R. Sim- ilarly, we can derive limx→−∞ V (x + ∫ t 0 c(s;ω, µ)ds, t;ω) = v∗(t;ω) uniformly in t ∈ R. a2(θtω)− 2c2(θtω)v∗(t;ω) + b2(θtω)v∗(t;ω) for t ∈ R. Under the assumptions (H1)–(H3), one of the most interesting dynamical prob- lems is to study the existence of random transition front (generalized traveling wave) solutions connecting (u∗(t;ω), 0) and (0, v∗(t;ω)) for (1.1). keywords: lim; θt0ω cache: ejde-429.pdf plain text: ejde-429.txt item: #370 of 601 id: ejde-43 author: Fereidooni, Amin; Moameni, Abbas; Grewal, Anant title: Existence of solutions to steady Navier-Stokes equations via a minimax approach date: 2023 words: 4536 flesch: 78 summary: = ∆v −∇pv ∀x ∈ Ω, in a weak sense; then there exists ū ∈ K such that Λū+ f(x) Since v is the unique minimizer of I, we can conclude that v̄(x) = v(x); therefore, there exits v ∈ K such that equation (3.2) is satisfied for a fixed u ∈ K. Step 5: Note that the existence of v ∈ K (as proved above) satisfies condition (i) of Theorem 1.1; therefore, a solution of the Navier-Stokes equations exist in the set K; i.e., there exists ū ∈ K that satisfies the following equations: Λū+ f(x) keywords: navier; stokes cache: ejde-43.pdf plain text: ejde-43.txt item: #371 of 601 id: ejde-430 author: Begout, Pascal title: Finite time extinction for a damped nonlinear Schrodinger equation in the whole space date: 2020 words: 8164 flesch: 87 summary: By (4.17), (4.22), (4.23), Remark 2.9 and Hölder’s inequality (recalling that 2m < m+ 1 < 2), we obtain u ∈ L∞loc ( [0,∞);H2(RN ) ) ∩ L∞loc ( [0,∞);L2m(RN ) ) , (4.24) u ∈ C ( [0,∞);L2(RN ) ) ∩ L∞loc ( [0,∞);L2m(RN ) ) ↪→ C ( [0,∞);Lm+1(RN ) ) . (4.25) Recalling that u ∈ W 1,∞ loc ( [0,∞);L2(RN ) ) , by (4.24) and the embedding 3) of Lemma A.4, we have u ∈ C ( [0,∞);H1(RN ) ) . keywords: bégout; l2(rn; solution cache: ejde-430.pdf plain text: ejde-430.txt item: #372 of 601 id: ejde-431 author: Ildefonso Diaz, Jesus; Padial,  Juan Francisco; Tello, Jose Ignacio; Tello, Lourdes title: Complex Ginzburg-Landau equations with a delayed nonlocal perturbation date: 2020 words: 7646 flesch: 77 summary: (2.10) Then by Gronwall’s lemma, we obtain that f(t) ≤ K1 for t ∈ (0, τ). By integrating inequality (2.6) over (0, t), for t ∈ (0, τ) we obtain∫ t 0 d dt ‖u(s)‖2L2(Ω)ds ≤ keywords: solution cache: ejde-431.pdf plain text: ejde-431.txt item: #373 of 601 id: ejde-432 author: Jimenez, Johana; Llibre, Jaume; Medrado, Joao C. title: Crossing limit cycles for a class of piecewise linear differential centers separated by a conic date: 2020 words: 15989 flesch: 76 summary: − b22(λ2 + λ3 − n2 − n3)ψ1 + 2b2(−λ4ψ2 +m2n2 −m3n3) ) + k1 ( − ψ2 + b22(λ2 + λ3 − n2 − n3)ψ1 + 2b2(λ1ψ2 −m2n2 +m3n3) )) + k2 ( (λ1 − λ4)(ψ2 − b22(λ1 + λ4 − n2 − n3)ψ1)− (k21 − k24)ψ1 − 2b2 ( λ2(−λ4ψ2 − (k1 − k4)ψ1) + (λ1 − λ2)(λ1 + λ2 − λ3 − l4)l1) ) , γ = 1 8(−(k1 − k4)(λ2 − λ3) + (k2 − k3)(λ1 − λ4)) keywords: centers; crossing limit; cycles; differential centers; limit cycles; linear differential; piecewise linear; points; system; theorem cache: ejde-432.pdf plain text: ejde-432.txt item: #374 of 601 id: ejde-433 author: Wang, Fei; Hao, Jianghao title: Decay of energy for viscoelastic wave equations with Balakrishnan-Taylor damping and memories date: 2020 words: 6264 flesch: 87 summary: = 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + 1 2 ‖∇ut‖22 + J(t), (2.10) where (g1 ◦ ∇u)(t) = ∫ Ω a1(x) ∫ t 0 g1(t− s)|∇u(t)−∇u(s)|2 ds dx, (g2 }∇u)(t) = ∫ Ω a2(x) ∫ ∞ 0 g2(s)|∇u(t)−∇u(t− s)|2 ds dx. Lemma 2.3. E(t) is a non-increasing function for t ≥ 0, and E′(t) = ∫ Ω a1(x) ∫ t 0 g′1(t− s)|∇u(t)−∇u(s)|2 ds dx, (g′2 }∇u)(t) keywords: e(t; g1(t−; ∇u)(t; − ∫; ∫ t; ∫ ω; ∫ ∞ cache: ejde-433.pdf plain text: ejde-433.txt item: #375 of 601 id: ejde-434 author: Li, Dandan; Du, Jiayin title: µ pseudo rotating-periodic solutions for differential equations date: 2020 words: 4217 flesch: 80 summary: In this article, we combine rotating periodic functions with µ er- godic functions to obtain a new class of functions called µ pseudo rotating periodic functions. Recently, many researchers have studied rotating periodic functions and obtained a series of results; see [14, 15, 20, 22, 23, 24, 25]. keywords: function; periodic; pseudo cache: ejde-434.pdf plain text: ejde-434.txt item: #376 of 601 id: ejde-435 author: Meng, Fengjuan; Zhang, Fubao; Zhang, Yuanyuan title: Multiple positive solutions for biharmonic equation of Kirchhoff type involving concave-convex nonlinearities date: 2020 words: 6434 flesch: 80 summary: Note that 〈I ′λ(un), un〉=0 and 〈I ′λ(un), un〉 − 〈I ′λ(u), u〉 = 〈I ′λ(un)− I ′λ(u), u〉 − 〈I ′λ(un), un − u〉 → 0, as n→∞, (2.12) we have 〈I ′λ(u), u〉 = 0, which implies u ∈ Nλ. = 〈I ′λ(un)− I ′λ(u), un − u〉 = ‖un − u‖2 + b ∫ RN |∇un|2dx ∫ RN |∇(un − u)|2dx − b (∫ RN |∇u|2dx− ∫ RN |∇un|2dx )∫ RN ∇u∇(un − u)dx − λ ∫ RN f1(x)(|un|q−2un − |u|q−2u)(un − u)dx − ∫ RN f2(x)(|un|p−2un − |u|p−2u)(un − u)dx = ‖un − u‖2 + b ∫ RN |∇un|2dx ∫ RN |∇(un − u)|2dx+ o(1) keywords: lemma cache: ejde-435.pdf plain text: ejde-435.txt item: #377 of 601 id: ejde-438 author: Su, Si; Zhang, Guo-Bao title: Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity date: 2020 words: 6540 flesch: 87 summary: ‖(u, v)‖ < r}, ∂Kri = {u ∈ Ki : ‖u‖ = ri}, ∂Pr = {(u, v) ∈ P : ‖(u, v)‖ = r}, Kri = {u ∈ Ki : ‖u‖ 6 ri}, ∀ri > 0, Pr = {(u, v) ∈ P : u ∈ keywords: lim; system; t∈[0,1 cache: ejde-438.pdf plain text: ejde-438.txt item: #378 of 601 id: ejde-44 author: Bostan, Mihai title: Periodic solutions for evolution equations date: 2002 words: 12911 flesch: 87 summary: |x0|+ ∫ T 0 |f(t)− g(t, x1)|dt (32) = |x0|+ ∫ T 0 |f(t)− g(t, x0)|dt, t ∈ = ∫ T 0 f(t)dt, τ ∈]0, T keywords: existence; lim; periodic; solutions; t 0; x(t; ∈ r cache: ejde-44.pdf plain text: ejde-44.txt item: #379 of 601 id: ejde-441 author: Su, Si; Zhang, Guo-Bao title: Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity date: 2020 words: 7746 flesch: 85 summary: − η, 0)dη + ∫ t 0 e−β(t−s) ∫ ∞ −∞ G2(η, t− s) lim ξ→+∞ P2(U10(ξ − η − cτ, s− τ)) dη ds = e−βtU20(∞, 0) ∫ ∞ −∞ G2(η, t)dη + ∫ t 0 e−β(t−s)P2(U10(∞, s− τ)) = e−αt ∫ ∞ −∞ G1(η, t)U10(ξ − η, 0)dη + ∫ t 0 e−α(t−s) ∫ ∞ −∞ G1(η, t− s)P1(U20(ξ − η − cτ, s− τ)) dη ds, U2(ξ, t) = e−βt ∫ ∞ −∞ G2(η, t)U20(ξ − η, 0)dη + ∫ t 0 e−β(t−s) ∫ ∞ −∞ G2(η, t− s)P2(U10(ξ − η − cτ, s− τ)) dη ds (3.3) for t ∈ [0, τ ], where Gi(η, t) is the heat kernel Gi(η, t) keywords: stability; waves cache: ejde-441.pdf plain text: ejde-441.txt item: #380 of 601 id: ejde-442 author: Lan, Yongyi; Tang, Biyun; Hu, Xian title: Positive solutions of Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity date: 2020 words: 3714 flesch: 83 summary: In this article, we study the nonlinear Schrödinger-Poisson system −∆u+ u− µ u |x|2 + l(x)φu = k(x)|u|p−2u x ∈ R3, −∆φ = l(x)u2 x ∈ R3, where k ∈ C(R3) and 4 < p < 6, k changes sign in R3 and lim sup|x|→∞ k(x) More precisely, f(x, u) = k(x)|u|p−1u + µh(x)u, where 4 < p < 6 and µ > 0, k(x) ∈ C(R3), k changes sign in R3, and lim|x|→∞ k(x) keywords: h1(r3; schrödinger cache: ejde-442.pdf plain text: ejde-442.txt item: #381 of 601 id: ejde-446 author: Llibre, Jaume; Pereira, Weber F.; Pessoa, Claudio title: Phase portraits of Bernoulli quadratic polynomial differential systems date: 2020 words: 9894 flesch: 78 summary: Now as eα + f = eβ + f = 0 and α 6= β, we obtain e = 0. First we suppose that eα+f = 0, so the eigenvalues associated with singular points p1 = (α, 0) are λ1 = 0 and µ1 = α−β. keywords: node; phase; points; saddle; singular; system cache: ejde-446.pdf plain text: ejde-446.txt item: #382 of 601 id: ejde-447 author: Briozzo, Adriana C. title: Supercooled Stefan problem with a Neumann type boundary condition date: 2020 words: 5141 flesch: 80 summary: Free boundary problems with diffusion coefficient given by (1.7) or with temper- ature dependent conductivity were considered in [2, 4, 6, 7, 16, 19, 23, 30]. Free boundary problems which involves the freezing of a supercooled liquid can be seen in [12, 13, 14, 17, 18, 22, 24]. keywords: boundary; solution cache: ejde-447.pdf plain text: ejde-447.txt item: #383 of 601 id: ejde-448 author: Calatayud, Julia; Caraballo, Tomas; Cortes, Juan Carlos; Jornet, Marc title: Mathematical methods for the randomized non-autonomous Bertalanffy model date: 2020 words: 8673 flesch: 71 summary: [t0, T ], ω ∈ Ω, x(t0, ω) = x0(ω), ω ∈ Ω. (1.1) In (1.1), we are also considering the stochastic processes a = {a(t, ω) : t ∈ [t0, T ], ω ∈ Ω}, b = {b(t, ω) : keywords: density; process; solution; stochastic; theorem; x(t cache: ejde-448.pdf plain text: ejde-448.txt item: #384 of 601 id: ejde-449 author: Tao, Kai title: Non-perturbative positivity and weak Holder continuity of Lyapunov exponent of analytic quasi-periodic Jacobi cocycles defined on a high dimension torus date: 2020 words: 5298 flesch: 82 summary: There exists an N0 := N0(λv, a) such that for any N > N0, E ∈ E , x2 ∈ T and DN ω1, it holds meas { x1 ∈ T : 1 N | N∑ j=1 Fixing x2, E ∈ E and λ > λ0 with κ = 1 100 , we expand uan into its Fourier series of x1 and denote the Fourier coefficient as ûan(k, x2, E, λ), i.e., uan(x,E, λ) = ∑ k∈Z ûan(k, x2, E, λ)e2πikx1 , ûan(k, x2, E, λ) = ∫ x1∈T uan(x1, x2, E, λ)e−2πikx1dx1. keywords: log cache: ejde-449.pdf plain text: ejde-449.txt item: #385 of 601 id: ejde-45 author: Escobedo, Miguel; Mischler, Stephane; Valle, Manuel A. title: Homogeneous Boltzmann equation in quantum relativistic kinetic theory date: 2003 words: 31013 flesch: 80 summary: = ∫ ∞ 0 [(1 + F ) ln(1 + F )− F lnF − εF ]ε2dε. (4.52) = ∫ R3 ( (1 + F ) ln(1 + F )− F lnF − Fβ0E1(p) ) dp (4.21) and DBQ(F ) keywords: = ∫; boltzmann; boltzmann equation; bose; case; collision; cross; ejde–2003; entropy; equation; escobedo; e−ε′; fermi; function; mischler; mon; non; particles; problem; q(f; quantum; section; − ∫; ∫ r3; ∫ ∞; ∫ ∫ cache: ejde-45.pdf plain text: ejde-45.txt item: #386 of 601 id: ejde-452 author: Wang, Xiaohui; Zhao, Peihao title: Existence of weak solutions to superlinear elliptic systems without the Ambrosetti-Rabinowitz condition date: 2020 words: 8139 flesch: 81 summary: We first consider the p-Laplacian equation −∆pu = λf(x, u) in Ω, u = 0 on ∂Ω, (1.1) where p > 1, λ > 0, Ω ⊂ Rn is a bounded domain, f : Ω × R → R is a continuous function, and for 1 < p <∞, the p-Laplacian operator is ∆pu = div(|Du|p−2Du) for u ∈W 1,p(Ω). |t|p = +∞ a.e. in Ω, or lim t→−∞ F (x, t) |t|p = +∞ a.e. in Ω. Our first objective is to establish the existence of the nontrivial weak solution for the p-Laplacian superlinear elliptic equation (1.1) under the weaker condition than the AR condition in this paper. keywords: condition; lim; superlinear cache: ejde-452.pdf plain text: ejde-452.txt item: #387 of 601 id: ejde-453 author: Salako, Rachidi B.; Shen, Wenxian title: Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source date: 2020 words: 8402 flesch: 81 summary: = c in the interval (0,min{ √ a, √ λ+τa (1−τ)+ }). Parabolic chemotaxis system; logistic source; traveling wave solution; minimal wave speed. keywords: wave cache: ejde-453.pdf plain text: ejde-453.txt item: #388 of 601 id: ejde-456 author: Riva, Lorenzo; Pennington, Nathan title: Low regularity of non-L^2(R^n) local solutions to gMHD-alpha systems date: 2020 words: 5794 flesch: 80 summary: ︸ ︷︷ ︸ RHS of (3.4) = γ−3 − r0 + n p0 − n p1 ≥ 0, the list reduces to γ−3 − 1 ≤ r0 ≤ γ−3 ≤ r1, r0 < n p1 , γ−1 > 1− 2r0 + r1 + 2n p0 − n p1 . 3.3. With this new bound on W1(u, v), we come back to J1 and see that J1 ≤ sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 ‖W1(u, v)‖γ−3 −1,π1 ds ≤ C sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 ‖u‖r0,p0‖u‖r1,p1ds = C sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 s−a1‖u‖r0,p0sa1‖u‖r1,p1ds ≤ C‖u‖0;r0,p0‖u‖a1;r1,p1 sup (0,T ) ∫ t 0 (t− s)−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 s−a1ds < CM2T 1−(r0−(γ−3 −1)+n/π1−n/p0)/γ−1 −a1 , where the last inequality holds by Proposition 2.3, if γ−1 > r0 − (γ−3 − 1) + n π1 − n p0 + γ1a1 = r0 − (γ−3 − 1) + n ( 1 p0 + 1 p1 − r0 n ) keywords: sup; γ−3 cache: ejde-456.pdf plain text: ejde-456.txt item: #389 of 601 id: ejde-457 author: Oliveira, Regilene; Valls, Claudia title: Global dynamics of the May-Leonard system with a Darboux invariant date: 2020 words: 8682 flesch: 72 summary: Moreover, the f1(x, y, z) = 0, f2(x, y, z) = 0 and f3(x, y, z) = 0 have cofactors, 1 − x − αy − βz, 1 − βx − y − αz and 1 − αx − βy − z, respectively. Denote by X̄ the vector field D(f ◦ X) defined on S2 \ S1, where S1 = {y ∈ S2 : y3 = 0} is identified with the infinity of R2. keywords: orbits; system cache: ejde-457.pdf plain text: ejde-457.txt item: #390 of 601 id: ejde-458 author: Li, Guofa; Cheng, Bitao; Huang, Yisheng title: Positive solutions for asymptotically 3-linear quasilinear Schrodinger equations date: 2020 words: 6481 flesch: 88 summary: (3) Given y ∈ RN and setting uy(x) := u(x− y), we have β(uy) = β(u) + y. Lemma 4.10. = ∫ t 0 h(s)ds. keywords: h1(rn; lemma; rn v; ∫ rn cache: ejde-458.pdf plain text: ejde-458.txt item: #391 of 601 id: ejde-459 author: Barreira, Luis; Llibre, Jaume; Valls, Claudia title: Linear type global centers of cubic Hamiltonian systems symmetric with respect to the x-axis date: 2020 words: 7017 flesch: 74 summary: Thus on the local chart U2 we obtain u′ = −3a12uv − 3a30u 3v − cu2v2 − αu4 − ω2 c v2, v′ = −v(a12v − a21x2 − 2a12xy − 3a03y 3 − 3αµx2y − αy3, ẏ = cx+ dy + 3a30x 2 + 2a21xy + a12y 2 + 3αµxy2. 6 L. BARREIRA, J. LLIBRE, C. VALLS EJDE-2020/57 Since this system must be invariant under the transformation (x, y, t) 7→ (x,−y,−t) we must have d = a21 = a03 = 0 keywords: linear; origin; points; singular cache: ejde-459.pdf plain text: ejde-459.txt item: #392 of 601 id: ejde-46 author: Wang, Hwai-chiuan title: Palais-Smale approaches to semilinear elliptic equations in unbounded domains date: 2004 words: 57992 flesch: 93 summary: Then for r > 0, x ∈ Ω exists such that B̃N (x, r) ⊂ Ω̃, which means that for any ỹ ∈ B̃N (x, r), y ∈ Ω exists and ỹ is the projection of y. By Lemma 10.9, λ > 0 exists such that {ỹ − χt : t ≥ λ} ⊂ p |x− y|θ a.e. for x, y ∈ Ω and |β| = k. In particular, Wm,p(Ω) ↪→ Ck,θ(Ω). keywords: approaches; ar 0; chiuan; domain; domain ω; ejde-2004; equation; exists; following; h1 0; hwai; iii; jh(u0; lemma; lim; mon; o(1; palais; proof; sequence; smale; solution; state; subsequence; symmetric; t ∈; theorem; u ∈; u(x; wang; x(ω; z ∈; α(ar; ∈ ar; ∈ h1; ∈ r; − ∫; ∫ ω cache: ejde-46.pdf plain text: ejde-46.txt item: #393 of 601 id: ejde-460 author: He, Ze-Rong; Zhou, Nan title: Controllability and stabilization of a nonlinear hierarchical age-structured competing system date: 2020 words: 6402 flesch: 80 summary: ∂t = −α ∫ t 0 bui i (τ)Mi(t− τ, t) exp { − ∫ t τ Mi(v − τ, v)dv } dτ + αbui i (t)− αp0i (a− t) exp { − ∫ t 0 Mi(a− t+ τ, τ)dτ } − α ∫ a−t 0 p0i (v)Mi(v + t, t) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv − p0i (A− t) exp { − ∫ t 0 Mi(A− t+ τ, τ)dτ } + p0i (a− t) exp { − ∫ t 0 Mi(a− t+ τ, τ)dτ } − ∫ A−t a−t p0(v)Mi(v + t, t) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv + α ∫ t 0 exp { − ∫ t v Mi(θ − t, θ)dθ } ui(v − t, v)dv − α ∫ t 0 ∫ s 0 Mi(t− s, t) exp { − ∫ t v Mi(θ − s, θ)dθ } ui(v − s, v)dvds − α ∫ t 0 exp { − ∫ t v Mi(θ + a− t, θ)dθ } ui(v + a− t, v)dv + α ∫ a−t 0 [ui(t+ s, t) − ∫ t 0 ui(v + s, s)Mi(t+ s, t) exp { − ∫ t v Mi(θ + s, θ)dθ } ]ds − ∫ t 0 exp { − ∫ t v Mi(θ +A− t, θ)dθ } ui(v +A− t, v)dv + ∫ t 0 exp { − ∫ t v Mi(θ + a− t, θ)dθ } ui(v + a− t, v)dv EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 7 + ∫ A−t a−t + ∫ t 0 Ki(t, s;P )bui i (t− s;P )ds, t ∈ (0, T ), (3.7) EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 5 where Fi(t;P ) keywords: system; − ∫ cache: ejde-460.pdf plain text: ejde-460.txt item: #394 of 601 id: ejde-461 author: Floridia, Giuseppe title: Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science date: 2020 words: 11812 flesch: 69 summary: Integrating by parts, recalling that u−(·, t) ∈ H1 a(−1, 1) for every t ∈ (0, T ),and using Proposition 3.1 we deduce∫ 1 −1 (a(x)ux)xu − dx = [a(x)uxu −]1−1 − ∫ 1 −1 a(x)ux(u−)x dx = [a(x)uxu −]1−1 + ∫ 1 −1 a(x)u2x dx . (3.2) If β1γ1 6= 0, keeping in mind the boundary conditions, for t ∈ (0, T ) we have [a(x)uxu −]1−1 = a(1)ux(1, t)u−(1, t)− a(−1)ux(−1, t)u−(−1, t) = −γ0 γ1 (u+(1, t)− u−(1, t))u−(1, t) For a.e. x ∈ (−1, 1), from the equation ut(·, t) = αεj(·) T − T1 u(·, t) + ((a(·)ux(·, t))x + f(·, t, u)) t ∈ (T1, T ), by the classical variation constants technique, we obtain a representation formula of the solution u(x, t) of (3.24), that computed at time T , for x ∈ (−1, 1), becomes u(x, T ) keywords: a(−1; case; controllability; equations; function; l2(−1; proposition; solution; t t1 cache: ejde-461.pdf plain text: ejde-461.txt item: #395 of 601 id: ejde-465 author: Huang, Chuangxia; Wang, Jiafu; Huang, Lihong title: Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure date: 2020 words: 6380 flesch: 79 summary: Obviously, according to the biological interpretation of Nicholson’s blowflies models in [22, 17], it is necessary to relax the above technical conditions as follows: M lim sup t→+∞ γij(t) ≤ κ̃, for all i ∈ Q, j ∈ I, (1.8) sup t∈[t0,+∞) {−aii(t) + n∑ j=1,j 6=i aij(t) + 1 eM m∑ j=1 βij(t) γij(t) } < 0, i ∈ Q, (1.9) lim inf t→+∞ {−aii(t) + n∑ j=1,j 6=i aij(t) + m∑ j=1 βij(t) γij(t) e−κ} > 0, i ∈ Q. (1.10) τhi0j(t))e −γh i0j(t)xi0 (t−τh i0j(t)), for t ∈ [t0, t̄i0), we obtain 0 keywords: j=1; j=1,j; lim cache: ejde-465.pdf plain text: ejde-465.txt item: #396 of 601 id: ejde-467 author: Ivorra, Benjamin; Ngom, Diene; Ramos, Angel M. title: Stability and sensitivity analysis of the epidemiological model Be-CoDiS predicting the spread of human diseases between countries date: 2020 words: 13306 flesch: 67 summary: [34] Z. Shuai, P. Van den Driessche; Global stability of infectious disease models using lyapunov functions, SIAM Journal on Applied Mathematics 73 (2013), no. 4, 1513–1532. Here, we have considered the functions (see [27]): mI,i(t) = mH,i(t) = mD,i(t) = exp ( − κi max(t− λi, 0) ) , (2.2) where κi in [0,+∞) (day−1) simulates the efficiency of the control mea- sures (greater value implies lower value of disease contact rates) and λi in R ∪ {+∞} (day) denotes the first day of application of those control mea- sures. keywords: countries; country; day−1; disease; epidemic; equilibrium; model; parameters; people; rate; state; system; time cache: ejde-467.pdf plain text: ejde-467.txt item: #397 of 601 id: ejde-469 author: Benes, Michal title: Global weak solutions to degenerate coupled transport processes in partially saturated deformable elastic-inelastic porous media date: 2020 words: 9601 flesch: 82 summary: In (1.1)–(1.9), p : ΩT → R, ϑ : ΩT → R, σ : ΩT → R4, εp` : ΩT → R4 and α : ΩT → Rd, d ∈ N, are the unknown functions. [L2(Ω)]4 and αn−1 N ∈ [L2(Ω)]d, n = 1, . . keywords: n n; ϑn−1 cache: ejde-469.pdf plain text: ejde-469.txt item: #398 of 601 id: ejde-47 author: Squassina, Marco title: Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems date: 2006 words: 85791 flesch: 87 summary: ON A CLASS OF QUASI-LINEAR ELLIPTIC PROBLEMS 143 µj ≥ Sσ p p∗ j , (6.111) where δxj denotes the Dirac measure at xj ∈ Ω and S denotes the best Sobolev constant for the embedding W 1,p 0 (Ω) ↪→ Lp ∗ (Ω) (see e.g. [138]). Let x0 ∈ Ω and δ > 0 and consider the functions Tδ,x0 as in (6.113). keywords: + ∞; + ∞.; 1,p; a.e; c ∈; case; class; condition; dsl; ejde-2006; equations; existence; f(u; following; function; h dx; h ∫; j=1 ∫; js(x; lemma; lim; linear; marco; mon; p ∫; palais; point; problems; proof; results; s ∈; satisfies; sequence; smale; solution; squassina; sup; t ∈; that∫ ω; theorem; u dx; u k; u ∈; u,∇u; uh,∇uh; x ∈; ν ∫; ξ ∈; ω aij(x; ω b; ω g(x; ω jξ(x; ω l∞(x,∇u; ω n∑; ω |u+; ω φp−1; ω ϕ; ω ∇ξl; ϕ ∈; → +; ∈ h1; ∈ l1(ω; ∈ r; − n∑; − ∫; ∫ rn; ∫ ω; ≤ ∫; ≥ ∫ cache: ejde-47.pdf plain text: ejde-47.txt item: #399 of 601 id: ejde-470 author: Castilho, Cesar; Gondim, Joao A. M.; Marchesin, Marcelo; Sabeti, Mehran title: Assessing the efficiency of different control strategies for the COVID-19 epidemic date: 2020 words: 5804 flesch: 68 summary: The second one evaluates different quarantine strategies by comparing their relative total number of deaths. In Section 5, different quarantine strategies for different age classes are considered and compared. keywords: age; class; control; epidemic; figure; model; number; parameters; quarantine; seir; strategies cache: ejde-470.pdf plain text: ejde-470.txt item: #400 of 601 id: ejde-471 author: Soriano Hernandez, Lorena; Siciliano, Gaetano title: Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations date: 2023 words: 7654 flesch: 79 summary: We study the existence and multiplicity of solutions for the Schrödinger-Bopp-Podolsky system −∆u+ φu = ωu in Ω a2∆2φ−∆φ = u2 in Ω u = φ = ∆φ = 0 on ∂Ω∫ Ω u2 dx = 1 where Ω is an open bounded and smooth domain in R3, a > 0 is the Bopp- Podolsky parameter. × H is a weak solution of (1.1) if∫ Ω ∇ua∇v dx+ ∫ Ω φauav dx = ωa ∫ Ω uav dx for all v ∈ H1 0 (Ω) (1.2) and a2 ∫ Ω ∆φa∆v dx+ ∫ Ω ∇φa∇v dx keywords: solutions; theorem cache: ejde-471.pdf plain text: ejde-471.txt item: #401 of 601 id: ejde-472 author: Wang, Guiyun; Zheng, Shenzhou title: Boundedness on generalized Morrey spaces for the Schrodinger operator with potential in a reverse Holder class date: 2023 words: 6213 flesch: 72 summary: Schrödinger operators; reverse Hölder class; generalized Morrey space; vanishing generalized Morrey space; BMOθ(ρ) coefficients. This is done in in generalized Morrey spaces, and in vanishing generalized Morrey spaces. keywords: morrey; spaces cache: ejde-472.pdf plain text: ejde-472.txt item: #402 of 601 id: ejde-475 author: Besalu, Mireia; Binotto, Giulia; Rovira, Carles title: Convergence of delay equations driven by a Holder continuous function of order 1/3 date: 2020 words: 11008 flesch: 87 summary: t | ≤ N∆̃β y ≤ T ∆̃β−1 y + ∆̃β y . (5.42) 18 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65 By Proposition 4.2 we have A2 ≤ K‖σ‖∞ Φβ′(a,b)(y − y·−r, y) +K ( ‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′(a,b)T λβ′ ) Φβ′(a,b)(x̂ r, y − y·−r, y)T β ′ = K‖y‖β′ ( ‖σ‖∞ + ( ‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′Tλβ ′ ) keywords: b)(σ; sup; t s; β′(a cache: ejde-475.pdf plain text: ejde-475.txt item: #403 of 601 id: ejde-476 author: Gialelis, Nikolaos title: Inviscid limit of linearly damped and forced nonlinear Schrodinger equations date: 2020 words: 7999 flesch: 83 summary: [0, T ]× U, (1.1) where λ ∈ R∗ and α > 0, γ > 0 and u = u(t, x; γ), f = f(t, x; γ) are complex-valued functions for t ∈ [0, T ], then, following the notation of, e.g., [11] and [23], we associate with u the mapping u : [0, T ] → F(U ;C), defined by [u(t)](x) := u(t, x), for every x ∈ U and t ∈ keywords: 0,2,u; equations cache: ejde-476.pdf plain text: ejde-476.txt item: #404 of 601 id: ejde-48 author: Gorban, Alexander N. title: Singularities of transition processes in dynamical systems: Qualitative theory of critical delays date: 2004 words: 31094 flesch: 79 summary: For a given parameter value k ∈ K and an initial state x ∈ X, the ω-limit set ω(x, k) is the set of all limit points of f(t, x, k) as t→∞: y is in ω(x, k) if and only if there exists a sequence ti ≥ 0 such that ti →∞ and f(ti, x, k) → y. Examples of ω-limit points are stationary (fixed) points, points of limit cycles and so on. [0,∞)×X ×K → X (1.1) be a continuous mapping for any t ≥ 0, k ∈ K; let mapping f(t, ·, k) : X → X be homeomorphism of X into subset of X and under every k ∈ K let these homeomorphisms form monoparametric semigroup: f(0, ·, k) keywords: limit; motion; point; proof; proposition; relaxations; sequence; set; sets; slow; system; theorem; ω(x cache: ejde-48.pdf plain text: ejde-48.txt item: #405 of 601 id: ejde-480 author: Nguyen, Tu title: Lower bounds at infinity for solutions to second order elliptic equations date: 2023 words: 3425 flesch: 85 summary: They proved that if u satisfies (1.1) then there exists C > 0 such that∫ B(x,1) u2 ≥ exp(−C|x|4/3 log |x|) ∀|x| ≥ 10. Then there exists C2 > 0 such that if |x| = R ≥ 10, then∫ B(x,τR) u2 ≥ e−C2R α (3.4) and ∫ B(x,1) u2 ≥ e−C2R α logR. (3.5) Proof. keywords: |x| cache: ejde-480.pdf plain text: ejde-480.txt item: #406 of 601 id: ejde-482 author: Zhang, Xuping; Chen, Pengyu; Li, Yongxiang title: Monotone iterative method for retarded evolution equations involving nonlocal and impulsive conditions date: 2020 words: 11252 flesch: 77 summary: We mention that in 2012, Chuong and Ke [18] studied the retarded evolution inclusions involving nonlocal and impulsive conditions u′(t) +Au(t) ∈ F (t, u(t), ut), t ∈ If a function u ∈ PC([−r, a], X) ∩ C1(I ′′, X) ∩ C(I ′, X1) satisfies u′(t) +Au(t) ≤ f(t, u(t), ut), t ∈ keywords: banach; equations; function; solution; v(0; w(0 cache: ejde-482.pdf plain text: ejde-482.txt item: #407 of 601 id: ejde-483 author: Danecek, Josef; Viszus, Eugen title: Holder continuity for vector-valued minimizers of quadratic functionals date: 2020 words: 7340 flesch: 84 summary: Case Aαβij = Aαβij (u). Case Aαβij = Aαβij (x, u). keywords: aαβij cache: ejde-483.pdf plain text: ejde-483.txt item: #408 of 601 id: ejde-484 author: Mushayabasa, Steady; Losio, Anthony A. E.; Modnak, Chairat; Wang, Jin title: Optimal control analysis applied to a two-patch model for Guinea worm disease date: 2020 words: 9173 flesch: 67 summary: In particular we will investigate heterogeneity on (i) disease transmission rates, with the assumption that β2 = 6β1 (ii) bounds of the controls a2 < a1 (iii) both disease transmission rates and upper bounds of the controls. As we can observe, an increase in disease transmission rate for the risk patch (patch 2) will lead to an increase on the total number of new infections and total cost over a period of 120 months. keywords: control; disease; e∗i; model; number; patch; total; worm cache: ejde-484.pdf plain text: ejde-484.txt item: #409 of 601 id: ejde-486 author: Zhao, Yihan; Xia, Yuanpei; Yang, Zhichun title: Asymptotic behavior of stochastic three-species predator-prey systems with white and Levy noise date: 2020 words: 6816 flesch: 82 summary: = (r1 − β1 − a11eu1(t) Combining inequality (3.5), (3.2) and Lemma 2.6, we can deduce that a22x2(t)∗ ≥ lim inf t→∞ { − r2 − β2 − ln(x2(t)/x2(0)) keywords: stochastic; system; −r2 cache: ejde-486.pdf plain text: ejde-486.txt item: #410 of 601 id: ejde-487 author: Qiu, Kee; Wang, Jinrong title: Representation of solutions of a second order delay differential equation date: 2020 words: 7335 flesch: 85 summary: ∫ 0 −τ1 V (x− 2τ1 − s)φ(s)(ds)α + A2B2 Γ(1 + α) ∫ 0 −τ1 V (x− τ1 − τ2 − s)φ(s)(ds)α + A2B2 Γ(1 + α) ∫ 0 −τ2 V (x− τ1 − τ2 − s)φ(s)(ds)α + B4 Γ(1 + α) ∫ 0 −τ2 V (x− 2τ2 − s)φ(s)(ds)α − A2 Γ(1 + α) ∫ keywords: a(x−; γ(1 cache: ejde-487.pdf plain text: ejde-487.txt item: #411 of 601 id: ejde-488 author: Fama, Alessio; Restuccia, Liliana title: Propagation of coupled porosity and fluid-concentration waves in isotropic porous media date: 2020 words: 7004 flesch: 62 summary: = C1 = L1 = L2, D2 = B3 = C2 = L3, D3 = B4 = B5 = C3 = C5 = L4 = L5 = L7 = L10, D4 = B6 = C4 = C8 = L6 = L13, D5 = B7 = C6 = L8 = L11, D6 = B8 = B9 = C7 = C9 = L9 = L12 = L14 = L15, (5.18) where we have used expressions (5.13) and (5.16). +B4(δikδjl + δilδjk)δmn +B5(δikδjm + δimδjk)δln +B6(δikδjn + δinδjk)δlm +B7(δilδjm + δimδjl)δkn +B8(δilδjn + δinδjl)δkm +B9(δimδjn + δinδjm)δkl; (5.12) with B1 = L1, B2 = L2, B3 = L3, B4 = L4 = L7, B5 = L5 = L10, B6 = L6 = L13, B7 = L8 = L11, B8 = L9 = L14, B9 = L12 = L15. (5.13) keywords: concentration; equations; field; fluid; flux; isotropic; porosity; propagation; tensor; waves cache: ejde-488.pdf plain text: ejde-488.txt item: #412 of 601 id: ejde-489 author: Zhan, Huashui; Feng, Zhaosheng title: Stability of anisotropic parabolic equations without boundary conditions date: 2020 words: 4442 flesch: 84 summary: if x ∈ ∂Ω, and ai(x) > 0 if x ∈ Ω, without any boundary conditions. When a(x) ∈ C1(Ω), and a(x) > 0, x ∈ Ω and a(x) keywords: ai(x cache: ejde-489.pdf plain text: ejde-489.txt item: #413 of 601 id: ejde-49 author: Hafstein, Sigurdur Freyr Hafstein title: An algorithm for constructing Lyapunov functions date: 2007 words: 41571 flesch: 78 summary: ∈ J for all x ∈ I \ C. It is clear from elementary calculus, that if g : I → R is a function from a nonempty open subset I ⊂ R into R and y ∈ I, then all four Dini derivatives D+g(y), D+g(y), D−g(y), and D−g(y) of g at the point y exist. keywords: autonomous; define; definition; equilibrium; function; j+1; linear; lyapunov; lyapunov function; mon; origin; p ∈; problem; programming; set; solution; system; theorem; y ∈; y(z; σ ∈; ∈ sp; ∈ u cache: ejde-49.pdf plain text: ejde-49.txt item: #414 of 601 id: ejde-490 author: Tuan Duy, Nguyen; Nguyen, Huy Bac title: Cylindrical Hardy inequalities on half-spaces date: 2020 words: 4674 flesch: 84 summary: [26] Goldstein, J. A.; Kombe, I.; Yener, A.; A unified approach to weighted Hardy type inequalities on Carnot groups. [4] Barbatis, G.; Filippas, S.; Tertikas, A.; A unified approach to improved Lp Hardy inequalities with best constants. keywords: hardy; inequalities; |y| cache: ejde-490.pdf plain text: ejde-490.txt item: #415 of 601 id: ejde-491 author: Schulz-Baldes, Hermann; Urban, Liam title: Space versus energy oscillations of Prufer phases for matrix Sturm-Liouville and Jacobi operators date: 2020 words: 10620 flesch: 73 summary: Let us note that SE1 = E − V1 , SE2 = (E − V1)T−1 2 (E − V2)T−1 2 (E − V1)− (E − V1) , and that there is a recurrence relation SEn = (φEn ) Even though the particular form of matrix Sturm-Liouville operator may not be of great importance, let us spell it out explicitly anyhow. keywords: eigenvalues; energy; jacobi; liouville; matrix; phases; prüfer; sin; sturm cache: ejde-491.pdf plain text: ejde-491.txt item: #416 of 601 id: ejde-493 author: Naumkin, Pavel I.; Sanchez-Suarez, Isahi title: KdV type asymptotics for solutions to higher-order nonlinear Schrodinger equations date: 2020 words: 12100 flesch: 91 summary: Let the weights P ∈ C1(R \ 0) and Q ∈ C2(R \ 0) be such that ∂kηP (η) = O(|η|α1−k), k = 0, 1, and ∂kξQ(ξ) = O(|ξ|α2−k), k = 0, 1, 2. Suppose that 8 P. I. NAUMKIN, I. SÁNCHEZ-SUÁREZ EJDE-2020/77 h(ξ) ∈ C4(R \ 0) is such that |∂kξ h(ξ)| ≤ C|ξ|α3−k for ξ ∈ R \ 0, 0 ≤ k ≤ 4. |ξ̂|6〈ξ̂〉−6 iξ Λ′′(ξ) |ϕ̂|2ϕ̂+O(|ξ̂|〈ξ̂〉−−1−ν‖ϕ̂‖3Y ) holds for all t ≥ 1 and ξ ∈ R, where ϕ̂(t) = FU(−t)u(t), ν > 0 is small. keywords: ejde-2020/77; equation; estimate; lemma; naumkin; nonlinear; operator; order; proof; schrödinger; µ(xt−2/3; φ(0 cache: ejde-493.pdf plain text: ejde-493.txt item: #417 of 601 id: ejde-494 author: Wang, Wenbo; Li, Quanqing title: Existence and concentration of positive ground states for Schrodinger-Poisson equations with competing potential functions date: 2020 words: 7350 flesch: 86 summary: 1 2 W. WANG, Q. LI EJDE-2020/78 potential and double parameters perturbation: −ε2∆u+ V (x)u+ φu = u5 + f(u), x ∈ R3, −ε2∆φ = u2, u(x) > 0, x ∈ R3. They multiply the nonlinearity by a potential b(x), that is, −ε2∆u+ V (x)u+ φu = u5 + b(x)f(u), x ∈ R3, −ε2∆φ = u2, u(x) > 0, x ∈ R3. keywords: lemma; lim; schrödinger cache: ejde-494.pdf plain text: ejde-494.txt item: #418 of 601 id: ejde-496 author: Wang, Shaoqing; Yang, Jiazhong title: Period functions and critical periods of piecewise linear system date: 2020 words: 4097 flesch: 76 summary: Some orbits of piecewise linear system X in Section 3 In more detail, the trajectories of X = (X1, X2) in Figure 5 show that A6B4, B3B4, B3A3, A4B2, B1B2 and B1A1 all consist of sliding or escaping points (see the dashed lines in Figure 6), consequently, the closed orbits can only intersect three lines: PA5A6, A3A4 and A1A2Q. , nm, there exist two types of piecewise linear systems: one has a period annulus possessing exactly n critical periods; the other has m period annuli possessing exactly n1, n2, . . . keywords: linear; period; piecewise; system cache: ejde-496.pdf plain text: ejde-496.txt item: #419 of 601 id: ejde-499 author: D'Onofrio, Luigi title: G-convergence of elliptic operators in non divergence form in R^n date: 2023 words: 1594 flesch: 72 summary: The aim of this note is to prove a characterization of the G-limit of a sequence of elliptic operators in non-divergence form. The Dirichlet problem (1.1) has unique solution in the plane, but differently from elliptic operator in divergence form, we need extra assumptions to guarantee the solvability of (1.1). keywords: convergence; operators cache: ejde-499.pdf plain text: ejde-499.txt item: #420 of 601 id: ejde-50 author: Brooks, Robert M.; Schmitt, Klaus title: The contraction mapping principle and some applications date: 2009 words: 32542 flesch: 81 summary: αi ≤ ξi ≤ βi, 1 ≤ i ≤ N}, (B = ∏N i=1[αi, βi]), where the numbers αi, βi, 1 ≤ i ≤ N , are fixed real numbers (for each box). A positive mapping T is called homogeneous of degree p, p ≥ 0, whenever T (λu) = λpT (u), ∀λ > 0, u ∈ K. A positive mapping is called monotone provided that u, v ∈ K, u ≤ v, imply T (u) ≤ T (v). keywords: case; constant; contraction; contraction mapping; d(t; define; ejde-2009; following; function; mapping; mapping t; metric; mon; norm; point; principle; proof; sequence; set; solution; space; theorem; ∈ e cache: ejde-50.pdf plain text: ejde-50.txt item: #421 of 601 id: ejde-500 author: Pantha, Buddhi; Agusto, Folashade B.; Elmojtaba, Ibrahim M. title: Optimal control applied to a visceral leishmaniasis model date: 2020 words: 9945 flesch: 58 summary: Visceral leishmanisis; PKDL; vaccination; canine reservoir; optimal control. And we seek to find optimal controls, u∗1, u ∗ 2 and u∗3, such that J(u∗1, u ∗ 2, u ∗ 3) = min (u1,u2,u3)∈U {J(u1, u2, u3)} (4.3) where the admissible set is U = {(u1, u2, u3) ∈ (L∞(0, T ))3 : 0 ≤ ui ≤Mi;Mi ∈ R+, i = 1, 2, 3}. keywords: canine; control; disease; humans; leishmaniasis; model; number; parameters; population; rate; reservoir; sandflies; values cache: ejde-500.pdf plain text: ejde-500.txt item: #422 of 601 id: ejde-501 author: Chhetri, Maya; Girg, Petr; Hollifield, Elliott title: Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments date: 2020 words: 10929 flesch: 74 summary: We prove the existence of positive weak solution for classes of sublin- ear nonlinearities including logistic type. Next, using the method of sub- and supersolutions we establish the existence of positive weak solutions to (1.1) for classes of nonlinearities: sublinear at infinity, weighted logistic problems, and logistic problems with constant yield harvesting. keywords: a.e; fractional; solutions; theorem cache: ejde-501.pdf plain text: ejde-501.txt item: #423 of 601 id: ejde-502 author: Iagar, Razvan Gabriel; Munoz, Ana I.; Sanchez, Ariel title: Qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction date: 2023 words: 11740 flesch: 68 summary: It is also shown in [26] that such solution lies below any solution (and supersolution) to (1.5) and consequently also below any solution to (1.1) (which is a strict supersolution to (1.5)). To state our results concerning the qualitative theory of solutions to (1.1), we first have to introduce the notion of weak solution that will be used throughout the paper. keywords: cauchy; problem; reaction; solutions cache: ejde-502.pdf plain text: ejde-502.txt item: #424 of 601 id: ejde-504 author: de Lima, Henrique F.; Ramalho, Andre F. A.; Velasquez, Marco Antonio L. title: Solutions to mean curvature equations in weighted standard static spacetimes date: 2020 words: 7972 flesch: 75 summary: = 〈N,∇XY 〉 = 〈N,∇X∗Y 〉 − 〈X,Y 〉 ρ2 〈N,∇Y Y 〉 = 1 ρ 〈X,∇ρ〉〈N,Y 〉 − 1 ρ 〈X,Y 〉〈N,∇ρ〉. The metric induced on Pn from the Lorentzian metric (2.1) via Σ(z) is given by 〈, 〉z = 〈, 〉P − ρ2dz2. keywords: curvature; function; spacelike cache: ejde-504.pdf plain text: ejde-504.txt item: #425 of 601 id: ejde-505 author: Han, Bang-Sheng; Chang, Meng-Xue; Yang, Yinghui title: Spatial dynamics of a nonlocal bistable reaction diffusion equation date: 2020 words: 10305 flesch: 82 summary: satisfies − u′′ − cu′ = ku(u− u0)(u− − u) + τku2(u− φ ∗ u) ≤ ku2 ≤ kMu. Then ∣∣∣∣∣∣ b− dl2 − λ − 1 2 (k − 2b+ √ k2 − 4kb) − 1 2 (k − 2b+ √ k2 − 4kb) 3a2 σ2 − a2 σ2 − l2 0 keywords: equation; solution; u(x cache: ejde-505.pdf plain text: ejde-505.txt item: #426 of 601 id: ejde-506 author: Jin, Kun-Peng; Liang, Jin; Xiao, Ti-Jun title: Stability of initial-boundary value problem for quasilinear viscoelastic equations date: 2020 words: 5053 flesch: 83 summary: Clearly, we can rewrite the first equation in (1.1) as |ut|ρutt −∆utt − ( 1− ∫ t 0 g(s)ds ) ∆u− ∫ t 0 g(t− s) (∆u(t)−∆u(s)) = ( 1− ∫ t 0 g(s)ds )∫ Ω u∆udx+ ∫ Ω u(t) ∫ t 0 g(t− s) (∆u(t)−∆u(s)) ds dx + ∫ Ω |∇ut|2dx+ 1 ρ+ 1 ∫ Ω |ut|ρ+2dx = − ( 1− ∫ t 0 g(s)ds )∫ Ω |∇u|2dx − ∫ Ω ∇u(t) · ∫ t 0 g(t− s) (∇u(t)−∇u(s)) keywords: ∫ t; ∫ ω cache: ejde-506.pdf plain text: ejde-506.txt item: #427 of 601 id: ejde-507 author: Zhu, Jiazhen; Zhou, Jiazheng; Lin, Zhigui title: Dynamics of a diffusive competitive model on a periodically evolving domain date: 2020 words: 7441 flesch: 78 summary: Denote V 43 = M − V 42 . Noticing that V3 = M − V2, we have V 3 = M − V 2 = M = M − V 1 = V 3. keywords: problem; solution; ω(0 cache: ejde-507.pdf plain text: ejde-507.txt item: #428 of 601 id: ejde-508 author: Bhuyan, Ajit Kumar; Padhy, Laxmi Narayan; Rath, Radhanath title: Oscillatory behavior for nonlinear homogeneous neutral difference equations of second order with coefficient changing sign date: 2020 words: 6108 flesch: 79 summary: As of now, many researchers all over the world are engaged to find necessary or sufficient conditions for oscillation or non oscillation for neutral difference equations, because of its important applications in different fields of science and technology. Application to neutral difference equations with oscillating coefficients keywords: proof; solution cache: ejde-508.pdf plain text: ejde-508.txt item: #429 of 601 id: ejde-510 author: Bouhoufani, Oulia; Messaoudi, Salim A.; Zahri, Mostafa title: Existence and decay of solutions to coupled systems of nonlinear wave equations with variable exponents date: 2023 words: 7360 flesch: 80 summary: A pair of functions (u, v) is said to be a weak solution of (1.1) on [0, T ), if u, v ∈ L∞((0, T ), H1 0 (Ω)), ut, vt ∈ L∞((0, T ), L2(Ω)), ut ∈ Lm(·) α (Ω× (0, T )), vt ∈ Lr(·)β (Ω× (0, T )) and (u, v) satisfies∫ Ω utφdx− ∫ Ω u1φdx+ ∫ t 0 ∫ Ω α(τ)|ut|m(x)−2utφdx dτ + ∫ t 0 ∫ Ω ∇u.∇φdx dτ + ∫ t 0 ∫ Ω |u|p(x)−2u|v|p(x)φdx dτ = 0 and ∫ Ω vtψ dx− ∫ Ω v1ψ dx+ ∫ t 0 ∫ Ω β(τ)|vt|r(x)−2vtψ dx dτ + ∫ t 0 ∫ Ω ∇v.∇ψ dx dτ + ∫ t 0 ∫ Ω |v|p(x)−2v|u|p(x)ψ dx dτ = 0, for all φ, ψ ∈ H1 0 (Ω) and all t ∈ (0, T ), with (u(·, 0), v(·, 0)) = {w : Ω× (0, T )→ R : ∫ T 0 ∫ Ω β(τ)|w(x, τ)|r(x) dx dτ < +∞}. keywords: t s cache: ejde-510.pdf plain text: ejde-510.txt item: #430 of 601 id: ejde-511 author: Bao, Xiongxiong; Li, Ting title: Existence and stability of traveling waves for a competitive-cooperative recursion system date: 2020 words: 7724 flesch: 81 summary: If a2−b1 a1a2−b1c2 > 0, a1−c2 a1a2−b1c2 > 0 and a1a2 66= b1c2, then there is a nonnegative equilibrium (ŭ+, v̆+, 0) = ( a2 − b1 a1a2 − b1c2 , a1 − c2 a1a2 − b1c2 , 0 ) . = (Φ1(x − cn),Φ2(x − cn),Φ3(x − cn)) with speed c satisfies Φ(−∞) keywords: 1+r1; system; wave cache: ejde-511.pdf plain text: ejde-511.txt item: #431 of 601 id: ejde-512 author: Boutaous, Fatiha title: Fractional-power approach for the study of elliptic second-order boundary-value problems with variable-operator coefficients in an unbounded domain date: 2020 words: 6276 flesch: 83 summary: = − 1 2πi ∫ Γ e− √ −zx(K − zI)−1ϕdz + 1 4πi ∫ Γ ∫ x 0 e− √ −z(x−s) √ −z (1− e−2 √ −zs)(K − zI)−1f(s) ds dz + 1 4πi ∫ Γ ∫ +∞ x e− √ −z(s−x) √ −z (1− e−2 √ −zx)(K − zI)−1f(s)dsdz = exKϕ− 1 2 ∫ It follows that ‖Aλ(x)m0(x, g∗)‖X ≤ C ∫ Γ (∫ +∞ 0 e−C0(x+s)|z|ds ) ‖g∗‖C∞([0,∞);X)d|z| ≤ C (∫ Γ e−C0x|z| |z| d|z| ) ‖g∗‖C∞([0,∞);X) < keywords: + ∞; kλ(x)−; ∫ + cache: ejde-512.pdf plain text: ejde-512.txt item: #432 of 601 id: ejde-514 author: Manna, Utpal; Ashirbad Panda, Akash title: Local existence and blow-up criterion for the two and three dimensional ideal magnetic Benard problem date: 2020 words: 10077 flesch: 81 summary: For R′ > R, using the property of Fourier truncation operator provided 0 < ε < s− 1, the first term of (3.19) becomes∣∣∣((SR − SR′)[(uR · ∇)uR],uR − uR ′)∣∣∣ ≤ ‖(SR − SR′)[(uR · ∇)uR]‖L2 σ ‖uR − uR ′ ‖L2 σ ≤ C Rε ‖(uR · ∇)uR‖Hεσ‖u R − uR ′ ‖L2 σ = C Rε ‖∇ · (uR ⊗ uR)‖Hεσ‖u R − uR ′ ‖L2 σ ≤ C Rε ‖uR ⊗ uR‖Hsσ‖u R − uR ′ ‖L2 σ ≤ C Rε ‖uR‖2Hsσ‖u R − uR ′ ‖L2 σ . (3.20) ∇)uR],uR − uR ′ ) + ( SR′ [(uR ′ · ∇)(uR − uR ′ )],uR − uR ′ ) . keywords: σ(rn; ‖2l2 cache: ejde-514.pdf plain text: ejde-514.txt item: #433 of 601 id: ejde-515 author: da Silva, Severino Horacio title: Asymptotic behavior for a non-autonomous model of neural fields with variable external stimuli date: 2020 words: 6784 flesch: 71 summary: Let us fix ε > 0 and t ∈ R. Thus choose τ ∈ R, τ ≤ t, such that distH(TS0 (t, τ)B(0, R),AS0 (t)) Moreover if |f(t, x)− f(t, y)| ≤ C2(t)(1 + |x|p−1 + |y|p−1)|x− y|, (2.9) for any (x, y) ∈ RN×RN , t ∈ R, and for some strictly positive function C2 : R→ R, then, for any 1 ≤ p <∞, the function F is locally Lipschitz continuous on bounded sets with respect to the second variable. keywords: function cache: ejde-515.pdf plain text: ejde-515.txt item: #434 of 601 id: ejde-517 author: Mendoza, Renier; Keeling, Stephen title: Existence of solution for a segmentation approach to the impedance tomography problem date: 2020 words: 11701 flesch: 82 summary: Suppose Ω ⊆ Rn is a bounded domain with a sufficiently smooth bound- ary. In the forward EIT problem, given the boundary currents f ∈ L2(∂Ω) and the conductivity distribution σ ∈ L∞(Ω) satisfying σ(x) ≥ σ > 0, for all x ∈ Ω, the electric potential φ in Ω and the boundary voltage V = φ ∣∣ ∂Ω are solved. keywords: h1(ω; inequality; lemma; problem; δχ1; δχδ1; χδ1 cache: ejde-517.pdf plain text: ejde-517.txt item: #435 of 601 id: ejde-519 author: Hao, Jianghao; Lv, Mengxian title: Energy decay for variable coefficient viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary date: 2020 words: 4888 flesch: 75 summary: In this article, we study a variable coefficients viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally re- acting boundary. Variable coefficients; viscoelastic wave equation; acoustic boundary conditions; nonlocally reacting boundary. keywords: boundary; decay; equation; wave cache: ejde-519.pdf plain text: ejde-519.txt item: #436 of 601 id: ejde-52 author: Belaidi, Benharrat; Biswas, Tanmay title: Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order date: 2023 words: 6098 flesch: 82 summary: [2] B. Beläıdi; Estimation of the hyper-order of entire solutions of complex linear ordinary dif- ferential equations whose coefficients are entire functions. Since γ(r +R0) ∼ γ(r) as r → +∞, it follows that σ(α(log),β,γ)[f ′] = lim sup r→+∞ α(log[3]M(r, f ′)) β(log γ(r)) ≤ lim sup r→+∞ (α(log[3]M(r + 1, f)) keywords: β(log; γ(r; γ)[f; σ(α(log),β cache: ejde-52.pdf plain text: ejde-52.txt item: #437 of 601 id: ejde-520 author: Kurima, Shunsuke title: Time discretization of an abstract problem from linearized equations of a coupled sound and heat flow date: 2020 words: 9497 flesch: 86 summary: h2(Φλϕλ, ϕλ)H = (g, ϕλ)H − h2(Lϕλ − L0, ϕλ)H − h2(L0, ϕλ)H − ηh2(B2(I + hA1)−1ϕλ, ϕλ)H ≤ cL 2 ‖ϕλ‖2H + 1 2cL ‖g‖2H + CLh 2‖ϕλ‖2H + ‖L0‖2H 2 h2 + 1 2 h2‖ϕλ‖2H + ηCA1,B2 (h+ h2)‖ϕλ‖2H , whence the conditions (A2) and (A3), the monotonicity of B1 and Φλ imply that there exists h1 ∈ (0,min{1, h̃}) such that for all h ∈ (0, h1) there exists a constant C2 = C2(h) > 0 satisfying ‖ϕλ‖2V ≤ C2 (3.3) for all λ > 0. (4.17) Condition (A6) and Lemma 4.1 mean that there exists a constant C1 = C1(T ) > 0 such that − h (Φϕn+1 − Φϕn h , zn+1 ) H ≤ CΦh(1 + ‖ϕn+1‖pV + ‖ϕn‖qV )‖vn+1‖V ‖zn+1‖H ≤ C1h‖vn+1‖V ‖zn+1‖H (4.18) for all h ∈ (0, h2). keywords: lemma; m−1∑ cache: ejde-520.pdf plain text: ejde-520.txt item: #438 of 601 id: ejde-521 author: Barbosa, Pricila S.; Pereira, Antonio L. title: Continuity of attractors for C^1 perturbations of a smooth domain date: 2020 words: 13056 flesch: 80 summary: − w2‖Xη‖Φ‖X1/2 , where K1 is the embedding constant of X1/2 into Lq(Ω), K2 is the embedding constant of Xη in L∞(Ω) and w1(x) ≤ ξx ≤ w2(x) or w2(x) ≤ ξx ≤ w1(x). ≤ ξx ≤ w2(x). keywords: attractors; bounded; lemma; problem; theorem; ε u; ∗−1 cache: ejde-521.pdf plain text: ejde-521.txt item: #439 of 601 id: ejde-523 author: Biswas, Reshmi; Tiwari, Sweta title: Nehari manifold approach for fractional p(.)-Laplacian system involving concave-convex nonlinearities date: 2020 words: 11101 flesch: 87 summary: (5.2) Set fm(x, t) := |um − tu|α(x)−2(um − tu)|vm|β(x) gm(x, t) := |um − u|α(x)|vm − tv|β(x)−2(vm − tv). − u|α(x)|vm − v|β(x)dx = lim m→∞ ∫ Ω |um|α(x)|vm|β(x)dx− ∫ Ω |u|α(x)|v|β(x)dx (5.1) 26 R. BISWAS, S. TIWARI EJDE-2020/98 For t ∈ (0, 1), we note that∫ Ω ∫ 1 0 α(x)|um − tu|α(x)−2(um − tu)u|vm|β(x) dx dt − ∫ Ω ∫ 1 0 β(x)|um keywords: + +; lemma; m→∞; − ∫ cache: ejde-523.pdf plain text: ejde-523.txt item: #440 of 601 id: ejde-524 author: Wei, Yuanhong; Tian, Jian title: Asymptotically linear and superlinear elliptic equations with gradient terms date: 2020 words: 5343 flesch: 82 summary: Introduction This article concerns the existence of solutions for nonlinear elliptic equations with a gradient term, −∆u = f(x, u,∇u) in Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ Rn, n ≥ 1, is bounded, smooth and open with the boundary ∂Ω, f : The existence of solution was established while f satisfies the classical condition by Ambrosetti-Rabinowitz [2]: (AR) there exist ν > 2 and t0 > 0 such that 0 < νF (x, s, ξ) ≤ sf(x, s, ξ), x ∈ Ω, t ≥ t0, ξ ∈ Rn, where F (x, s, ξ) = ∫ keywords: lemma cache: ejde-524.pdf plain text: ejde-524.txt item: #441 of 601 id: ejde-526 author: Benali, Aharrouch; Jaouad, Bennouna title: Nonlinear degenerate elliptic equations in weighted Sobolev spaces date: 2020 words: 5382 flesch: 87 summary: − Tk(u)| ≤ η}; and since {x ∈ Ω : |uε α 0 φ(0, s)− φ(k, s)ds = 1 α ∫ |u| 0, 1 ≤ i ≤ N , with N ∈ N and N ≥ 2, associated with boundary conditions ∂u1 ∂x (0, t)− h0u1(0, t) ∂ui ∂x ) + ∫ t 0 gi(t− s) ∂ ∂x ( µ̄i(x, s) ∂ui ∂x (x, s) ) keywords: i=1; n∑ i=1 cache: ejde-530.pdf plain text: ejde-530.txt item: #444 of 601 id: ejde-531 author: Dore, Giovanni title: Dirichlet problem for second-order abstract differential equations date: 2020 words: 6373 flesch: 85 summary: = c1 sin (( 2`n + 1 2 ) (π − t) ) + c2 sin (( 2`n + 1 2 ) t ) . bn − bn+1 + ∞∑ n=n+1 (bn+1 − bn) = bn − b1 + bn − bn+1 + lim n→∞ bn − bn+1 ≤ 2bn − 2bn+1 ≤ 4 k − (1/4) . keywords: problem cache: ejde-531.pdf plain text: ejde-531.txt item: #445 of 601 id: ejde-532 author: Deng, Jin; Xia,  Aliang; Yang, Jianfu title: Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential date: 2020 words: 7953 flesch: 87 summary: = Eβn(un, vn) ≤ I∞ for all n ∈ N. So, after passing to a subsequence, there exist u∞, v∞ ∈ H such that (un, vn) ⇀ (u∞, v∞) weakly in H, (un, vn)→ (u∞, v∞) strongly in L4(R2)× L4(R2), (un, vn)→ (u∞, v∞) a.e. in R2 × R2. ∇v − v∆θ = 0, x ∈ RN , u, v ≥ 0, x ∈ RN . (1.4) If we assume u(x) = u(|x|) and choose the angular coordinate in R2 as phase function, see [3, 4], that is, θ(x) :=  arctan x2 x1 , if x1 > 0, π + arctan x2 x1 , if x1 < 0, π/2, if x1 = 0 and x2 > 0, −π/2, if x1 = 0 and x2 < 0, (1.5) we obtain ∆θ = 0, ∇θ · ∇u = 0, |∇θ|2 = 1 |x|2 , EJDE-2020/108 SCHRÖDINGER SYSTEM WITH SINGULAR POTENTIAL 3 and the system reduces to −∆u+ λ1u+ k2 0 u |x|2 = µ1u 3 + βuv2, x ∈ R2, −∆v + λ2v + k2 0 v |x|2 = µ2v 3 + βu2v, x ∈ R2, u, v ≥ 0, x ∈ R2. keywords: |x|2 cache: ejde-532.pdf plain text: ejde-532.txt item: #446 of 601 id: ejde-534 author: Ahmad, Bashir; Alsaedi, Ahmed; Berbiche, Mohamed; Kirane, Mokhtar title: Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations date: 2020 words: 11024 flesch: 87 summary: = N 2 γ1( 1 r1 − 1 s1 ), σ1 + γ1 − N 2 γ1 ( p s2 − 1 s1 ) − pσ2 = 0, σ1 + γ1 − N 2 γ1 ( p s2 − 1 s1 ) + ( γ2 − N 2 γ2 ( q s1 − 1 s2 ) − qσ1 ) p = 0, σ1 + γ1 − γ1δ + (γ2 − γ2δ − qσ1)p = 0. pq pq − 1 + 4 γ1 (γ1 − 1), (−4 1 q + ( 4 γ1 + 2N) 1 p′q − 4 + ( 4 γ1 + 2N) 1 q′ ) pq pq − 1 + 4 γ1 (γ1 − 1) } , and δ2 = max { (− 4 γ1 γ2 + ( 4 γ1 + 2N) 1 p′ − 4 1 p + ( 4 γ1 + 2N) 1 pq′ ) pq pq − 1 + 4 γ1 (γ2 − 1), (−4 + ( 4 γ1 + 2N) 1 p′ − 4 1 p + ( 4 γ1 + 2N) 1 pq′ ) pq pq − 1 + 4 γ1 (γ2 − 1) } . keywords: solutions; time cache: ejde-534.pdf plain text: ejde-534.txt item: #447 of 601 id: ejde-535 author: Dahan Kassim, Mohammed; Eddine Tatar, Nasser title: Convergence of solutions of fractional differential equations to power-type functions date: 2020 words: 5278 flesch: 84 summary: Asymptotic behavior; boundedness; fractional differential equation; Caputo fractional derivative; Riemann-Liouville fractional derivative. [21] M. Medveď; Asymptotic integration of some classes of fractional differential equations, Tatra Mt. Math. keywords: fractional cache: ejde-535.pdf plain text: ejde-535.txt item: #448 of 601 id: ejde-536 author: Ma, Luyi; Niu, Hong-Tao; Wang, Zhi-Cheng title: Pyramidal traveling fronts in the Belousov-Zhabotinskii reaction-diffusion systems in R^3 date: 2020 words: 11238 flesch: 89 summary: = (u1(1 − r − u1 + ru2), bu1(1 − u2)). v−2 (x) = U2 ( c s (x3 − y3 + h(x′ − y′)) ) keywords: fronts; i=1,2; j=1; lemma; lim; pyramidal; sup; vi(x; v−2; x))βi cache: ejde-536.pdf plain text: ejde-536.txt item: #449 of 601 id: ejde-537 author: Qiu, Haijing; Wang, Yan title: Continuous dependence of recurrent solutions for stochastic differential equations date: 2020 words: 3762 flesch: 75 summary: [6] and Ji et al [12] for periodic solutions for SDEs, see Halanay [10], Da Prato and Tudor [6] F. Chen, Y. Han, Y. Li, X. Yang; Periodic solutions of Fokker-Planck equations, J. Differen- tial Equations, 263 (2017), 285–298. keywords: equations; solutions; sup cache: ejde-537.pdf plain text: ejde-537.txt item: #450 of 601 id: ejde-539 author: Pardo, Rosa; Sanjuan, Arturo title: Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth date: 2020 words: 6717 flesch: 87 summary: (i) From (2.5), Lemma 2.1, and (4.1) with t = T and r = 2∗ − 1, we have y′α(T ) [ln(e+ u)]α , p∗ = Np N − p , α > p (N − p) ; see [7]. keywords: lemma; lim; log(e+; n−2 cache: ejde-539.pdf plain text: ejde-539.txt item: #451 of 601 id: ejde-540 author: Heidarkhani, Shapour; Gharehgazlouei, Fariba; Imbesi, Maurizio title: Existence and multiplicity of homoclinic solutions for a difference equation date: 2020 words: 4836 flesch: 75 summary: = 1 p ‖u‖p − ∑ k∈Z H(u(k)) ∀u ∈ X, (2.2) Ψ(u) := ∑ k∈Z F (k, u(k)) ∀ u ∈ lp (2.3) where F (k, t) = ∫ t 0 f(k, ξ)dξ for t ∈ R and k ∈ Z, H(t) = ∫ t 0 h(ξ)dξ for t ∈ R. Let Iλ : X → R be the energy functional associated to the problem (1.1) defined by Iλ(u) = λf(k, u(k)) + sin4( u(k) 2 ) ∀k ∈ Z, u(k)→ 0 as |k| → ∞. For all (k, t) ∈ Z× R put f(k, t) keywords: solutions; theorem cache: ejde-540.pdf plain text: ejde-540.txt item: #452 of 601 id: ejde-541 author: Adhikari, Dhruba R.; Stachura, Eric title: General p-curl systems and duality mappings on Sobolev spaces for Maxwell equations date: 2020 words: 9706 flesch: 79 summary: (2.7) Thus, we see that W 0 N = W 1,p 0 (curl,Ω) ∩W Thus, we conclude that u ∈ W 1,p 0 (curl,Ω) (note that ũ(k) ∈ (C∞0 (Ω)) 3 for each k). keywords: 1,p; curl; domain; theorem cache: ejde-541.pdf plain text: ejde-541.txt item: #453 of 601 id: ejde-543 author: Ding, Yuanlin; Feckan, Michal; Wang, Jinrong title: Stability for conformable impulsive differential equations date: 2020 words: 7331 flesch: 86 summary: = g(t, y(t)), t ∈ I := [a, b]\{t1, . . . ,m, y(t) = ξy(t−k ), t ∈ (tk, sk], k = 1, 2, . . . keywords: a)β; tn(a cache: ejde-543.pdf plain text: ejde-543.txt item: #454 of 601 id: ejde-544 author: Yang, Lu; Liu, Xiangqing; Zhou, Jianwen title: Concentration of nodal solutions for semiclassical quadratic Choquard equations date: 2023 words: 8015 flesch: 91 summary: j,ε,p|1 ≤ j ≤ k} such that Γ(λ) ε,p (u (λ) j,ε,p) = cj(ε, p, λ) ≤ c̃k, By Corollary 4.3, we have p− 1 2 ∫ R3 ( 1 | · | ∗ (|un,p|p))|un,p|p−2ϕ2 dx ≤ m∑ i=1 c ∫ R3 ( 1 | · | ∗ e−c|x−yn,i|)ϕ2 dx and p 2 ∫ R3 ( 1 | · | ∗ (|un,p|p−2un,pϕ))|un,p|p−2un,pϕdx ≤ m∑ i=1 c ∫ R3 ( 1 | · | ∗ (e−c|x−yn,i|ϕ) ) e−c|x−yn,i|ϕdx + m∑ i 6=j c ∫ R3 ( 1 | · | ∗ (e−c|x−yn,j |ϕ) ) e−c|x−yn,i|ϕdx ≤ m∑ i=1 c ∫ R3 ( 1 | · | ∗ (e−c|x−yn,i|ϕ) ) e−c|x−yn,i|ϕdx+ o(1) keywords: lemma; ∫ r3 cache: ejde-544.pdf plain text: ejde-544.txt item: #455 of 601 id: ejde-545 author: Rani, Anu; Goyal, Sarika title: Polyharmonic systems involving critical nonlinearities with sign-changing weight functions date: 2020 words: 11556 flesch: 86 summary: = (2− β − γ)t1−β−γ‖(u, v)‖2 − (r − β − γ)tr−β−γ−1Qλ,µ(u, v), we obtain ξ ′ (u,v)(t) Moreover, ξ(u,v)(tmax) = ( (2− r)‖(u, v)‖2 (β + γ − r) ∫ Ω h(x)|u|β |v|γdx ) 2−r β+γ−2 ‖(u, v)‖2 − ( (2− r)‖(u, v)‖2 (β + γ − r) ∫ Ω h(x)|u|β |v|γdx ) β+γ−r β+γ−2 ∫ Ω h(x)|u|β |v|γdx = ‖(u, v)‖r( 2− r β + γ − r ) 2−r β+γ−2 (β + γ − 2 β + γ − r )( ‖(u, v)‖β+γ∫ Ω h(x)|u|β |v|γdx ) 2−r β+γ−2 ≥ ‖(u, v)‖r( 2− r β + γ − r ) 2−r β+γ−2 (β + γ − 2 β + γ − r )(S β+γ 2 |h|∞ ) 2−r β+γ−2 . keywords: 2−r; λ,µ; − r cache: ejde-545.pdf plain text: ejde-545.txt item: #456 of 601 id: ejde-546 author: Devi, Darshana; Chutia, Duranta; Haloi, Rajib title: Rothe's method for solving semi-linear differential equations with deviating arguments date: 2020 words: 3933 flesch: 79 summary: = 1 Γ(α) ∫ t 0 u(s) (t− s)1−α ds+ f(t), t ∈ (0, T ], u(0) = u0, where 0 < α < 1, -A is the infinitesimal generator of a C0-semigroup of contractions, f is a given map from [0, T ] to X, and the initial point u0 ∈ D(A) ⊂ X, the domain of A. Dubey = f(t, u(t), ut), t ∈ (0, T ], h(u0) = φ on [−τ, 0], where 0 < T < ∞, φ ∈ C0 := C([−τ, 0];X), τ > 0, the nonlinear operator A is single-valued and m-accretive defined from the domain D(A) ⊂ X to X, the nonlinear map f is defined from [0, T ]×X×C0 to X, the map h is defined from C0 to C0. keywords: method cache: ejde-546.pdf plain text: ejde-546.txt item: #457 of 601 id: ejde-547 author: Mansouri, Sabeur; Tebou, Louis title: Stabilization of coupled thermoelastic Kirchhoff plate and wave equations date: 2020 words: 6167 flesch: 79 summary: Consider the coupled thermoelastic Kirchhoff plate/wave system ytt − γ∆ytt + a∆2y + α∆θ + µz = 0 in Ω× (0,+∞), θt − σ∆θ − β∆yt = 0 in Ω× (0,+∞), ztt − η∆z + µy = 0 in Ω× (0,+∞), y = ∂νy = 0, θ = z = 0, on Γ× (0,+∞), y(x, 0) = y0, yt(x, 0) = y1, θ(x, 0) = θ0 in Ω, z(x, 0) = z0, zt(x, 0) = z1 in Ω, (1.1) where a, η, γ, σ are positive physical constants representing respectively, the flexural stiffness of the plate, wave speed, rotational force constant, and thermal conductiv- ity, while µ denotes the coupling parameter, and is a nonzero real number. We introduce the Hilbert space over the field C of complex numbers Hγ := H2 0 (Ω)×H1 0 (Ω)× L2(Ω)×H1 0 (Ω)× L2(Ω), EJDE-2020/121 THERMOELASTIC KIRCHHOFF PLATE AND WAVE EQUATIONS 3 equipped with the norm ‖U‖2Hγ := a‖∆u‖2 + γ‖∇v‖2 + ‖v‖2 + α β ‖θ‖2 + η‖∇y‖2 + ‖z‖2 + 2µ ∫ Ω Re(uy)dΩ, (1.3) for all U = (u, v, θ, y, z) ∈ Hγ . keywords: equations; plate; system; thermoelastic; ‖z‖1/2γ cache: ejde-547.pdf plain text: ejde-547.txt item: #458 of 601 id: ejde-548 author: Yan, Jianlu; Li, Yuxiang title: Existence and boundedness of solutions for a Keller-Segel system with gradient dependent chemotactic sensitivity date: 2020 words: 5571 flesch: 84 summary: Finally, for arbitrary non-negative ψ ∈ C∞0 (Ω̄× [0,∞)), multiplying the second equation in (2.7) by ψ, and integrating by parts, we have∫ Ω v0ψ(·, 0) + ∫ T 0 ∫ Ω vεψt = ∫ T 0 ∫ Ω ∇vε · ∇ψ + ∫ T 0 ∫ Ω vεψ − ∫ T 0 ∫ Ω uεψ (3.19) for all ε ∈ (0, 1). − χ∇ · ( u∇v√ 1 + |∇v|2 ) , 0 = ∆v −M + u, (1.5) where M = 1 |Ω| ∫ Ω u0(x)dx, n ≥ 2 and χ < 1. keywords: ∫ ω cache: ejde-548.pdf plain text: ejde-548.txt item: #459 of 601 id: ejde-55 author: Bu, Zhen-Hui; Wang, Chen-Lu; Zhang, Xin-Tian title: Pyramidal traveling fronts of a time periodic diffusion equation with degenerate monostable nonlinearity date: 2023 words: 8914 flesch: 85 summary: In addition, there exist positive constants L1, L2, L3, β1 such that L1e Λ2ξ ≤ Ψ(ξ, t), Ψξ(ξ, t), |Ψξξ(ξ, t)| ≤ L2e Λ2ξ, ∀ξ < 0, t ∈ R, (1.4) |Ψ(ξ, t)− 1|, Ψξ(ξ, t), |Ψξξ(ξ, t)| ≤ L3e −β1ξ, ∀ξ > 0, t ∈ R. (1.5) Since limϑ→−∞Ψ(ϑ, t) = 0 uniformly for t ∈ keywords: fronts; i=1; periodic; ψ(x cache: ejde-55.pdf plain text: ejde-55.txt item: #460 of 601 id: ejde-550 author: Mahdi, Achache; Hossni, Tebbani title: Maximal regularity for non-autonomous Cauchy problems in weighted spaces date: 2020 words: 8836 flesch: 91 summary: (4.7) Integrating by parts, we obtain for t ∈ [0, τ ] and f ∈W 1,2 β,0(0, τ,H) A(0)u(t) = A(0) ∫ t 0 e−(t−s)A(0)f(s) ds = f(t)− ∫ t 0 e−(t−s)A(0)ḟ(s) ds = u̇(t) +A(0)u(t)− ∫ t 0 e−(t−s)A(0)ḟ(s) ds. Let x ∈ H and t ∈ keywords: β(0; τ 0 cache: ejde-550.pdf plain text: ejde-550.txt item: #461 of 601 id: ejde-551 author: Cabanillas Lapa, Eugenio title: Global solutions for fractional viscoelastic equations with logarithmic nonlinearities date: 2020 words: 5414 flesch: 83 summary: = 1 2 (g′ � u)(t)− 1 2 g(t)‖u(t)‖2W0 − ‖ut(t)‖22 − a ∫ Ω |u(t)|r(x)−2u2 t (t) dx − ε‖u(t)‖2αW0 − ε‖u(t)‖2W0 + ε ∫ Ω uf(u) log |u(t)| dx + ε ∫ t 0 g(t− τ)〈u(τ), u(t)〉W0 dτ. (3.36) + ∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx ≤ Em(0). keywords: log cache: ejde-551.pdf plain text: ejde-551.txt item: #462 of 601 id: ejde-552 author: Bauer, Sean; Petrov, Nikola P. title: Existence of KAM tori for presymplectic vector fields date: 2020 words: 12145 flesch: 79 summary: = K∗θ ωθ, where K∗θ : TθTd+n → TK(θ)P is the derivative of K at θ ∈ Td+n and we consider ω ∈ Rd+n as ωθ ∈ TθTd+n = Rd+n. ∈ TK(θ)K ⊆ TK(θ)P is the value of Vλ̄ at K(θ) ∈ K, ωθ ∈ TθTd+n is the Diophantine vector ω considered as an element of TθTd+n = Rd+n, and K∗θ : keywords: basis; form; invariant; kam; kj(θ; lemma; matrix; section; td+n; vector cache: ejde-552.pdf plain text: ejde-552.txt item: #463 of 601 id: ejde-553 author: da Silva Almeida Junior, Dilberto; Araujo Ramos, Anderson de Jesus; Pantoja Fortes, Joao Carlos; de Lima Santos, Mauro title: Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations date: 2020 words: 11001 flesch: 83 summary: = ∫ L 0 xφxφt dx and from where results that Xφ(t) ∣∣T 0 + ∫ T 0 F (t)dt = α ∫ T 0 ∫ = ∫ L 0 xψxψt dx and taking into account the energy defined by (2.45), we obtain Xψ(t) ∣∣T 0 + ∫ T 0 keywords: j=0 ∫; observability; ∫ l; ∫ t cache: ejde-553.pdf plain text: ejde-553.txt item: #464 of 601 id: ejde-555 author: Li, Fengbai; Wang, Weike; Wang, Yutong title: Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms date: 2020 words: 8330 flesch: 85 summary: For Π2,1, let Ω = [0, t]× Rn, Ω1 = Ω ∩ { t2 < τ ≤ t}, Ω2 = Ω ∩ {0 ≤ τ ≤ t 2}. Since 1 + |x|2 t ≤ { 2, |x|2 ≤ t, 2 |x| 2 t , |x| 2 ≥ t, we have |Dα xG2(t, x)| ≤ Ce− t 2m0BN (t, |x|), which completes the proof. keywords: c(1; div; estimate; solution; wang cache: ejde-555.pdf plain text: ejde-555.txt item: #465 of 601 id: ejde-558 author: Cely, Liliana title: Stability of ground states of nonlinear Schrodinger systems date: 2023 words: 8546 flesch: 88 summary: = 1 r‖∂xφ‖L∞ ≤ Cε; (ii) Since φr is identically 1 on |x − yn| ≤ r and φr vanishes on |x − yn| ≥ 2r, then |φr|2 log |φr|2 vanishes on both |x − yn| ≤ r and |x − yn| ≥ 2r. [0, η]× [0, ζ] such that γ = η + ζ and J(µ1, µ2) + J(η − µ1, ζ − µ2) ≤ J(η, ζ). keywords: j(η; lemma; log cache: ejde-558.pdf plain text: ejde-558.txt item: #466 of 601 id: ejde-560 author: Ahmad, Israr; Nieto, Juan Jose; Rahman, Ghaus ur; Shah, Kamal title: Existence and stability for fractional order pantograph equations with nonlocal conditions date: 2020 words: 6239 flesch: 82 summary: − δ1 − µ1)| , K2 |λ2 − δ2 − µ2)| ) . − δ1 − µ1| ‖x1 − x̄1‖ , ‖H2x2 −H2x̄2‖ ≤ K2 |λ2 − (δ2 + µ2)| ‖x2 − x̄2‖ . keywords: differential; equations; fractional; |λ1 cache: ejde-560.pdf plain text: ejde-560.txt item: #467 of 601 id: ejde-561 author: Zhang, Jiangwei; Xie, Zhe; Xie, Yongqin title: Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter date: 2024 words: 10679 flesch: 78 summary: Assume that U(t, τ) is a process on {Mt}t∈R and it has a pull- back bounded absorbing set B̃ = {Bt}t∈R. U(t, τ) is called Mt-contractive process if for any given ε > 0, there exist T = T (ε) and φt T (·, ·) Hence, for each ε > 0, there exists δ = ε 4 such that 2∥U(t, T )z1T − U(t, T )z2T ∥2Mt < ε, (3.68) holds for any t ≥ T = T (ε) fixed. keywords: attractors; equation; lemma; memory; process; time; u(t; xie; ∫ ∞ cache: ejde-561.pdf plain text: ejde-561.txt item: #468 of 601 id: ejde-562 author: Perez-Lopez, Jhean E.; Rueda-Gomez, Diego A.; Villamizar-Roa, Elder J. title: Existence of global solutions for cross-diffusion models in a fractional setting date: 2023 words: 6817 flesch: 79 summary: For 1 ≤ p < N − λ, 1 α < κ and s∗ = N−λ p , we define the Banach spaces X1 and X2 by X1 = L̃∞([0,∞);N s∗−θ p,λ,∞) ∩ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ), X2 = L̃∞([0,∞);N s∗ p,λ,∞), (2.8) 6 J. E. PÉREZ-LÓPEZ, D. A. RUEDA-GÓMEZ, E. J. VILLAMIZAR-ROA EJDE-2023/77 endowed with the corresponding norms ‖x‖X1 = ‖x‖ L̃∞([0,∞);N s∗−θp,λ,∞) + ‖x‖ L̃κ([0,∞);N s∗−θ(1− 1 ακ ) p,λ,∞ ) , ‖x‖X2 = ‖x‖ L̃∞([0,∞);N s∗p,λ,∞) . N − λ, s∗ = N−λ p , and [n0,m0, v0] ∈ N s∗−θ p,λ,∞ × N s∗−θ p,λ,∞ × N s∗ p,λ,∞. keywords: fractional; lemma cache: ejde-562.pdf plain text: ejde-562.txt item: #469 of 601 id: ejde-563 author: Melo, Wilberclay G.; Rocha, Nata F.; Costa, Natielle dos Santos title: Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces date: 2023 words: 5554 flesch: 84 summary: Assume that f ∈ X sa,σ(R3). [−1, 0)× (1,+∞) ) ∪ ( [0,+∞)×{0}× [1,+∞) ) and u0 ∈ X sa,σ(R3). keywords: equations; x sa cache: ejde-563.pdf plain text: ejde-563.txt item: #470 of 601 id: ejde-564 author: Wang, Lulu; Ma, Qiaozhen title: Uniform attractors of non-autonomous suspension bridge equations with memory date: 2024 words: 6597 flesch: 81 summary: = 0, y ∈ (−l, l), t ≥ τ, uyy(x,±l, t) + = 0, x ∈ (0, π), t ≥ τ, (1.9) ηt(0, y, s) = ηtxx(0, y, s) = ηt(π, y, s) = ηtxx(π, y, s) = 0, y ∈ (−l, l), s ∈ R+, ηtyy(x,±l, s) + keywords: bridge; suspension cache: ejde-564.pdf plain text: ejde-564.txt item: #471 of 601 id: ejde-568 author: Song, Fei; Wang, Yuping; Akbarpoor, Shahrbanoo title: Inverse nodal problems for Dirac operators and their numerical approximations date: 2023 words: 5768 flesch: 78 summary: Inverse nodal problems consist in recovering the potential Q(x) and the coefficients α, β from the given subsets of the nodal points (zeros of eigenfunctions). This class of inverse nodal problems has been studied for the Sturm-Liouville operator keywords: inverse; nodal; solution cache: ejde-568.pdf plain text: ejde-568.txt item: #472 of 601 id: ejde-57 author: Treinen, Raymond title: Discussion of a uniqueness result in "Equilibrium Configurations for a Floating Drop" date: 2023 words: 4082 flesch: 74 summary: Then we use Chebyshev spectral methods to approximate solutions to certain boundary value problems used to check this hypothesis holds at least on a range of cases. Consider the intersection points of these curves with r = ρ0. keywords: liquid; problem cache: ejde-57.pdf plain text: ejde-57.txt item: #473 of 601 id: ejde-571 author: Chen, Wei title: Variety of solutions and dynamical behavior for YTSF equations date: 2023 words: 5922 flesch: 77 summary: [22] Z. Y. Yan; New families of nontravelling wave solutions to a new (3+1)-dimensional potential- YTSF equation, Phys. In this section we will study the interaction between two-lump solution and soliton wave solution 4 W. CHEN EJDE-2023/82 (a) u0 = −2 (b) u0 = −1.5 (c) u0 = −1 (d) u0 = 0 (e) u0 = 1 (f) u0 = 1.5 (g) u0 = −2,−1.5,−1 (h) u0 = 0, 1, 1.5 (i) u0 = −1.5, 1.5 Figure 1. keywords: equation; lump; solution; wave; ytsf cache: ejde-571.pdf plain text: ejde-571.txt item: #474 of 601 id: ejde-572 author: Jia, Man; Su, Youfeng; Chen, Hebai title: Global analysis on a continuous planar piecewise linear differential system with three zones date: 2023 words: 23028 flesch: 82 summary: = {(α, tc) ∈ R2 : α = −dc}, BE12 = {(α, tc) ∈ R2 : α = dc}. 4 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (b) Homoclinic bifurcation curves: HL11 = {(α, tc) ∈ R2 : −dc < α ≤ dc, tc = φ(α)}, HL12 = {(α, tc) ∈ R2 : α > dc, tc = ϕ(α)}. (c) Double limit cycle bifurcation curve: DL1 = {(α, tc) ∈ R2 : α > dc, tc = h(α)}, where the function tc = φ(α) is continuous, monotonic and satisfies max{t∗c ,−2 √ dc} < φ(α) < 0 for − dc < α < dc, max{t∗c ,−2 √ dc} < φ(α) < < α < α∗ < dc, tc = φ(α)}, HL112 = {(α, tc) ∈ R2 : −dc keywords: cycle; homoclinic; limit; linear; system; φ(α; ∈ r2 cache: ejde-572.pdf plain text: ejde-572.txt item: #475 of 601 id: ejde-573 author: Li, Yang; Chen, Guiling title: Existence of periodic solutions and stability for a nonlinear system of neutral differential equations date: 2024 words: 6647 flesch: 87 summary: ∫ t 0 K−1(s) +G(s, x(s), x(s− τ2(s)))] ds + ∫ t 0 K−1(T )K−1(s)A(s) keywords: s−τ1(s; t−τ1(t; x(t−; − ∫; ∫ t cache: ejde-573.pdf plain text: ejde-573.txt item: #476 of 601 id: ejde-575 author: Tu, Jin; Wei, Huizhen title: Form of solutions to quadratic trinomial partial differential equations with two complex variables date: 2024 words: 10109 flesch: 88 summary: Similarly, we obtain f(t, s) = ∫ t 0 [ α2k1 2 √ 2(1 + α) + (α2 + 4)k2 2 √ 2(1− α) ] eg(z)/2dt+ ϕ0(s), (5.14) where ϕ0(s) is a finite order transcendental entire function in s = z2 − z1. Similarly, we obtain f(t, s) = ∫ t 0 1√ 2 [ (A1 −A2 +A1B12)e (B11+B12)t+B12s+β1 + (A2 −A1 +A2B22)e (B21+B22)t+B22s+β2 ] dt+ ϕ0(s), (5.28) where ϕ0(s) is a transcendental entire function with finite order in s = z2 − z1. keywords: + √; b21; −a2 + cache: ejde-575.pdf plain text: ejde-575.txt item: #477 of 601 id: ejde-58 author: Jia, Xiaobiao; Ma, Shanshan title: Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space date: 2023 words: 5251 flesch: 83 summary: − z|1+α ≤ dα(y, z) ≤ C|y − z|, (2.1) dα(y, z) ∼ |y − z| if y, z ∈ B+ 1 ∩ { xn ≥ 1 8 } . ,∣∣∣u( 1 2k h 1 2(1+α) en ) − u ( 1 2k+1 h 1 2(1+α) en )∣∣∣ ≤ C2−k−1h 1 2(1+α) , This implies that∣∣u( 1 2h 1 2(1+α) en ) − u(0) ∣∣ h 1 2(1+α) ≤ ∞∑ k=1 ∣∣u( 1 2k h 1 2(1+α) en ) keywords: |x′|2; βx2 cache: ejde-58.pdf plain text: ejde-58.txt item: #478 of 601 id: ejde-580 author: Avila, Jake; Cabarrubias, Bituin title: Periodic unfolding method for domains with very small inclusions date: 2023 words: 15806 flesch: 82 summary: | uδ,ε1 ∈ V δ,εp , uδ,ε2 ∈W 1,p(Ωδ,ε2 ) } , (3.4) equipped with the norm, ‖uδ,ε‖p Hδ,εγ,p = ‖∇uδ,ε1 ‖ p Lp(Ωδ,ε1 ) + ‖∇uδ,ε2 ‖ p Lp(Ωδ,ε2 ) + εγ‖uδ,ε1 − u δ,ε 2 ‖ p Lp(Γδ,ε) . Then there exists a subsequence (still denoted by ε), u1 ∈ H1 0 (Ω) and û1 ∈ L2(Ω;H1 per(Y1)) such that T ε1 (uε1)→ u1 strongly in L2(Ω;H1(Y1)), (2.2) T ε1 (∇uε1) ⇀ ∇u1 +∇yû1 weakly in L2(Ω× Y1), (2.3) with MΓ(û1) = 0 for almost every x keywords: t δ; theorem cache: ejde-580.pdf plain text: ejde-580.txt item: #479 of 601 id: ejde-581 author: Wu, Wanping; Zhang, Yinghui title: Global low-energy weak solutions for compressible magneto-micropolar fluids with discontinuous initial data in R^3 date: 2023 words: 12504 flesch: 82 summary: = sup 1≤s≤t ( ‖∇u‖2L2 + ‖w‖2L2 + ‖∇w‖2L2 + ‖∇H‖2L2 ) + sup 1≤s≤t ( ‖u̇‖2L2 + ‖ẇ‖2L2 + ‖∇W1‖2L2 + ‖∇W2‖2L2 + ‖Ht‖2L2 ) + ∫ t 1 ( ‖u̇‖2L2 + ‖ẇ‖2L2 + ‖∇W1‖2L2 + ‖∇W2‖2L2 + ‖Ht‖2L2 ) ds + ∫ t 1 (‖∇u̇‖2L2 + ‖∇ẇ‖2L2 + ‖∇Ht‖2L2) ds, (3.3) Oq(t) = sup 0≤s≤t (‖u‖qLq + ‖w‖qLq + ‖H‖qLq ) + ∫ t 0 ∫ R3 ( |u|q−2|∇u|2 + |w|q−2|∇w|2 + |H|q−2|∇H|2 ) dxds + ∫ t 0 ∫ R3 ( |u|q−4|∇(|u|2)|2 + |w|q−4|∇(|w|2)|2 + |H|q−4|∇(|H|2)|2 ) Later, to deal with the terms 2ζ ∫ t 0 ∫ R3 ϑ rotw · u̇dx ds and 2ζ ∫ t 0 ∫ R3 ϑ rotu ·ẇ dxds, we use integration by parts to obtain 2ζ ∫ t 0 ∫ R3 ϑ rotw · u̇dxds+ 2ζ ∫ t 0 ∫ R3 ϑ rotu · keywords: dxds; energy; q−2; solutions; sup; − ∫; ∫ r3; ∫ t; ∫ t1 cache: ejde-581.pdf plain text: ejde-581.txt item: #480 of 601 id: ejde-587 author: Hao, Jianghao; Wang, Dingkun title: Asymptotic stabilization for Bresse transmission systems with fractional damping date: 2023 words: 16392 flesch: 81 summary: The system is written as ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) + γ1(−∂xx)θϕt = 0 in (0, L)× R+, ρ2ψtt [40] got that when there are two locally distributed feedbacks on the shear angular displacement and longitudinal displacement, ρ1ϕtt − κ(ϕx + ψ + lw)x − κ0l(wx − lϕ) = 0 in (0, L)× R+, ρ2ψtt keywords: + 4θ; c‖e; eα∗; system; ψ̃‖2; ϕ̃‖2; ‖f‖2; ‖f‖‖u‖+ cache: ejde-587.pdf plain text: ejde-587.txt item: #481 of 601 id: ejde-589 author: Lu, Shuaishuai; Yang, Xue title: Stability and rate of decay for solutions to stochastic differential equations with Markov switching date: 2024 words: 6682 flesch: 78 summary: = 4k22t ∧ 4k23t, we obtain ‖Vx(x, t, i)g(x, t, i)‖2 ≥ h2(t)V 2(x, t, i) for all i ∈ {1, 2}. Therefore, (2.1) can be rewritten as the result of the following N equations: dx(t) = f(x(t), t, i)dt+ g(x(t), t, i)dW (t), t ≥ 0, 1 ≤ i ≤ N. (2.2) keywords: decay; stability; x(t cache: ejde-589.pdf plain text: ejde-589.txt item: #482 of 601 id: ejde-59 author: Chen, Ye-Jun; Ding, Hui-Sheng title: Pseudo almost periodicity for stochastic differential equations in infinite dimensions date: 2023 words: 5802 flesch: 79 summary: A family of measurable mappings on the sample space, θt : Ω→ Ω, t ∈ R, is called a measurable dynamical system if the following conditions are satisfied: Let X : R× Ω→ B be a θp-almost periodic random process, and let J be a compact interval of R. Then (i) the set LJ = {X (s+ t, θ−t·) : s ∈ J, t ∈ R} is relatively compact in Lp(Ω,B), (ii) the set S = {law(X(t, ·)) : t ∈ R} is uniformly tight, that is, for each ε > 0, there exists a compact subset K of B such that sup t∈R P ({ω ∈ Ω : X(t, ω) /∈ keywords: periodic; pseudo; stochastic cache: ejde-59.pdf plain text: ejde-59.txt item: #483 of 601 id: ejde-590 author: Lou, Zhaowei; Wu, Youchao title: A KAM theorem for degenerate infinite-dimensional reversible systems date: 2024 words: 8376 flesch: 79 summary: The map Φ = ΦtF |t=1 defined above transforms X into X+ = Φ∗X = N+ + P+ on D(s− 5σ, ηr), where N+ = N + N̂ , P+ = (Φ1 F )∗(P −R) + ∫ 1 0 (ΦtF )∗[R(t), F ]dt, with R(t) − iΩ(ξ)z̄ ∂ ∂z̄ , (5.13) P = ∑ w∈{θ,I,z,z̄} Pw(θ, I, z, z̄; ξ) ∂ ∂w (5.14) with ωb = λjb − 1 4 n∑ k=1 λ−1 jk ajkjkjbjbξk, Ωj = λj − 1 4 n∑ k=1 λ−1 jk ajkjkjjξk (5.15) P (θb) = −1 4 n∑ k=1 λ−1 jk ajkjkjbjbIk − 1 4 ∑ l∈N1 λ−1 l alljbjb |zl|2 + (Q̂(qjb ) +K(qjb )) keywords: d(s; field; kam; lemma; systems; vector cache: ejde-590.pdf plain text: ejde-590.txt item: #484 of 601 id: ejde-592 author: Diaz, Jesus Ildefonso; Shaposhnikova, Tatiana A.; Podolskiy, Alexander V. title: Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size date: 2024 words: 12889 flesch: 75 summary: and we have ε−γ ∫ T 0 ∫ Sε ∂tuε(φ− uε) ds dt+ ∫ T 0 ∫ Ωε ∇uε∇(φ− uε) ds dt = ∫ T 0 ∫ Ωε f(φ− uδε) dx dt. keywords: \gj ε; j ε; j ε/4; l2(0; problem; t j; δ ε; ε−γ ∫; ∫ t; ∫ ∂gj cache: ejde-592.pdf plain text: ejde-592.txt item: #485 of 601 id: ejde-593 author: Xu, Hong Yan; Haldar, Goutam title: Entire solutions to Fermat-type difference and partial differential-difference equations in C^n: System of Fermat-type difference equations in $ \mathbb{C}^n $ date: 2024 words: 9037 flesch: 88 summary: − ( ia3e −ik − a1 ) e−ike2ip1(z) = a1e −ik + ia3. (3.26) Let us choose a1 = a2 = a3 = 1, a4 = −1, L(z) = i(z1 − z2), Φ(t) = i(c2z1−c1z2) 5, A = 3, and c = (c1, c2) ∈ C2 such that c1+2c2 = (2m−1/2)π, m being an integer. keywords: difference; equations; fermat; ia2; p1(z; type cache: ejde-593.pdf plain text: ejde-593.txt item: #486 of 601 id: ejde-594 author: Li, Yuequn; Liu, Hui; Guo, Fei title: Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion date: 2024 words: 10909 flesch: 89 summary: (4.9) Substituting (4.7) into (3.3) gives fs − y 2 fy − 1 2 f = g, s > 0, y ∈ R, e−s b2(t(s)) (gs − 3 2 g − y 2 gy) + g = (a(yes/2)fy)y + α(s)(a0(yes/2)ϕ′0(y))y + h(s, y), s > 0, y ∈ R, f(0, y) = v(0, y)− α(0)ϕ0(y), y ∈ R, g(0, y) = w(0, y)− α̇(0)ϕ0(y)− α(0)ψ0(y), y ∈ R, (4.10) where h(s, y) = e−s b2(t(s)) ( −2α̇(s)ψ0(y) + α(s)( y 2 ψ′0(y) + 3 2 ψ0(y)) ) + r(s, y)− ϕ0(y) ∫ R r(s, y)dy, (4.11) here we have used (4.5) and a(x) = ã(x) + a0(x). keywords: a(yes/2; b2(t(s; dα(s; e−s; lemma; proof; solution; ∫ r cache: ejde-594.pdf plain text: ejde-594.txt item: #487 of 601 id: ejde-596 author: Calamai, Alessandro; Spadini, Marco title: Caratheodory periodic perturbations of degenerate systems date: 2024 words: 6698 flesch: 76 summary: We concentrate on the family of systems where the constraining manifold is of the form M = M × N , the cartesian product of two smooth boundaryless manifolds M ⊆ Rk and N ⊆ Rs, and G : M ×N → Rk ×Rs is of the form (0, g), i.e., the first component is identically zero and g : Moreover, by the assumptions on the sequences {fn}, {gn}, {hn} we have, for a.a. t ∈ keywords: set cache: ejde-596.pdf plain text: ejde-596.txt item: #488 of 601 id: ejde-597 author: Osawa, Satoshi; Takaoka, Hideo title: Global well-posedness for Cauchy problems of Zakharov-Kuznetsov equations on cylindrical spaces date: 2024 words: 10169 flesch: 88 summary: | |ξ| ∼ K, |(ξ1, q1)| ∈ IN1 , |(ξ − ξ1, q − q1)| ∈ IN2 , |τ1 − σ(ξ1, q1)| ∈ IL1 , |τ − τ1 − σ(ξ − ξ1, q − q1)| ∈ IL2 } . Here Γξ1,q1,τ1ξ,q,τ = |ξ|〈ζ〉 〈ζ1〉〈ζ − ζ1〉〈τ − σ(ζ)〉1/2−〈τ1 − σ(ζ1)〉1/2+〈τ − τ1 − σ(ζ − ζ1)〉1/2+ . keywords: case; equation; ql2 cache: ejde-597.pdf plain text: ejde-597.txt item: #489 of 601 id: ejde-598 author: Iaia, Joseph title: Existence of two infinite families of solutions for singular superlinear equations on exterior domains date: 2024 words: 6193 flesch: 89 summary: (2.47) 10 J. IAIA EJDE-2024/06 Evaluating (2.47) at t = R1+M1,a 2 we see v1−pa (R1 +M1,a 2 ) ≥ (p− 1)f4R −α̃1 1 2 (R1 −M1,a 2 )2 and therefore vp−1a (R1 +M1,a 2 ) (2.48) By (2.45) we see then for large a that va (R1 +M1,a 2 ) ≤ ( 32Rα̃1−2 1 (p− 1)f4 ) 1 p−1 . keywords: m1,a cache: ejde-598.pdf plain text: ejde-598.txt item: #490 of 601 id: ejde-6 author: Yu, Xiu-Fang; Wang, Jun-Min; Zhang, Han-Wen title: Internal stabilization of interconnected heat-wave equations date: 2023 words: 10059 flesch: 82 summary: η 0 |Gnξ (ξ, s)|ds+ |a|d ∫ η 0 |Hn ξ (ξ, s)|ds ≤ ( |a| 2 + |a|d ) MKn (n+ 1)! (ξ + η)n+1 ≤ ( |a| 2 + |a|d )MKn n! (ξ + η)n, |Hn+1 ηη (ξ, η)| ≤ |a| 2 ∫ ξ 0 |Gnη (τ, η)|dτ + |a|d ∫ ξ 0 |Hn η (τ, η)|dτ ≤ ( |a| 2 + |a|d )MKn n! (ξ + η)n, and |Gn+1(ξ, η)| ≤ α2 4 ∫ ξ η ∫ η 0 |Gn(τ, s)| ds dτ + |a| 2 ∫ ξ η ∫ η 0 |Hn ξξ(τ, s)| ds dτ + |a| ∫ ξ η ∫ η 0 |Hn ξη(τ, s)| ds dτ + |a| 2 ∫ ξ η ∫ η 0 |Hn ηη(τ, s)| ds dτ ≤ α2 4 MKn n! ∫ ξ η ∫ η 0 (τ + s)n ds dτ + 2|a|MKn (n− 1)! ∫ ξ η ∫ η 0 (τ + s)n−1 ds dτ = α2 4 MKn (n+ 1)! ∫ ξ η [(τ + η)n+1 − τn+1]dτ According to boundary conditions of the last line in (3.7), we obtain c1 + c2 = 0, c3 + c4 = 0, e √ λc1 + e− √ λc2 − pλeλ+αc3 − pλe−(λ+α)c4 = 0, p √ λe √ λc1 keywords: heat; sinh; system; wave; η 0; ξ η; ∫ η cache: ejde-6.pdf plain text: ejde-6.txt item: #491 of 601 id: ejde-60 author: Clark, Jason; Misiats, Oleksandr; Mogylova, Viktoriia; Stanzhytskyi, Oleksandr title: Asymptotic behavior of stochastic functional differential evolution equation date: 2023 words: 7818 flesch: 80 summary: dt ≤ C2 ∫ T 0 dt ∫ t 0 ( 1 + E‖Φs‖pBρ1 ) ds ≤ C3 + C2 ∫ T 0 ∫ t 0 E (∫ 0 −h ‖Φ(s+ θ, ·)‖2Bρ0 dθ )p/2 ds dt ≤ C3 + C4E ∫ T −h ‖Φ(t, ·)‖p Bρ0 dt <∞. (3.2) To estimate I3, we use [9, Lemma 7.2] and (2.10). EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 11 I4 ≤ Cpρ (T )Lp ∫ t 0 E (∫ s 0 ‖Φτ (·)− Φ̃τ (·)‖Bρ1 dτ )p ds ≤ C(ρ, T, p) ∫ t 0 ∫ keywords: bρ0; differential; equations; theorem cache: ejde-60.pdf plain text: ejde-60.txt item: #492 of 601 id: ejde-600 author: Belinskiy, Boris P.; Smith, Tanner A. title: Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesign date: 2024 words: 7400 flesch: 75 summary: Finally, we introduce a set of solvability conditions on the S-L problem data, confirming that the corre- sponding critical points represent meaningful solutions we refer to as designs. − 2 (∫ 1 0 q(x) sin2( √ |λ1|g(x) + z′)dx− φ sin2(z′) )] , (3.6) subject to solvability conditions which we omit here. keywords: conditions; mass; problem; q(s; sinh(2 cache: ejde-600.pdf plain text: ejde-600.txt item: #493 of 601 id: ejde-602 author: Motreanu, Dumitru; Tornatore, Elisabetta title: Dirichlet problems with anisotropic principal part involving unbounded coefficients date: 2024 words: 5036 flesch: 75 summary: . , N , and a Carathéodory function F : Ω × R × RN → R (i.e., F (·, t, ξ) is measurable on Ω for each (t, ξ) ∈ R×RN and F (x, ·, ·) is continuous on R × RN for almost all x ∈ Ω). , pN ) and denote by W 1,−→p 0 (Ω) the completion of the set of smooth functions with compact support C∞c (Ω) with respect to the norm ‖u‖ := N∑ i=1 ‖∂iu‖Lpi . keywords: problem cache: ejde-602.pdf plain text: ejde-602.txt item: #494 of 601 id: ejde-604 author: Zhang, Guoping; Aburamyah, Ghder title: Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential date: 2024 words: 8077 flesch: 75 summary: By Theorem 4.2, if u0 ∈ u0 ∈ DΘ∩B, then u(t) = S(t)u0 is a global classical solution of the initial value problem (1.1)-(1.2). [19] A. Pankov, G. Zhang; Standing wave solutions for discrete nonlinear Schrödinger equations with unbounded potentials and saturable nonlinearities, J. Math. keywords: n∈zd; pθ(zd; solution; u(t cache: ejde-604.pdf plain text: ejde-604.txt item: #495 of 601 id: ejde-605 author: Boccardo, Lucio; Diaz, Jesus Ildefonso; Gomez-Castro, David title: Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence date: 2024 words: 6850 flesch: 82 summary: We focus our efforts on the particular case E = −ϕ−1−γ 1 ∇ϕ1, for some γ > 0, (6.1) and f ∈ L∞c (Ω), the space of bounded functions with compact support in Ω. The aim of this section is to prove the following theorem. Let E be given by (6.1), M = I and f ∈ L∞c (Ω). keywords: theorem cache: ejde-605.pdf plain text: ejde-605.txt item: #496 of 601 id: ejde-606 author: Severo, Uberlandio B.; Ribeiro, Bruno H. C.; Germano, Diogo de S. title: Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity date: 2024 words: 6050 flesch: 82 summary: Introduction and main result In this work we consider the quasilinear equation − div(g2(u)∇u) + g(u)g′(u)|∇u|2 + V (x)u = f(x, u) + λ|u|p−2u+ h(x)g(u) (1.1) in R2, where g : R → R+ is a function of class C1, V : R2 → R is a potential that can change sign, f : R2 × R → R is a measurable function, which may have exponential critical growth of Trudinger-Moser type, λ ∈ R is a parameter, p ≥ 2 and h ∈ Lq(R2) for some 1 < q ≤ 2. When g(s) ≡ 1 and λ = 0, equation (1.1) becomes the nonhomogeneous semilin- ear Schrödinger equation −∆u+ V (x)u = f(x, u) + h(x) in R2, (1.2) which has been studied by several researchers, see for example keywords: equations; lemma cache: ejde-606.pdf plain text: ejde-606.txt item: #497 of 601 id: ejde-608 author: Ding, Yuanlin; Liu, Kui title: Properties of the solutions to periodic conformable non-autonomous non-instantaneous impulsive differential equations date: 2024 words: 6972 flesch: 87 summary: = δl(t)β(t + l ), t ∈ (tl, sl], l ∈ N, β(s+l ) = (E+ Pl)β(t − l ) +Ql, l ∈ N := {1, 2, . . . }, β(t) = δl(t)β(t + l ), t ∈ (tl, sl], l ∈ N, β(s+l ) keywords: pl)φ(tl; sς(c; λ(t; ς(c cache: ejde-608.pdf plain text: ejde-608.txt item: #498 of 601 id: ejde-609 author: Munoz Rivera, Jaime E.; Baldez, Carlos A. da Costa; Cordeiro, Sebastiao M. S. title: Signorini's problem for the Bresse beam model with localized Kelvin-Voigt dissipation date: 2024 words: 8696 flesch: 83 summary: − κ0(ζx − lv)x − lK(ζxt − lvt)x + lκ(vx + y + lζ) + lK(vxt + yt + lζt) = 0. (4.6) EJDE-2024/17 SIGNORINI’S PROBLEM FOR BRESSE BEAMS 15 Multiplying (4.4) by vt, (4.5) by wt, and integrating over [0, `], we obtain d dt ‖Um(t)‖2H + ∫ ` `0 K|vxt + yt + lζt|2 +B|yxt|2 +K|ζxt − lvt|2 dx+ ε|ut|2 + ε|zt|2 = −S̃m(`−0 , t)ϕt(` − 0 , t)− M̃m(`−0 , t)ψt(` − 0 , t)− Ñm(`−0 , t)wt(` − 0 , t) (4.7) where Um(t) = − byxx − (Byxt)x + κ(vx + y + lζ) +K(vxt + yt + lζt) = 0, (4.5) ρ1ζtt keywords: ejde-2024/17; l2(0; problem cache: ejde-609.pdf plain text: ejde-609.txt item: #499 of 601 id: ejde-61 author: Li, Sheng-Jie; Chai, Shugen title: Stabilization of semilinear wave equations with time-dependent variable coefficients and memory date: 2022 words: 7818 flesch: 80 summary: dt ≤ C2 ∫ T 0 dt ∫ t 0 ( 1 + E‖Φs‖pBρ1 ) ds ≤ C3 + C2 ∫ T 0 ∫ t 0 E (∫ 0 −h ‖Φ(s+ θ, ·)‖2Bρ0 dθ )p/2 ds dt ≤ C3 + C4E ∫ T −h ‖Φ(t, ·)‖p Bρ0 dt <∞. (3.2) To estimate I3, we use [9, Lemma 7.2] and (2.10). EJDE-2023/35 STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 11 I4 ≤ Cpρ (T )Lp ∫ t 0 E (∫ s 0 ‖Φτ (·)− Φ̃τ (·)‖Bρ1 dτ )p ds ≤ C(ρ, T, p) ∫ t 0 ∫ keywords: bρ0; differential; equations; theorem cache: ejde-61.pdf plain text: ejde-61.txt item: #500 of 601 id: ejde-610 author: Wang, Lixia; Zhao, Pingping; Zhang, Dong title: Existence of high energy solutions for superlinear coupled Klein-Gordons and Born-Infeld equations date: 2024 words: 4980 flesch: 73 summary: By (1.3) and the Gateaux derivative of I, we can obtain that ‖un − u‖2 = 〈I ′(un)− I ′(u), un − u〉+ V0 ∫ R3 (un − u)2 dx+ 2ω ∫ R3 (φunun − φuu)(un − u) dx + ∫ R3 (h(x, un)− h(x, u))(un − u) dx+ ∫ R3 (φ2 un un − φ2 uu)(un − u) dx By an easy computation, we obtain that 〈 [(φ2 un un − φ2 uu)(un − u) dx ∣∣ ≤ |φ2 un un − φ2 uu|3/2|un − u|3 ≤ (|φ2 un un|3/2 + |φ2 uu|3/2)|un − u|3 → 0, as n→ +∞. By Proposition 2.3 and un → u in Ls(R3) for 2 ≤ s < 6, we have∫ R3 (h(x, un)− h(x, u))(un − u) keywords: klein cache: ejde-610.pdf plain text: ejde-610.txt item: #501 of 601 id: ejde-611 author: Zhang, Bo; Zhang, Wei title: Localized nodal solutions for semiclassical Choquard equations with critical growth date: 2024 words: 14055 flesch: 91 summary: In this article, we study the existence of localized nodal solutions for semiclassical Choquard equation with critical growth −ε2∆v + V (x)v = εα−N (∫ RN |v(y)|2∗α |x− y|α dy ) |v|2 ∗ α−2v + ϑ|v|q−2v, x ∈ RN , where ϑ > 0, N ≥ 3, 0 < Introduction In this article, we study localized nodal solutions of the nonlinear Choquard equation with critical exponent −ε2∆v + V (x)v = εα−N (∫ RN |v(y)|2∗α |x− y|α dy ) |v|2 ∗ α−2v + ϑ|v|q−2v, x ∈ RN , v(x)→ 0 as |x| → ∞, (1.1) where ϑ > 0, N ≥ 3, 0 < keywords: 2∗α; b(y; choquard; dx dy; equations; lemma; rn χε(x)u2; solutions; y|α; y|α dx; zhang; |x−; ∫ rn cache: ejde-611.pdf plain text: ejde-611.txt item: #502 of 601 id: ejde-615 author: Zhang, Jichao; Bu, Shangquan title: Maximal regularity for fractional difference equations of order 2 date: 2024 words: 6758 flesch: 85 summary: − 1)α[z1−α(z − 1)α − T ]−1 : |z| = 1, z ̸= 1 } is R-bounded [18]. In the case 1 < α ≤ 2, 1 < p < ∞ and X is a UMD space, Lizama and Arcila showed that (1.1) with the initial conditions u(0) = u(1) = 0, has the ℓp-maximal regularity if and only if {z2−α(z − 1)α : |z| = 1, z ̸= 1} ⊂ ρ(T ), and the set { z2−α(z − 1)α[z2−α(z − 1)α − T ]−1 : |z| = 1, z ̸= 1 } is R-bounded [19]. keywords: regularity cache: ejde-615.pdf plain text: ejde-615.txt item: #503 of 601 id: ejde-619 author: Faria, Luiz F. O.; Montenegro, Marcelo title: Positive solution for a nonlinear elliptic equation on symmetric domains date: 2024 words: 9204 flesch: 83 summary: Then there exists λ∗ > 0 such that for every λ ∈ (0, λ∗) problem (1.14) possesses at least one positive radially symmetric solution uλ ∈ W 1,p 0 (RN\BR). Let u ∈ W 1,p 0,r (RN ) with ∥u∥W 1,p(RN ) keywords: 1,p; 1,p(rn cache: ejde-619.pdf plain text: ejde-619.txt item: #504 of 601 id: ejde-620 author: Ayachi, Moez; Abbas, Syed title: P-mean (mu1,mu2)-pseudo almost periodic processes and application to integro-differential stochastic evolution equations date: 2024 words: 9826 flesch: 80 summary: = U(t, a)Z(a) + ∫ t a U(t, s)F1(s, Z(s))ds + ∫ t a U(t, s) ∫ s a Q(s− ζ)F2(ζ, Z(ζ))dζds + ∫ t a U(t, s) ∫ [Z(0)− Z∗(0)] + ∫ t 0 U(t, s) keywords: lp(p; mean; stochastic; µ1,2; ∫ +; ∫ t cache: ejde-620.pdf plain text: ejde-620.txt item: #505 of 601 id: ejde-622 author: Gao, Yingchun; Liu, Kai; Qi, Xiaoguang title: Crossed differential systems of equations and Clunie lemma date: 2024 words: 6665 flesch: 79 summary: By the addition and subtraction of two equations in (3.6), we have ff ′+ gg′ = 2− f ′− g′ and ff ′− gg′ = f ′− g′. Integrating the above two equations, we have 1 2 f2 − 1 2 g2 = f − g +A1, and 1 2 f2 + 1 2 g2 = 2z − f − g +A2, thus we have T (r, f) If L(z, f) = L(z, g) implies that f = g, where f(z) and g(z) are two meromorphic functions, then L(z, f) is called a unique differential polynomial of meromorphic functions (UDPM). keywords: differential; equations; f(z; g(z; s(r; solutions cache: ejde-622.pdf plain text: ejde-622.txt item: #506 of 601 id: ejde-623 author: Gao, Yue; Yang, Xue title: Periodic solutions in distribution for stochastic lattice differential equations date: 2024 words: 5147 flesch: 82 summary: By Hölder inequality, [20, Theorem 1.7.2], Assumption 2.1, and (3.1), we obtain E ( sup 0≤s≤t ∥u(t)∥pρ ) = E ( sup 0≤s≤t ∥u0 + ∫ s 0 [−νAu(s)− λu(s) + f(u(s)) + g(r)]dr + ∫ s 0 σ(r, u(r))dW (r)∥pρ ) ≤ 3p−1E∥u0∥pρ + (12t)p−1E[ ∫ t 0 ∥ − νAu(s)− λu(s) + f(u(s)) = ∫ t 0 [−νA(u(s)− ũ(s))− λu(s) + λũ(s)) + f(u(s))− f(ũ(s))]ds+ ∫ t 0 keywords: distribution; equations; l2ρ; periodic; solutions cache: ejde-623.pdf plain text: ejde-623.txt item: #507 of 601 id: ejde-624 author: Webb, Jeffrey title: Nonexistence results for fractional differential inequalities date: 2024 words: 8823 flesch: 82 summary: We can integrate from 1 to t ≤ T to obtain g1−p(t) ≤ v1−p 0 − (p− 1) Γ(α) (t1−γ − 1) 1− γ , for γ < 1, g1−p(t) ≤ v1−p 0 − (p− 1) Γ(α) u ∈ AC[0, T ] if and only if u′ ∈ L1[0, T ], u′(t) exists for almost every (a.e.) t ∈ keywords: fractional; solution cache: ejde-624.pdf plain text: ejde-624.txt item: #508 of 601 id: ejde-627 author: Alshanti, Waseem Ghazi title: Solutions of linear and non-linear partial differential equations by means of tensor product theory of Banach space date: 2024 words: 2667 flesch: 77 summary: 5. Conclusions This article has introduced a new analytical method for handling non-separable, linear and non-linear partial differential equations via atomic solutions method. Partial differential equations; tensor product of Banach spaces; atomic solution. keywords: differential; equations; solution cache: ejde-627.pdf plain text: ejde-627.txt item: #509 of 601 id: ejde-629 author: Liu, Qiang; Zhu, Wanyu; Ye, Hailong title: Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative date: 2024 words: 5339 flesch: 77 summary: However, to the best of our knowledge, the well-posedness for the solutions of the time fractional thin film growth equation is not clear, which is the main motivation of the present work. Ct αγ 4β2 − α 2β R(t)1+β j , and similarly ∥∇2uj+1∥ βN 2−β ≤ ∥∇2Eα(−tαA)φ∥ βN 2−β + ∫ t 0 (t− s)α−1∥∇2Eα,α(−(t− s)αA)∇ · f(∇uj)∥ βN 2−β ds ≤ ∥∇2u0∥ βN 2−β + C ∫ t 0 (t− s) α 2 −1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ ds ≤ ∥∇2u0∥ βN 2−β + CR(t)1+β j ∫ t 0 (t− s) α 2 −1−αγ 4β s−α+αγ 4β ds (3.6) ≤ ∥∇2u0∥ βN 2−β + Ct− α 2 R(t)1+β j . Combining (3.5) and (3.6), for any fixed T > 0, we have R(T )j+1 ≤ R(T )0 + CR(T )1+β j , where C > 0 is independent of T . keywords: 2−β cache: ejde-629.pdf plain text: ejde-629.txt item: #510 of 601 id: ejde-636 author: Ma, Zhouji; Chang, Xiaojun; Feng, Zhaosheng title: Normalized ground state of a mixed dispersion nonlinear Schrodinger equation with combined power-type nonlinearities date: 2024 words: 9218 flesch: 85 summary: = 1 2 ∥∆u∥22 + 1 2 ∥∇u∥22 − µ 2 ∥u∥qq − 1 2 ∥u∥pp = −1 2 ωc, ∂K ∂t (1, 1) = ∥∆u∥22 + 1 2 ∥∇u∥22 − µγq∥u∥qq − γp∥u∥pp = 0, ∂2K ∂t2 (1, 1) = ∥∆u∥22 − µγq( N(q − 2) 4 − 1)∥u∥qq − γp( N(p− 2) 4 − 1)∥u∥pp < 0, which yields for δt small enough and δλ > 0, K(1 + δλ, 1 + δt) < K(1, 1) for ω > 0. (4.8) + 1 2 ∥∇u∥22 − µγq∥u∥qq − γp∥u∥pp = 0. keywords: q(c cache: ejde-636.pdf plain text: ejde-636.txt item: #511 of 601 id: ejde-637 author: Han, Qian-Qian; Huan, Song-Mei title: Global dynamics of a special class of planar sector-wise linear systems date: 2024 words: 11129 flesch: 79 summary: = −2 det(A) · y+e · (x∗ − x− e + x+ e 2 ) And by easy computation, we obtain that 2µ2 + x+ e − µ6 = λ1 − λ2 λ1 − a11 x+ e + 2a12λ1 a11(λ1 − a11) y+e , which implies 2µ2 + x+ e < µ6 and 2µ2 + x+ e > µ6 both may be true. keywords: cycle; e =; linear; orbit; point; systems; x− e cache: ejde-637.pdf plain text: ejde-637.txt item: #512 of 601 id: ejde-64 author: Liu, Zhenhai; Papageorgiou, Nikolaos S. title: A weighted (p,2)-equation with double resonance date: 2023 words: 7657 flesch: 81 summary: The reaction (right-hand side) of (1.1), is a Carathéodory function f(z, x) (that is, for all x ∈ R, z → f(z, x) is measurable and for a.a.z ∈ Ω, x → f(z, x) is continuous) which exhibits (p − 1) sublinear growth as x → ±∞ and resonance can occur with respect to the principal eigenvalue of (−∆a1 p ,W 1,p 0 (Ω)) EJDE-2023/30 A WEIGHTED (p, 2)-EQUATION WITH DOUBLE RESONANCE 13 Note that ∇yγ(z, y) = a1(z)|y|p−2[id+ (p− 2) y ⊗ y |y|2 ] + a2(z)id ∀z ∈ Ω, ;∀y ∈ RN ⇒ (∇yγ(z, y)β, β)RN ≥ ĉ|β|2 for all y, β ∈ RN . keywords: 1,p cache: ejde-64.pdf plain text: ejde-64.txt item: #513 of 601 id: ejde-644 author: Thorel, Alexandre title: Solvability of transmission problems with generalized diffusion equation in L^p-spaces date: 2024 words: 14190 flesch: 87 summary: edL+)ψ1 − (I − edL+)ψ2 + φ̃+ 1 ] α+ 2 = − 1 2r+ (L+ +M)U−1 + [ M(I + edM )ψ1 − (I − edM )ψ2 + φ̃+ 2 ] α+ 3 = 1 2r+ (L+ +M)V −1 + [ L+(I − edL+)ψ1 − (I + edL+)ψ2 + φ̃+ 3 ] α+ 4 = − 1 2r+ (L+ +M)V −1 + [ M(I − edM )ψ1 − (I + edM )ψ2 + φ̃+ 4 ] , (5.20) with φ̃1 + = −L+ ( I + edL+ ) φ+ 1 + ( I − edL+ ) ( F ′ +(b) + F ′ +(γ)− φ+ 2 ) φ̃2 + = −M ( I + edM ) [ L+(I − edL+)ψ1 − (I + edL+)ψ2 + φ̃+ 3 ] = 2M−1R2, Finally, using (5.2), (5.12), (5.13), (5.14) and (5.15), we obtain that the previous system writes as system (5.17). keywords: + m; + m)2; + √; ecm; k+(l+ +; r +; y +; z + cache: ejde-644.pdf plain text: ejde-644.txt item: #514 of 601 id: ejde-647 author: Jin, Lingyu; Wei, Suting title: A global compactness result for quasilinear elliptic problems with critical Sobolev nonlinearities and Hardy potentials on R^N date: 2024 words: 8946 flesch: 86 summary: (A4) There exists a constant θ ∈ (0, p∗ − p) such that t ∂ ∂tf(x, t) ⩾ Moreover, we extend f(x, t) ≡ 0 for all t ∈ (−∞, 0), x ∈ RN . (A3) There exists a constant q ∈ (p, Np N−p ) such that lim t→+∞ f(x,t) tq−1 = 0 and lim t→0+ f(x,t) tp−1 = 0 uniformly in x ∈ RN . keywords: 1,p(rn; o(1; |x|p cache: ejde-647.pdf plain text: ejde-647.txt item: #515 of 601 id: ejde-649 author: Caqui, Eduardo H.; Lima, Sandra M. de S.; Pereira, Fábio R. title: Multiplicity results for critical fractional Ambrosetti-Prodi type system with nonlinearities interacting with the spectrum date: 2025 words: 12401 flesch: 85 summary: − U∥2Y − (α+ β) ∫ Ω F (Un − U) dx+ ∫ Ω (∇F (U + UT ), UT )R2 dx + ∫ Ω (∇F (Un + UT ), UT )R2 dx+ o(1). Also (w, z) = ( (λ1,s − c)t+ br det(λ1,sI −A) ϕ1,s, bt+ (λ1,s − a)r det(λ1,sI −A) ϕ1,s ) is the unique solution of the system (− −→ ∆)sU = AU + Tϕ1,s in Ω, U = 0 in RN \ Ω. Consequently, if uT = (λ1,s − c)t+ br det(λ1,sI −A) ϕ1,s + u0, vT = bt+ (λ1,s − a)r det(λ1,sI −A) ϕ1,s + v0, then UT = (uT , vT ) is a solution of the system (− −→ ∆)sU = AU + Tϕ1,s + F1 in Ω, U = 0 in RN \ Ω. Clearly if uT and vT are negative in Ω, we deduce also that UT is a solution of (1.2). keywords: + ut cache: ejde-649.pdf plain text: ejde-649.txt item: #516 of 601 id: ejde-657 author: Zhao, Fengxiang; Tang, Haotian; Zheng, Jiashan; Li, Kaiqiang title: Existence and boundedness of solutions for a parabolic-parabolic predator-prey model date: 2025 words: 7168 flesch: 81 summary: To deal with u, multiplying both sides of the second equation in (1.1) by ur̃−1 and integrating by parts, for any small ε ∈ (0, 1), we derive from Young’s inequality that 1 r̃ d dt ∫ Ω ur̃ + (r̃ − 1) ∫ Ω ur̃−2|∇u|2 = − r̃ − 1 r̃ χ ∫ Ω ur̃∆w + λ1 ∫ Ω ur̃ − µ1 ∫ Ω ur̃+r1−1 + a ∫ Ω ur̃v ≤ r̃ − 1 r̃ κ ∫ Ω ur̃|∆w|+ λ1 ∫ Ω ur̃ − µ1 ∫ Ω ur̃+r1−1 + ε ∫ Ω ur̃+r1−1 + C3 ∫ Ω v r̃+r1−1 r1−1 ∀t ∈ (0, Tmax), (3.29) where C3 = r̃ + r1 − 1 r1 − 1 (ε r̃ + r1 − 1 r̃ )− r̃ r1−1 a r̃+r1−1 r1−1 . − r̃ r1−1 × ( r̃ − 1 r̃ κ ) r̃+r1−1 r1−1 ∫ Ω |∆w| r̃+r1−1 r1−1 = λ0 ∫ Ω ur̃+r1−1 + Ã1λ − r̃ r1−1 0 κ r̃+r1−1 r1−1 ∫ Ω |∆w| r̃+r1−1 r1−1 ∀t ∈ (0, Tmax), (3.32) where Ã1 = r1 − 1 r̃ + r1 − 1 ( r̃ + r1 − 1 r̃ )− r̃ r1−1 ( r̃ − 1 r̃ ) r̃+r1−1 r1−1 . keywords: r1−1; r2 −; r2−1; ∫ ω cache: ejde-657.pdf plain text: ejde-657.txt item: #517 of 601 id: ejde-66 author: Asso, Oumarou; Cuesta, Mabel; Doumate, Jonas Tele; Leadi, Liamidi title: Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights date: 2023 words: 11989 flesch: 83 summary: Published June 19, 2023. 1 2 O. ASSO, M. CUESTA, J. T. DOUMATÈ, L. LEADI EJDE-2023/38 In this article, we study the conditions under which the principal eigenvalues of the following homogeneous Dirichlet problem exist (−∆p) su+ V |u|p−2u = λm(x)|u|p−2u in Ω, u = 0 in RN \ Ω, (1.1) where Ω is a bounded regular domain of RN , V and m are indefinite sign-changing functions and satisfying the following conditions: (C1) V , m ∈ Lr(Ω) with r ∈ (1, +∞) ∩ (Nsp , +∞), (C2) m+ = max(m, 0) 6≡ 0. Let us consider the homogeneous problem (−∆p) su+ V ′|u|p−2u = 0 in Ω, u = 0 in RN \ Ω, (3.4) where V ′ satisfies condition (C1). keywords: eigenvalues; k(1−; p(rn; p(ω; w̃ s; λ1(v cache: ejde-66.pdf plain text: ejde-66.txt item: #518 of 601 id: ejde-664 author: Castillo, Ricardo; Guzman, Omar; Loayza, Miguel; Zegarra, Maria title: Global solution for coupled parabolic systems with degenerate coefficients and time-weighted sources date: 2024 words: 7870 flesch: 84 summary: = ∫ RN Γ(x, y, t)v0(y) dy + ∫ t 0 ∫ RN Γ(x, y, t− σ)h2(σ)u(y, σ) q dy dσ <∞, (1.5) for almost all x ∈ RN and t ∈ (0, T ). Suppose that (4.6) holds for some n ∈ N. Then, by (2.2) we have ∥un+1(t)∥∞ ≤ ∥u1(t)∥∞ + ∫ t 0 σr∥S(t− σ)vn(σ) p∥∞ dσ ≤ c1∥u0∥∞ + c1 ∫ t 0 σr∥vn(σ)∥p∞ dσ ≤ c1∥u0∥∞ + c1[2c1(∥u0∥∞ + ∥v0∥∞)]p ∫ t 0 σr dσ, (4.7) 14 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 for t ∈ (0, T ). keywords: 2−α; γ(x cache: ejde-664.pdf plain text: ejde-664.txt item: #519 of 601 id: ejde-665 author: Baig, Ayesha; Li , Zhouxin title: Existence of positive solutions for systems of quasilinear Schrodinger equations date: 2025 words: 8006 flesch: 84 summary: (1.8) where the functions W,V : RN → R are Hölder continuous satisfying W (x), V (x) ≥ α > 0 in RN and the condition: (5) There exists an open and bounded set Λ ⊂ RN , with x0 ∈ Λ and ρ > 0, such that W (x), V (x) ≥ ρ, for all x ∈ ∂Λ and W (x0), V (x0) < Within this class of potentials, V satisfies (A1), (A2) and (A4) there exists a domain Λ ⊂ RN where ∇V (x) ̸= 0 for all x ∈ ∂Λ. Given that V falls into either Class 1 or Class 2 and taking into account certain conditions met by the nonlinearity, the author demonstrated the existence of a positive solution for ε > 0 sufficiently small. Alves keywords: lemma cache: ejde-665.pdf plain text: ejde-665.txt item: #520 of 601 id: ejde-666 author: Bandyopadhyay, Shalmali; Lewis, Thomas; Mavinga, Nsoki title: Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions date: 2025 words: 10506 flesch: 77 summary: ∈ ∂Ω. Throughout this article we assume that each fi satisfies the quasimonotonicity condition (A1) the functions fi are quasimonotone nondecreasing in the sense that f1(x, u1, u2) is nonde- creasing in u2 for all fixed x ∈ ∂Ω, u1 ∈ R, and f2(x, u1, u2) is nondecreasing in u1 for all fixed x ∈ ∂Ω, u2 ∈ R. In this article, we establish the existence of maximal and minimal weak solutions for (1.1). We define the map T : J → (H1(Ω))2 by T (U) =W , where J := {U = (u1, u2) ∈ (H1(Ω))2 : U ≤ U ≤ U} and W = (w1, w2) is the unique weak solution of the decoupled system −∆wi + wi = 0 in Ω; ∂wi ∂η + kwi = fi(x, u1, u2) + kui on ∂Ω, i = 1, 2, (2.1) where k = k1 + k2 ≥ 0. keywords: existence; f1(x; solutions cache: ejde-666.pdf plain text: ejde-666.txt item: #521 of 601 id: ejde-67 author: El Attaouy, Meryem; Ezzinbi, Khalil; ˜N'Guerekata, Gaston Mandata title: Reduction principle for partial functional differential equation without compactness date: 2023 words: 6665 flesch: 73 summary: In this work we are interested in investigating the existence of almost automor- phic and almost periodic solutions for the partial functional differential equation x′(t) = Ax(t) + L(xt) + f(t) for t ∈ R, (1.3) whereA is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T (t) on a Banach space X. x(t) ∈ X, L is a bounded linear opera- tor from C([−r, 0], X) to X with C([−r, 0], X) is the space of continuous functions from [−r, 0] to X endowed with the uniform norm topology and r > 0. dξ ] (0), for t ∈ R, is a mild solution of (1.3) on R. As a consequence of the above, we establish the following fundamental reduction principle which allows us to prove the existence of an almost automorphic and almost periodic solution of the (1.3). keywords: function; periodic; semigroup; solution cache: ejde-67.pdf plain text: ejde-67.txt item: #522 of 601 id: ejde-673 author: Aramaki, Junichi title: Eigenvalue problems for Kirchhoff-type equations in variable exponent Sobolev spaces date: 2025 words: 13097 flesch: 88 summary: We say that a pair (u, λ) ∈ Y × R is a weak solution of (1.1), if M (∫ Ω A(x,∇u(x)) Let p ∈ C+(Ω) and let u, un ∈ Y (n = 1, 2, . . .). keywords: eigenvalue; function; proposition; space cache: ejde-673.pdf plain text: ejde-673.txt item: #523 of 601 id: ejde-675 author: Jiang, Qiaoyun; Li, Lin; Chen, Shangjie; Siciliano, Gaetano title: Ground state solutions for the nonlinear Schrödinger-Bopp-Podolsky systems with nonperiodic potentials date: 2024 words: 10740 flesch: 88 summary: (ii) ∥un − u∥ss = ∥un∥ss − ∥u∥ss + on(1), where s ∈ (2, 6]. (iii) ∥un − u∥s−2(un − u) = By a direct calculation, for p ∈ [4, 6), J (un) = J (un)− 1 4 ⟨J ′(un), un⟩ = 1 4 ∥un∥2 + 1 12 ∫ |un|6 dx+ λ( 1 4 − 1 p ) ∫ |un|p dx ≥ 1 4 ∥un∥2. keywords: lemma; on(1 cache: ejde-675.pdf plain text: ejde-675.txt item: #524 of 601 id: ejde-681 author: Silva, Joao Pablo Pinheiro da; Silva, Edcarlos Domingos da title: Existence of semi-nodal solutions for elliptic systems related to Gross-Pitaevskii equations date: 2024 words: 4200 flesch: 85 summary: = (un(δ), vn(δ)) by (u(δ), v(δ)) := ( t(δ)[u− δz], r(δ)[v − δw]+ − s(δ)[v − δw]− ) ∈ Mλµ. Recall also that Iλµ ∈ C1(H,R), where H = H1 0 (Ω)×H1 0 (Ω). − s′(0)v− − w∥ as δ → 0+. keywords: solutions cache: ejde-681.pdf plain text: ejde-681.txt item: #525 of 601 id: ejde-682 author: Hua, Yang; Lin, Xiaojie; Liu, Jiang; Lu, Haixia title: Dynamics of traveling waves for predator-prey systems with Allee effect and time delay date: 2024 words: 7741 flesch: 74 summary: Comparing the coefficients of ε and ε2, one has f1 = 0, f2 = 2UV α+ U + βV − 2(σU − σU2 − η)U, h1 = 0, h2 = cX − (σU − σU2 − η)U + UV α+ U + βV , g1 = X, g2 = 0, r1 = 0, r2 = 0. Furthermore, under certain conditions, see [7] for more details, the system (2.3) admits the fourth equilibrium P∗(u∗, v∗, 0, 0), where u∗ is a positive root of the cubic equation σγβu3 − σγβu2 + (γβη + γ − 1)u− α = 0, (2.4) and v∗ = γu∗ − u∗ − α β > 0. keywords: + ∞; delay; system cache: ejde-682.pdf plain text: ejde-682.txt item: #526 of 601 id: ejde-683 author: Sousa, Jose Vanterler da C.; Pigossi, Mariane; Nyamoradi, Nemat title: Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance date: 2024 words: 7525 flesch: 83 summary: 4 J. V. D. C. SOUSA, M. PIGOSSI, N. NYAMORADI EJDE-2024/34 For the first eigenfunctions φ1(a) > 0, if we let V = span{φ1(a)}, then V ⊥ = { ξ ∈ Hα,β,ψp : ∫ T 0 (φ(a))p−1ξdx = 0 } . such that∫ T 0 (∣∣∣HDα,β,ψ 0+ ξ(x) ∣∣∣p − a(x)|ξ|p ) dx ⩾ λ(a) ∫ T 0 |ξ|pdx (1.10) for any ξ ∈ V ⊥. Similarly, we can define λ1(b), φ1(b) and λ(b). keywords: fractional cache: ejde-683.pdf plain text: ejde-683.txt item: #527 of 601 id: ejde-685 author: Hoang, Luan title: Behavior near the extinction time for systems of differential equations with sublinear dissipation terms date: 2025 words: 11419 flesch: 84 summary: This expression and properties (5.15), (5.18) imply, as t → T− ∗ , |y(t)− (T∗ − t)1/αξ∗| = O ( (T∗ − t)1/α(|eh1(t) (6.24) Utilizing this estimate in (6.23) gives |Rλj v(t)|2 16 L. HOANG EJDE-2025/08 ≤ e−θµ ∫ t t̄ (T∗−τ)−1dτ |Rλjv(t̄)|2 + C6 ∫ t t̄ e−θµ ∫ t τ (T∗−s)−1ds(T∗ − τ)−1+2δ/αdτ = (T∗ − t)θµ (T∗ − t̄)θµ |Rλjv(t̄)|2 + C6(T∗ − t)θµ ∫ t t̄ (T∗ − τ)−1+2δ/α−θµdτ = (T∗ − t)θµ (T∗ − t̄)θµ |Rλj v(t̄)|2 + C6(T∗ − t)θµ 2δ/α− θµ ( (T∗ − t̄)2δ/α−θµ − (T∗ − t)2δ/α−θµ ) . keywords: equation; extinction; function; o((t∗; t)1; theorem; time; tmax; y(t cache: ejde-685.pdf plain text: ejde-685.txt item: #528 of 601 id: ejde-686 author: Ngai , Sze-Man; Zhang, Meng-Ke; Zhao, Wen-Quan title: Nodal sets and continuity of eigenfunctions of Krein-Feller operators date: 2025 words: 9801 flesch: 83 summary: Hence the Green function G(x,y) is symmetric on Ω × Ω i.e., there exists x0 ∈ Ω such that u1(x0) = 0. keywords: eigenfunctions; function; g(x; nodal; theorem cache: ejde-686.pdf plain text: ejde-686.txt item: #529 of 601 id: ejde-69 author: Yu, Xiaozhu; ing, Shiwen; Lian, Hairong title: Positive solutions for a class of phi-Laplacian differential systems with multiple parameters date: 2022 words: 5887 flesch: 88 summary: We prove the existence of positive solutions under the φ-super-linear condition by means of the Guo-Krasnosel’skii fixed point theorem and the topological degree. There are lots of im- portant results on positive solutions of nonlinear problems. keywords: solution cache: ejde-69.pdf plain text: ejde-69.txt item: #530 of 601 id: ejde-696 author: Gnanasekaran, Shanmugasundaram; Nithyadevi, Nagarajan title: Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity date: 2024 words: 7213 flesch: 75 summary: ′ ≤ 3 4 ∫ T 0 ∫ Ω ∣∣∇u1ϵ ∣∣4/3 + d41|Ω|T 4 + M1 2 ∫ T 0 ∫ Ω u2 1ϵ + M1 2 ∫ T 0 ∫ Ω |∇v1ϵ|2 + M2 2 ∫ T 0 ∫ Ω u2 1ϵ + M2 2 ∫ T 0 ∫ Ω |∇v2ϵ|2 + δ1 ∫ T 0 ∫ Ω u1ϵ + δ1 ∫ T 0 ∫ Ω u2 1ϵ + δ1a1 2 ∫ T 0 ∫ Ω u2 1ϵ + δ1a1 2 ∫ T 0 ∫ Ω u2 2ϵ + ϵ ∫ T 0 ∫ Ω uq 1ϵ ≤ C(T + 1). if u1 ∈ L2 loc ( (0,∞);L2(Ω) ) , u2 ∈ L2 loc ( (0,∞);L2(Ω) ) , v1 ∈ L2 loc ( (0,∞);W 1,2(Ω) ) , v1 ∈ L2 loc ( (0,∞);W 1,2(Ω) ) EJDE-2024/56 WEAK SOLUTION TO A TUMOR CHEMOTAXIS COMPETITION SYSTEM 5 and satisfies − ∫ ∞ 0 ∫ Ω u1ϕt = d1 ∫ Ω u10ϕ0 − ∫ ∞ 0 ∫ Ω ∇u1 · ∇ϕ+ ∫ ∞ 0 ∫ Ω χ1(v1)u1∇v1∇ϕ + ∫ ∞ 0 ∫ Ω χ2(v2)u1∇v2∇ϕ+ δ1 ∫ ∞ 0 ∫ Ω u1(1− u1 − a1u2)ϕ, − ∫ ∞ 0 ∫ Ω u2ϕt = d2 ∫ Ω u20ϕ0 − ∫ ∞ 0 ∫ Ω ∇u2 · ∇ϕ+ ∫ ∞ 0 ∫ Ω ξ1(v1)u2∇v1∇ϕ + ∫ ∞ 0 ∫ Ω ξ2(v2)u2∇v2∇ϕ+ δ2 ∫ ∞ 0 ∫ Ω u2(1− u2 − a2u1)ϕ, − ∫ ∞ 0 ∫ Ω v1ϕt = d3 ∫ Ω v10ϕ0 − ∫ ∞ 0 ∫ Ω ∇v1 · ∇ϕ+ α1 ∫ ∞ 0 ∫ Ω u1ϕ+ β1 ∫ ∞ 0 ∫ Ω u2ϕ − γ1 ∫ ∞ 0 ∫ Ω v1ϕ, − ∫ ∞ 0 ∫ Ω v2ϕt = d4 ∫ Ω v20ϕ0 − ∫ ∞ 0 ∫ Ω ∇v2 · ∇ϕ+ α2 ∫ ∞ 0 ∫ Ω u1ϕ+ β2 ∫ ∞ 0 ∫ Ω u2ϕ − γ2 ∫ ∞ 0 ∫ Ω v2ϕ, for all ϕ ∈ C∞ 0 (Ω× [0,∞)). keywords: system; u1ϵ; δ1 ∫; ω u2; ∫ t; ∫ ω cache: ejde-696.pdf plain text: ejde-696.txt item: #531 of 601 id: ejde-698 author: Ding, Xin; Zheng, Xiu Min Zheng title: Forms of entire solutions of partial differential difference equations with constant coefficients date: 2024 words: 6118 flesch: 83 summary: Some examples confirm the existence and the forms of transcendental entire solutions with finite order of such equations. Introduction In this article, we consider transcendental entire solutions of certain quadratic trinomial partial differential difference equations (PDDEs) in C2, related to the Fermat type functional equations with constant coefficients. keywords: equation; f(z1; solutions; ∂z1 cache: ejde-698.pdf plain text: ejde-698.txt item: #532 of 601 id: ejde-7 author: Díaz Palencia, José Luis title: Semigroup theory and asymptotic profiles of solutions for a higher-order Fisher-KPP problem in R^N date: 2023 words: 6373 flesch: 71 summary: − S(t)v‖2 ≤ ‖S(t)‖2 ‖S(τ)v − v‖2 ≤ mewt‖S(τ)v − v‖2. − v1)− v2(g − v2)]|2 } dξ = ∫ Γr Υ(ξ) { |(v1 − v2)(g − (v1 − v2))|2 + 4∑ k=1 k∑ i=1 ∣∣(k keywords: equation; fisher; function; operator; order; solutions cache: ejde-7.pdf plain text: ejde-7.txt item: #533 of 601 id: ejde-70 author: Guesmia, Aissa title: Decay rates for two Cauchy thermoelastic laminated Timoshenko problems of type III with interfacial sli date: 2022 words: 10464 flesch: 86 summary: − ε)|ẑ|2 + (k3λ3 − ε)|φ̂|2 + (λ2 − ε)|û|2 + (λ4 − λ3 − ε)|θ̂|2) − ξ2((k1λ5 − k1λ4 − k1λ2 − ε)|v̂|2 + ( |γ|λ0 − λ1 − λ5 − ε ) |ŷ|2 + (k4 − ε)|σ̂|2) + Cε,λ0,...,λ9 f̃(ξ)|η̂|2. (5.26) − λ4 − λ6 − λ7. keywords: 2ε0; case; equations; multiplying; timoshenko cache: ejde-70.pdf plain text: ejde-70.txt item: #534 of 601 id: ejde-700 author: Mennuni, Federica; Salvatore, Addolorata title: Radial bounded solutions for modified Schrodinger equations date: 2024 words: 11675 flesch: 83 summary: [a(x, un,∇un) · ∇un +At(x, un,∇un)un] dx + ∫ B+ k,n k un a(x, un,∇un) · ∇un dx+ ∫ B+ k,n |un|p−2un(un − k) dx − ∫ B+ k,n g(x, un)R + k un dx ≥ α1 ∫ B+ k,n a(x, un,∇un) · ∇un dx− ∫ B+ k,n g(x, un)R + k un dx. Hence, from the previous inequalities, (A5) and (A6) it follows that α0α1 η0 ∫ B+ k,n |∇un|p dx ≤ ⟨dJ (un), R + k un⟩+ ∫ B+ k,n g(x, un)R + k un dx. keywords: a(x; a.e; g(x cache: ejde-700.pdf plain text: ejde-700.txt item: #535 of 601 id: ejde-703 author: Cho, Namkyeong title: Global gradient estimates  for  shear thinning-type Stokes system on the non-smooth domains date: 2024 words: 12270 flesch: 86 summary: dx ≤ δ, (1.8) where δ is the same as the one chosen in (i). Then,∫ B+ 8 φ(η|d+Dw|) dx ≤ c ∫ B+ 8 η2|d+V (Dw)|2 dx+ c ∫ B+ 8 φ(|∇w|) dx, (3.5)∫ B+ 8 |πw|2 dx ≤ c ∫ B+ 8 φ(|∇w|) φ∗(1) + 1 dx. (3.6) 12 N. CHO EJDE-2024/47 Proof. keywords: c ∫; dx+ c; ejde-2024/47; equations; lemma; shear; solution; ε ∫; φ(|∇w|; ∫ −ω8 cache: ejde-703.pdf plain text: ejde-703.txt item: #536 of 601 id: ejde-71 author: Kurima , Shunsuke title: Existence for a nonlocal Penrose-Fife type phase field system with inertial term date: 2023 words: 6422 flesch: 92 summary: Therefore, since vh − vτ = vh − v̂h + v̂τ − vτ + v̂h − v̂τ , we can prove Lemma 5.1 by (5.3)-(5.7), the Schwarz inequality, the Young inequality, (2.11), (2.13), Lemmas 4.1, 4.3, 4.8. − 1 2 ‖θn‖2L2(Ω) + 1 2 ‖θn+1 − θn‖2L2(Ω) + h(−∆un+1, θn+1)L2(Ω) = h(fn+1, θn+1)L2(Ω) − h(vn+1, θn+1)L2(Ω). (4.11) Here, since un+1 = − 1 θn+1 , θn+1 > 0, and gn+1 ≤ 0, we have that h(−∆un+1, θn+1)L2(Ω) = h ∫ Ω ∇un+1 · ∇θn+1 + h ∫ ∂Ω un+1θn+1 − h ∫ ∂Ω gn+1θn+1 ≥ h ∫ Ω |∇ ln θn+1|2 − h|∂Ω|. keywords: l2(ω; lemma cache: ejde-71.pdf plain text: ejde-71.txt item: #537 of 601 id: ejde-718 author: Doresic, Tvrtko; Pazanin, Igor title: Curved-pipe flow with boundary conditions involving Bernoulli pressure date: 2024 words: 7258 flesch: 77 summary: − V 1 0 ∂V 1 0 ∂y2 = 0 in Ω , −ν∆y∗V 3 1 + ∂P2 ∂y3 − V 1 0 ∂V 1 0 ∂y3 = 0 in Ω , ∂V 2 1 ∂y2 + ∂V 3 1 ∂y3 = 0 in Ω, V 2 1 = V 3 1 = 0 on Γ . y∗)V 1 0 ∂V 1 0 ∂y3 − V 1 1 ∂V 1 0 ∂y3 − V 1 0 ∂V 1 1 ∂y3 = 0 in Ω , which, since V 2 1 = V 3 1 = 0 and V 1 0 = V 1 0 (y∗), reduces to the problem − ν ( ∆y∗V 2 2 + κ′V 1 0 cosα+ κτV 1 0 sinα ) + ∂P3 ∂y2 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y2 − V 1 1 ∂V 1 0 ∂y2 − V 1 0 ∂V 1 1 ∂y2 = 0 in Ω , − ν ( ∆y∗V 3 2 − κ′V 1 0 sinα+ κτV 1 0 cosα ) + ∂P3 ∂y3 − κ(eα · y∗)V 1 0 ∂V 1 0 ∂y3 − V 1 1 ∂V 1 0 ∂y3 − V 1 0 ∂V 1 1 ∂y3 = 0 in Ω , ∂V 1 1 ∂x1 + ( κ′ (eα · y∗) + κτ ( e⊥α · y∗ )) keywords: cosα; pipe; sinα; ε κ; ∂y2; ∂y3 cache: ejde-718.pdf plain text: ejde-718.txt item: #538 of 601 id: ejde-719 author: Gil, Michael title: Delay-dependent stability conditions for delay differential equations with unbounded operators in Banach spaces date: 2024 words: 2462 flesch: 77 summary: = ϕ(t) (−h ≤ t ≤ 0), (1.2) where ϕ ∈W ([−h, 0],X ) ∩D(A) is given. + ∫ t 0 eA(t−s)By(s− h)ds. keywords: differential; stability cache: ejde-719.pdf plain text: ejde-719.txt item: #539 of 601 id: ejde-72 author: Ye, Qin; Zhang, Yinghui title: Space-time decay rates of a two-phase flow model with magnetic field in R^3 date: 2023 words: 10490 flesch: 86 summary: Let (ρ− ρ̄, u, n− n̄, v, B) be the strong solution to the system (1.1)- (1.2) with initial data (ρ0 − ρ̄, u0, n0 − n̄, v0, B0) belonging to the Schwartz class S. Under the assumptions in Theorem 1.1, then there exists a large enough T such that ‖∇k(u− v)(t)‖L2 γ ≤ C(1 + t)− 5 4− k 2+γ , (1.15) EJDE-2023/41 SPACE-TIME DECAY RATES 7 for all t > T , 0 ≤ k ≤ `− 2 and γ ≥ 0, where C is a positive constant independent of t. Now, let us outline the strategies for proving Theorem 1.1 and 1.4, and explain the main difficulties in the process. γ−1 γ , (1.16) for t is large enough and γ > 3 2 , where E(t) := ‖(m,u, σ, v,B)‖2L2 γ and C0, C1, C2 are positive constants independent of t. Applying Lemma 2.5 for (1.16) and the interpolation trick, we show that the Theorem 1.1 holds for case k = 0. keywords: decay; inequality; lemma; space; v)‖2l2 γ; γ−1 cache: ejde-72.pdf plain text: ejde-72.txt item: #540 of 601 id: ejde-720 author: Karakostas, George L. title: Solving linear differential equations with mixed arguments date: 2024 words: 8678 flesch: 84 summary: Differential equations with mixed arguments; nonsingular matrices. 1 2 G. L. KARAKOSTAS EJDE-2024/54 In this article we present a type of differential equations and give some answers to these questions. keywords: differential; equation; interval; m=0; solution cache: ejde-720.pdf plain text: ejde-720.txt item: #541 of 601 id: ejde-73 author: Li, Cuicui; Liu, Fang title: Viscosity solutions to the infinity Laplacian equation with lower terms date: 2023 words: 9623 flesch: 80 summary: Then there exist x0 ∈ Ω and ϕ ∈ C2(Ω) such that ϕ(x0) = u(x0), u(x)− ϕ(x) ≥ 0, x ∈ Bρ(x0) ⊆ Ω, for some ρ > 0, but ∆h ∞ϕ(x0) > f(x0, u(x0)). We establish the existence and uniqueness of viscosity solutions to the Dirichlet problem ∆h ∞u = f(x, u), in Ω, u = q, on ∂Ω, where q ∈ C(∂Ω), h > 1, ∆h ∞u = |Du|h−3∆∞u. keywords: c(ω; existence; f(x; infinity; problem; solution; viscosity cache: ejde-73.pdf plain text: ejde-73.txt item: #542 of 601 id: ejde-734 author: Karuppusamy, Yadhavan; Lingeshwaran, Shangerganesh; Jeyaraj, Manimaran title: Solvability of an attraction-repulsion chemotaxis Navier-Stokes system with arbitrary porous medium diffusion date: 2024 words: 10842 flesch: 77 summary: ∥u∥1+α 1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22 ) + ∫ T 0 ( ∥∇u 1+α 2 ∥22 + ∥∇u 1+2α 2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22 ) and either one of the assumptions (2.5) or (2.59) holds by replacing R3 by Ω. Then for each T > 0, system (2.2) with boundary conditions (2.74) possesses a weak soluion (u, v, w, z) that satisfies sup 0≤t≤T (∫ Ω u| log u| dx+ ∥u∥1+α 1+α + ∥∇v∥22 + ∥∇w∥22 + ∥z∥22 ) + ∫ T 0 ( ∥∇u 1+α 2 ∥22 + ∥∇u 1+2α 2 ∥22 + ∥∆v∥22 + ∥∆w∥22 + ∥∇z∥22 ) keywords: + 2α; chemotaxis; existence; p+α; solutions; system; ∥22; ∫ r3 cache: ejde-734.pdf plain text: ejde-734.txt item: #543 of 601 id: ejde-741 author: de Carvalho, Pitágoras; Demarque, Reginaldo; Límaco, Juan; Viana, Luiz title: Null-controllability for 1-D degenerate quasilinear parabolic equations date: 2025 words: 9251 flesch: 74 summary: Moreover, if u0 ∈ H1 a , then u ∈ U := H1(0, T ;L2(0, 1)) ∩ L2(0, T ;H2 a) ∩ C0([0, T ];H1 a), and there exists a constant CT > 0 such that sup t∈[0,T ] (∥u(t)∥2H1 a ) + ∫ T 0 ( ∥ut|2L2(0,1) + ∥(aux)x∥2L2(0,1) ) ≤ CT ( ∥u0∥2H1 a + ∥g∥2L2(Q) + ∥h∥2L2(Qω) ) . ∥δ, that is, ∥f∥δ = (∫ T 0 ∫ 1 0 δf2 dx dt )1/2 for each f ∈ L2(Q; δ). keywords: controllability; equations; l2(q cache: ejde-741.pdf plain text: ejde-741.txt item: #544 of 601 id: ejde-745 author: Sikorska-Nowak, Aneta title: Existence of pseudosolutions for dynamic fractional differential equations date: 2024 words: 4217 flesch: 75 summary: In this article, we consider the existence of pseudosolutions for boundary value problem for fractional differential equations of the form C T ∆αx(t) = f(t, x(t)), for t ∈ Ia = [0, a] ∩ T, x(0) = x0, x0 ∈ E, where C T ∆αx(t), α ∈ (0, 1] denotes the Caputo fractional derivative, T denotes a time scale, and the function f is weakly-weakly sequentially continuous with values in a Banach space E and satisfies some boundary conditions and con- ditions expressed in terms of measures of weak non-compactness. In this paper, we consider the existence of a pseudosolution for the boundary value problem for fractional differential equations of the form C T∆ αx(t) = f(t, x(t)), for t ∈ Ia = [0, a] ∩ T, x(0) = x0, x0 ∈ E, (1.1) where C T∆ αx(t), α ∈ (0, 1] is the Caputo fractional derivative, T denotes a time scale. keywords: fractional cache: ejde-745.pdf plain text: ejde-745.txt item: #545 of 601 id: ejde-747 author: Zheng, Lan-Ling; Ding, Hui-Sheng title: Massera type theorems for abstract non-autonomous evolution equations date: 2024 words: 4532 flesch: 80 summary: As appli- cation, we present an existence result on periodic mild solutions to abstract nonautonomous semilinear evolution equations. As application of our Massera type theorems, in the last part of this paper, we establish an existence result on periodic mild solutions to (1.2). keywords: lim; periodic cache: ejde-747.pdf plain text: ejde-747.txt item: #546 of 601 id: ejde-75 author: Xu, Hong Yan; Haldar, Goutam title: Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2 date: 2023 words: 8547 flesch: 87 summary: In view of the fact that a1c1 + a2c2 = 4kπi, k ∈ Z, it follows from the second equation of (3.23) that∫ z1 0 [G0(z2 − βz1 + c2 − βc1)−G0(z2 − βz1)]dz1 +G1(z2 − αz1 + c2 − αc1)−G1(z2 − αz1) = 0. Let δ = η = 4, ξ = 5, c1 = 2, c2 = 3, a0 = 1, L(z) = z1 − z2 and g(z1, z2) keywords: equations; f(z1; order; solutions; view cache: ejde-75.pdf plain text: ejde-75.txt item: #547 of 601 id: ejde-754 author: Banerjee, Abhijit; Sarkar, Jhuma title: Existence and forms of entire solutions to system  of non-linear partial differential equations date: 2024 words: 6169 flesch: 86 summary: Let b10 = 1, b01 = 1, b11 = 1, b20 = 1, b02 = 1, d1 = 1, d2 = −1, c1 = c2 = 1, W1 = πi 4 , W2 = πi 4 . The main objective of this article is to explore the existence and forms of transcendental entire solutions of some systems of non-linear partial differential equations. keywords: s(r; solution cache: ejde-754.pdf plain text: ejde-754.txt item: #548 of 601 id: ejde-756 author: Jia, Yan-Na; Jin, Can; Yu, Xiu-Fang title: Output tracking for a 1-D wave equations with spatially varying coefficients and subject to unknown disturbances date: 2024 words: 5019 flesch: 78 summary: = −ma(1)ỹ2t (1, t)− ka(0)ỹ2t (0, t). Solving (2.34) gives Φ̂(t) = eΥtΦ̂(0) + ∫ t 0 eΥ(t−τ)K̂χ(τ)dτ. keywords: control; equation; error; output; system; tracking; wave cache: ejde-756.pdf plain text: ejde-756.txt item: #549 of 601 id: ejde-759 author: Banagere Erajikkappa, Manjunath; Waghamore, Harina P. title: Entire solutions for non-linear differential-difference equations date: 2025 words: 3921 flesch: 74 summary: Preliminaries To prove our results, we first give some Lemmas as follows: The first Lemma presents the difference analogs of the Logarithmic Derivative Lemma, a crucial tool in investigating complex difference equations. [2] Chen, M. F.; Cui, N.; On zeros and growth of solutions of complex difference equations, Adv. Difference Equ., 2021 (2021), 16. keywords: differential; equations; solutions cache: ejde-759.pdf plain text: ejde-759.txt item: #550 of 601 id: ejde-76 author: Hu, Rong; Sofonea, Mircea title: Duality arguments for well-posedness of history-dependent variational inequalities date: 2022 words: 4413 flesch: 77 summary: Problem P. Find a function u ∈ C([0, T ];V ) such that the following inequality holds: u(t) ∈ K(t) and (Au(t), v − u(t))V + (Su(t), v − u(t))V ≥ (f(t), v − u(t))V (3.1) for all v ∈ K(t) and t ∈ Under assumptions (H2)–(H4), the operator D : C([0, T ];V )→ C([0, T ];V ) defined by Du(t) = Au(t) + Su(t)− f(t) ∀u ∈ C([0, T ];V ), t ∈ keywords: c([0; problem cache: ejde-76.pdf plain text: ejde-76.txt item: #551 of 601 id: ejde-761 author: Diaz Palencia, Jose Luis title: Instability of energy solutions, travelling waves, and scaling invariance for a fourth-order p-Laplacian operator with superlinear reaction date: 2024 words: 8596 flesch: 66 summary: Solution profiles for low values of TW-speed. The analysis of problem (1.1) begins with the definition of energy solutions, as proposed for general diffusion in [26]. keywords: diffusion; energy; equation; lemma; operator; order; problem; solution cache: ejde-761.pdf plain text: ejde-761.txt item: #552 of 601 id: ejde-77 author: Penney, Richard C.; Urban, Roman title: Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces date: 2022 words: 5211 flesch: 82 summary: Then from (4.1) for 0 < s < t,∫ t 0 EaLσt−sβ v(s, x, σt−s) = Ea ∫ t 0 Lσt−sβ Uσ(0, s)f(x, σt−s) = ∫ t 0 Ea ∫ `2 ∞∑ j=1 e2λj(σt−s)βj∂ 2 xjv(x+ y)N2[β]Aσ(s,0)(dy) ds. Since ∣∣Ea ∫ t 0 e2λj(σt−s) ds ∣∣ = ∣∣Ea ∫ t 0 e2λj(σu) du ∣∣ = ∣∣Ea ∫ t 0 e2(σu)j du ∣∣ = ∣∣Ea ∫ t 0 e2(bu)j−αjt du ∣∣ ≤ ∣∣∣∣Ea ∫ t 0 e2(bu)j du ∣∣∣∣ = Ct we obtain (since β ∈ `1) that |Laβv(s, x, a)| ≤ ∣∣∣‖v‖2∞ ∞∑ j=1 βjEa ∫ t 0 e2λj(σt−s) ds ∣∣∣ ≤ ‖v‖2∞Ct ∣∣ ∞∑ j=1 βj ∣∣ ≤ Ct‖v‖∞‖β‖`1 . keywords: measure; poisson cache: ejde-77.pdf plain text: ejde-77.txt item: #553 of 601 id: ejde-771 author: Zhang, Xuping; Ding, Kaibo; Chen, Pengyu title: Existence of positive S-asymptotically omega-periodic solutions of time-space fractional  nonlocal reaction-diffusion equations date: 2025 words: 7154 flesch: 75 summary: [29] X. Shu, F. Xu, Y. Shi; S-asymptotically ω-positive periodic solutions for a class of neutral fractional differential equations. Tk 0 (Tk − s)α−1Kα,β(Tk − s)G(s, u(s))ds + ∫ t 0 (t− s)α−1Kα,β(t− s)G(s, u(s))ds. (3.2) Moreover, if u(t) ≥ θ for all t ≥ 0, then it is said to be a positive mild solution of nonlocal problem (3.1). keywords: fractional; k=1 cache: ejde-771.pdf plain text: ejde-771.txt item: #554 of 601 id: ejde-772 author: Khaider, Hassan; Azanzal, Achraf ; Raji, Abderrahmane title: Well-posedness of solutions for the 2D stochastic quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces date: 2024 words: 4414 flesch: 75 summary: Consequently, stochastic partial differential equa- tions (SDE) such as quasi-geostrophic equations (QG), stochastic Navier-Stokes equations are gaining more and more interest in fluid mechanics research. To examine how stochastic forces affect quasi-geostrophic equations, we first present the outcome of the deterministic quasi-geostrophic equations, or the case g = 0 in (1.1). keywords: equations; sup; ∥φj cache: ejde-772.pdf plain text: ejde-772.txt item: #555 of 601 id: ejde-774 author: Garg, Swati; Sardar, Bidhan Chandra Sardar title: Optimal control problem for Stokes systems: asymptotic analysis via unfolding method in a perforated domain date: 2023 words: 8584 flesch: 73 summary: [9] B. Cabarrubias; Homogenization of optimal control problems in perforated domains via pe- riodic unfolding method, Appl. MR 2563641 [26] I. Mishra; Homogenization of boundary optimal control problem, Electron. keywords: boundary; control; domain; l2(o; problem; stokes cache: ejde-774.pdf plain text: ejde-774.txt item: #556 of 601 id: ejde-778 author: Liu, Mengqian; Niu, Lei; Wu, Zhigang title: Existence and uniqueness of global strong solutions for 3D fractional compressible systems date: 2025 words: 10341 flesch: 76 summary: Specifically, |I22| = | − 3∑ i=1 ⟨[∆j , µ′ (κ+ 1 aρ) a − µ]DiΛ 2αu, Diuj⟩ | ≲ 3∑ i=1 cj2 −js∥ρ∥Hs+1∥Di∇Λ2α−2u∥Hs∥Diuj∥, |I23| = | − ⟨[∆j ,u · ∇]∇ρ,∇Λ2ρj⟩| ≲ cj2 −js∥∇u∥Hs∥∇ρ∥Hs∥∇Λ2ρj∥, |I24| = | − ⟨[∆j ,u · ∇]∇u,∇Λ2uj⟩| ≲ cj2 −js∥∇u∥Hs∥∇u∥Hs∥∇Λ2uj∥, |I25| = | − 1 a ⟨[∆j , ρ]∇divu,∇Λ2ρj⟩| ≲ cj2 −js∥∇ρ∥Hs∥ divu∥Hs∥∇Λ2ρj∥, |I26| = | − 1 a ⟨[∆j , ρ]∇2ρ,∇Λ2uj⟩| ≲ cj2 −js∥∇ρ∥Hs∥∇ρ∥Hs∥∇Λ2uj∥, |I27| = |⟨[∆j , ( µ′ (κ+ 1 aρ) a − µ)]∇Λ2αu,∇Λ2uj⟩| ≲ cj2 −js∥ρ∥Hs+1∥Λ2αu∥Hs∥∇Λ2uj∥. Inserting the above inequalities about I1 − I27 into (3.16), we have 1 2 d dt (∥ρj∥2 + ∥uj∥2 + 3∑ i=1 ∥Diρj∥2 + 3∑ i=1 ∥Diuj∥2 + ∥Λ2ρj∥2 + ∥Λ2uj∥2 + 2β1⟨∇ρj ,uj⟩+ 2β2 3∑ i=1 ⟨Di∇ρj , Diuj⟩) + ⟨ µ′ (κ+ 1 aρ) a Λαuj ,Λ αuj⟩ + 3∑ i=1 ⟨ µ′ (κ+ 1 aρ) a DiΛ αuj , DiΛ αuj⟩ + ⟨( µ′ (κ+ 1 aρ) a − µ)∇Λ1+αuj ,∇Λ1+αuj⟩ + µ∥Λ2+αuj∥2 + β1κ∥∇ρj∥2 + β2κ 3∑ i=1 ∥Di∇ρj∥2 ≲ β1κ∥divu∥2 + β2κ 3∑ i=1 ∥Di divu∥2 + 8ϵ1∥∇ρj∥2 + 5ϵ1∥Λ2ρj∥2 + 4ϵ1 3∑ i=1 ∥Di∇ρj∥2 + 5ϵ1∥Λαuj∥2 (3.17) + 4ϵ1 3∑ i=1 ∥ΛαDiuj∥2 + 7ϵ1∥ΛαΛ2uj∥2 + 2ϵ1 3∑ i=1 ∥Λα divDiuj∥2 + ϵ1∥∇Λ1+αuj∥2 + Cϵ1β 2 1∥Λ2αuj∥2 + Cϵ1β 2 2 3∑ i=1 ∥DiΛ 2αuj∥2 + 2−2jCϵ1(∥∆j(u · ∇ρ)∥2 + ∥∆j(ρdivu)∥2) + 2−2jαCϵ1(∥∆j(u · ∇u)∥2 + ∥∆j(ρ∇ρ)∥2) EJDE-2025/35 FRACTIONAL COMPRESSIBLE SYSTEMS 17 + Cϵ1 3∑ i=1 ( ∥Di∆j(u · ∇ρ)∥2 + ∥Di∆j(ρdivu)∥2 ) + 2−2jαCϵ1 3∑ i=1 ( ∥Di∆j(u · ∇u)∥2 + ∥Di∆j ( ρ∇ρ)∥2 + ∥∥∆j(Di( µ′ (κ+ 1 aρ) a − µ)Λ2αu) ∥∥2) + Cϵ1 ( ∥ div∆j(∇u · ∇ρ)∥2 + ∥ div∆j(∇ρdivu)∥2 ) + 2−2jαCϵ1∥∇u∥2L∞∥∇2uj∥2 + 2−2jαCϵ1∥ div u∥2L∞∥Λ2uj∥2 + 2−2jαCϵ1∥div∆j(∇u · ∇u)∥2 + 2−2jαCϵ1∥div∆j(∇ρ∇ρ)∥2 + Cϵ1∥∇u∥2L∞∥∇2ρj∥2 + Cϵ1∥ divu∥2L∞∥Λ2ρj∥2 + Cϵ1∥∇ρ∥2L∞∥∇ divuj∥2 + 2−2jαCϵ1∥∇ρ∥2L∞(∥∇2ρj∥2 + ∥Λ2ρj∥2) + 2−2jαCϵ1∥div∆j(∇( µ′ (κ+ 1 aρ) a − µ)Λ2αu)∥2 + 2−2jαCϵ1β 2 1(∥∇∆j(u · ∇ρ)∥2 + ∥∇∆j(ρdivu)∥2) + Cϵ1β 2 1 ( ∥∆j(ρ∇ρ)∥2 + ∥∆j(u · Similarly, |I3| = | − 3∑ i=1 ⟨Di∆j(u · ∇ρ), Diρj⟩ − 3∑ i=1 1 a ⟨Di∆j(ρdivu), Diρj⟩| ≲ Cϵ1 3∑ i=1 ∥Di∆j(u · ∇ρ)∥2 + Cϵ1 3∑ i=1 ∥Di∆j(ρ divu)∥2 + 2ϵ1∥∇ρj∥2, 14 M. LIU, L. NIU, Z. WU EJDE-2025/35 |I4| keywords: 3∑ i=1; ejde-2025/35; equations; ḣs−1; i=1; lemma; navier; solution; stokes; − µ)λ2αu; ⟨[∆j cache: ejde-778.pdf plain text: ejde-778.txt item: #557 of 601 id: ejde-78 author: Teles, Ricardo de Sa title: Pullback attractors for non-autonomous Bresse systems date: 2022 words: 7207 flesch: 86 summary: =  ϕ′ ψ′ w′ k ρ1 (ϕx + ψ + lw)x + k0l ρ1 (wx − lϕ) b ρ2 ψxx − k ρ2 (ϕx + ψ + lw) k0 ρ1 (wx − lϕ)x − kl ρ1 (ϕx + ψ + lw)  , with domain D(B1) = (H2(0, L) ∩ H1 0 (0, L))3 × H1 0 (0, L)3, and B2 : H → H, is given by B2y (3.23) Now we estimate Ψ′(t), Ψ′(t) = ρ1‖ϕt‖2 + ρ2‖ψt‖2 + ρ1‖wt‖2 − b‖ψx‖2 − k‖ϕx + ψ + lw‖2 − k0‖wx − lϕ‖2 − ∫ keywords: + ψ; − ∫ cache: ejde-78.pdf plain text: ejde-78.txt item: #558 of 601 id: ejde-784 author: Kryspin, Marek title: Oseledets decomposition on sub semiflow date: 2024 words: 6644 flesch: 78 summary: For any family of subspaces {W (ω)}ω∈Ω0 of X1 such that the equality U (1) ω (t)W (ω) = W (θtω) holds for all t ≥ 0 and all ω ∈ Ω0, there exists a family of subspaces {V (ω)}ω∈Ω0 of X2 such (i) iV (ω) = W (ω) for any ω ∈ Ω0, (ii) U (2) ω (t)V (ω) = V (θtω) for all t ≥ 0 and ω ∈ Ω0. U (1) ω (1) ∣∣ E (1) j (ω) ◦G(ω)−1, ω ∈ Ω0, generates a two-sided discrete-time linear skew-product dynamical system Φ̂ = ((Ûω(n)), (θn)) on Ω0 × Rl, with Ûω(n) := G(θnω) ◦ keywords: decomposition; oseledets; θ−1ω cache: ejde-784.pdf plain text: ejde-784.txt item: #559 of 601 id: ejde-786 author: Dong, Sitong; Zhang, Xin; Jin, Yuanfeng title: A second order convergent difference scheme for the initial-boundary value problem of Rosenau-Burgers equation date: 2024 words: 5088 flesch: 83 summary: − 1, 0 ≤ k ≤ N − 1, (3.8) V k i = δ2xU k i +Qk i , 1 ≤ i ≤M − 1, 0 ≤ k ≤ N, (3.9) there exist constants c1 and c2 such that |Rk+ 1 2 i | ≤ c1(τ 2 + h2), 1 ≤ i ≤M − 1, 0 ≤ k ≤ N − 1, |Qk i | ≤ i − ψ(uk+ 1 2 , uk+ 1 2 )i] = R k+ 1 2 i , 1 ≤ i ≤M − 1, 0 ≤ k ≤ N − 1, (5.1) fki = δ2xe k i +Qk i , 1 ≤ i ≤M − 1, 0 ≤ k ≤ N, (5.2) e0i = 0, 1 ≤ i ≤M − 1, (5.3) ek0 = 0, ekM = 0, 0 ≤ k ≤ N, (5.4) fk0 = 0, fkM = 0, 0 ≤ k ≤ N. (5.5) It follows from (5.2) that f k+ 1 2 i = δ2xe k+ 1 2 i +Q k+ 1 2 i , 1 ≤ i ≤M − 1, 0 ≤ k ≤ N − 1. keywords: scheme cache: ejde-786.pdf plain text: ejde-786.txt item: #560 of 601 id: ejde-788 author: Polyakov, Dmitry M. title: Asymptotic behavior of eigenvalues of fourth-order differential operators with spectral parameter in the boundary conditions date: 2024 words: 11170 flesch: 84 summary: Recall that z = λ1/4, z ∈ Z, λ ∈ C, where Z = { z ∈ C : arg z ∈ ( − π 4 , π 4 ]} , Z = { z ∈ C : arg z ∈ ( − π 4 , π 4 )} . × ( 1 + (ζσ,14 − ζσ,24 − ζσ,34 + ζσ,44)(1, z) +O(z−σ−1) ) keywords: + o(n−2; asymptotics; behavior; det; differential; eigenvalues; equation; form; matrix; order cache: ejde-788.pdf plain text: ejde-788.txt item: #561 of 601 id: ejde-79 author: Al Nazer, Safaa; Rosier, Carole; Tsegmid, Munkhgerel title: Mathematical analysis of a Dupuit-Richards model date: 2022 words: 10132 flesch: 75 summary: Dupuit approximation reads H̃ ' H|z=h− , the pressure P thus satisfies in Ω−t P (t, x, z) = ρ0g ( H̃(t, x)− z ) for t ∈ = {u(t, ·) ∈ H1(Ωt), t ∈ keywords: equation; flow; hbot; model; problem; richards cache: ejde-79.pdf plain text: ejde-79.txt item: #562 of 601 id: ejde-792 author: Gao, Siyu; Liu, Qingbo; Sun, Yingxin title: Local bifurcation structure and stability of the mean curvature equation in the static spacetime date: 2024 words: 3754 flesch: 77 summary: Bifurcation; mean curvature operator; stability. [6] C. Bereanu, P. Jebelean, J. Mawhin; The Dirichlet problem with mean curvature operator in Minkowski space-a variational approach, Adv. keywords: curvature; mean; λ1 h0 cache: ejde-792.pdf plain text: ejde-792.txt item: #563 of 601 id: ejde-795 author: Artes, Joan C.; Llibre, Jaume; Schlomiuk, Dana; Vulpe, Nicolae title: Abel quadratic differential systems of second kind date: 2024 words: 18843 flesch: 79 summary: Moreover we detect that the slope of the line is greater than the slope of the flow on the line, because we have SlL − f − √ 1− 4b √ f2 − 4b 2b = ( √ 1− 4b+ 1)( √ f2 − 4b− f) 4b > 0. = (h+ 1)xy, ẏ = − f2 (h− 1)2 + fy − x2 + hy2. keywords: case; condition; invariant; picture; siv; systems; t4 = cache: ejde-795.pdf plain text: ejde-795.txt item: #564 of 601 id: ejde-81 author: huy, Le Thi title: Asymptotic behavior of solutions to 3D Kelvin-Voigt-Brinkman-Forchheimer equations with unbounded delays date: 2022 words: 6719 flesch: 84 summary: − φ(θ‖ ≤ ‖Pmφ(θm) It is easy to check that if u, v, w ∈ V , then b(u, v, w) = −b(u,w, v), and in particular, b(u, v, v) = 0, ∀u, v ∈ V. (2.2) Using Hölder’s inequality and Ladyzhenskaya’s inequality, we can choose the best positive constant c0 such that |b(u, v, w)| ≤ c0‖u‖‖v‖|w|1/2‖w‖1/2, ∀u, v, w ∈ V. (2.3) From (2.3) and using Poincaré’s inequality (2.1), we obtain |b(u, v, w)| ≤ c0λ−1/4 1 ‖u‖‖v‖‖w‖, ∀u, v, w ∈ V. (2.4) 4 L. T. THUY EJDE-2022/07 We will assume that f ∈ L2(0, T ;V ′). keywords: equations; solution cache: ejde-81.pdf plain text: ejde-81.txt item: #565 of 601 id: ejde-811 author: Li, Chunhong; Zhou, Tiantian title: Discrete Stein-Weiss inequalities date: 2025 words: 6401 flesch: 90 summary: When we cut off f = (fi)i∈Zn N and g = (gi)i∈Zn N , inequality (1.3) is reduced to∑ |i|≤N,|j|≤N,i̸=j |fi||gj | |i− j|λ ≤ KN∥f∥lr(Zn N )∥g∥ls(Zn N ), ∀(f, g) ∈ lr(Zn N )×ls(Zn N ), (1.4) where KN ∈ (0,∞) is the best constant, and Zn N := {i ∈ Zn; |i| ≤ N}. Denote the extremal sequences of (1.13) by (fN , gN ) and |fN i1 | = max{|fN i |; |i| ≤ N}, |gNi2 | = max{|gNi |; |i| ≤ N} We write a(N) keywords: 0,n; j∈zn cache: ejde-811.pdf plain text: ejde-811.txt item: #566 of 601 id: ejde-816 author: Baladi, Houssam; Aglzim, Abdellatif; Filali, Mohammed; Tsouli, Najib title: Multiple solutions for p(x)-Kirchhoff type problems with extended Robin boundary conditions date: 2024 words: 9704 flesch: 83 summary: +G(x, u) ) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λh(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u+ g(x, u) = 0, x ∈ ∂Ω, (1.1) where Ω is a bounded domain in RN with smooth boundary ∂Ω, ∂u ∂ν is the outer normal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L1(∂Ω), β− := infx∈∂Ω β(x) > 0, g : ∂Ω × R → R is a measurable function, with G(x, t) :=∫ t 0 g(x, s) ds, p ∈ C+(Ω̄), 1 < p− := inf x∈Ω̄ p(x) ≤ p+ := max x∈Ω̄ p(x) < = o ( |t|αp+−1 ) , t→ 0, uniformly a.e. x ∈ Ω; (A5) There exists a constant µ > αp+ such that µF (x, t) := µ ∫ t 0 f(x, s) ds ≤ f(x, t)t, ∀(x, t) ∈ Ω× R; (A6) inf{x∈Ω;|t|=1} F (x, t) > 0. (A7) g(x, t) = o(|t|r1(x)−1) uniformly a.e. x ∈ ∂Ω, as t→ 0, where r1 ∈ C+(∂Ω), supx∈∂Ω r1(x) = r+1 < p− ≤ p(x) for all x ∈ ∂Ω; (A8) g(x, t) = o ( |t|r2(x)−1 ) , t→ +∞, uniformly a.e. x ∈ ∂Ω, where r2 ∈ C+(∂Ω), supx∈∂Ω r2(x) = r+2 < p− ≤ p(x) for all x ∈ ∂Ω; (A9) G(x, t) := ∫ t 0 g(x, s) ds ≥ 0, ∀(x, t) ∈ ∂Ω× R, where α and µ are given in (A1) and (A5). keywords: dx+ ∫; dσx; g(x; p(x; β(x; − ∫; ∫ ω cache: ejde-816.pdf plain text: ejde-816.txt item: #567 of 601 id: ejde-817 author: Shpakivskyi, Vitalii title: Construction of solutions to PDEs using holomorphic functions of several variables date: 2024 words: 5138 flesch: 70 summary: Holomorphic functions of several complex variables; holomorphic functions of several hypercomplex variables; harmonic algebra; elliptic PDE. (6.4) The general solution of equation (6.4) is the function f(φ,ψ, η) = ∫∫ g(φ,ψ, η)dφdφ+ φh1(ψ, η) + h2(ψ, η), where h1, h2 are arbitrary holomorphic functions in the domain Q2 := {(ψ, η) ∈ C2 : (x, y, z) ∈ Ω}. keywords: equation; functions; holomorphic; variables cache: ejde-817.pdf plain text: ejde-817.txt item: #568 of 601 id: ejde-82 author: Feng, Meiqiang; Chen, Haiping title: Existence and nonexistence of positive solutions for fourth-order elliptic problems date: 2023 words: 6199 flesch: 81 summary: New criteria for the existence and nonexistence of positive solution are established under some sublinear conditions which involve the principal eigenvalues of the corresponding linear problems. [23] used a variant version of Mountain Pass Theorem to demonstrate the existence and nonexistence of positive solution for the fourth-order elliptic prob- lem ∆2u = f(x, u) in Ω, u = ∆u = 0 on ∂Ω, where Ω denotes a smooth bounded domain in Rn (n > 4). keywords: g(x cache: ejde-82.pdf plain text: ejde-82.txt item: #569 of 601 id: ejde-823 author: Purushothaman, Ganesh ; Suresh, Kannan ; Thandapani, Ethiraju ; Tunc, Ercan title: Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations date: 2024 words: 4831 flesch: 76 summary: Assuming (3.1) and (3.2), we define G such that for t ≥ t2, 0 < G(t) if τ(t) ≤ t, 1− p(t)R12(τ(t)) R12(t) Q12(τ(t)) Q12(t) if τ(t) ≥ t, (3.3) Assuming (3.1), we define G1 such that for t ≥ t2, 0 < G1(t) keywords: differential; equations cache: ejde-823.pdf plain text: ejde-823.txt item: #570 of 601 id: ejde-83 author: Shen, Xuhui; Ding, Juntang title: Blow-up for parabolic equations in nonlinear divergence form with time-ependent coefficients date: 2022 words: 6277 flesch: 86 summary: xj − k(t)f(u) in Ω × (0, t∗), n∑ i,j=1 aij(x)uxiνj = g(u) on ∂Ω × (0, t∗), u(x, 0) = u0(x) ≥ 0 in Ω, where Ω is a bounded convex domain in Rn (n ≥ 2) with smooth boundary ∂Ω. By constructing suitable auxiliary functions and using a differential inequality technique, when Ω ⊂ Rn (n ≥ 2) Published January 25, 2022. 1 2 X. SHEN, J. DING EJDE-2022/08 ( h(u) ) t = n∑ i,j=1 ( aij(x)uxi ) xj − k(t)f(u) in Ω× (0, t∗), n∑ i,j=1 aij(x)uxi νj = g(u) on ∂Ω× (0, t∗), u(x, 0) = u0(x) ≥ 0 in Ω, (1.1) where Ω is a bounded convex domain in Rn (n ≥ 2) with smooth boundary ∂Ω, (aij(x))n×n is a differentiable positive definite matrix, ν is the outward normal vector to ∂Ω, u0(x) is the initial value, t∗ is the maximal existence time of u, and Ω is the closure of Ω. Set R+ = (0,+∞). keywords: p−1; ∫ ω cache: ejde-83.pdf plain text: ejde-83.txt item: #571 of 601 id: ejde-84 author: Ferreira, Jorge; Piskin, Erhan; Shahrouzi, Mohammad; Cordeiro , Sebastiao; Raposo, Carlos Alberto title: Existence of global weak solutions for a p-Laplacian inequality with strong issipation in noncylindrical domains date: 2022 words: 4880 flesch: 79 summary: Let f ∈ L2(0, T, L2(Ωt)), u0 ∈ W 1,p 0 (Ω0), u1 ∈ L2(Ω0) ∩K, with K being a convex and closed subset of W 1,p 0 (Ω), and 0 ∈ K. Lets us suppose that (H1) and (H2) are satisfied. To this end, let ũ0 ∈ W 1,p 0 (Ω), ũ1 ∈ L2(Ω), and f̃ ∈ L2(Q0) be the extensions to zero outside Ω0 of u0, u1, and f , respectively. keywords: 1,p cache: ejde-84.pdf plain text: ejde-84.txt item: #572 of 601 id: ejde-846 author: Corcho, Adan J.; Mallqui, Lindolfo P. title: L^2 Solutions for cubic NLS equation with higher order fractional elliptic/hyperbolic operators on R cross T and  R^2 date: 2025 words: 6614 flesch: 78 summary: First note that S̃2 = 4 ⌊(C+K) 1 2α ⌋∑ n=⌊C 1 2α ⌋+1 √ C +K − n2α ≤ 4 √ K + ∫ ⌊(C+K) 1 2α ⌋ ⌊C 1 2α ⌋+1 √ C +K − z2αdz ≤ 4 √ K + ∫ (C+K) 1 2α C 1 2α √ C +K − z2αdz ≤ 4 √ K + √ K ( (C +K) 1 2α − C 1 2α ) . Considering the nonlinear change of variables ρ2α = z2α − C, and using that α > 1, C > 0 and K ≥ 1, we have J2 = K ∫ ∞ K 1 2α ρ2α−1 ρα ( ρ2α + C )1− 1 2α dρ ≤ K ∫ ∞ K 1 2α ρ−αdρ = 1 α− 1 KK 1−α 2α ≲α K. (2.17) Then, inserting (2.17) in (2.16) one obtains S− 2 ≲α K. Therefore, from (2.14) and the estimates obtained for S− 1 and S− 2 we have m ( G− 1,K ) ≲α K. By the same way we have m ( G− 2,K ) ≲α K. So, m ( G− K ) keywords: case; |n|2α cache: ejde-846.pdf plain text: ejde-846.txt item: #573 of 601 id: ejde-85 author: Wen, Lan; Yang, Lu title: Dynamics of a non-autonomous stochastic weakly damped plate model with critical exponent date: 2022 words: 9013 flesch: 83 summary: (4.29) Let T = T1 = 4c45K0 σ1ε in (4.18), for τ − t ≤ r ≤ τ , we have∫ τ r ‖w1(s)‖q−1 2 ds ≤ ε 4c45 (τ − r) + = ( u(t+ τ, τ, θ−τω, ϕτ (θ−τω)) ut(t+ τ, τ, θ−τω, ϕτ ) + εu(t+ τ, τ, θ−τω, ϕτ )− h(x)z(θtω) ) over R and (Ω,F ,P, (θt)t∈R), where Φ(0, τ, ω)ϕτ (ω) = ϕτ (θ−τω) and Φ(t, τ − t, θ−tω)ϕτ−t(θ−tω) = ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω)). keywords: θ−τω cache: ejde-85.pdf plain text: ejde-85.txt item: #574 of 601 id: ejde-850 author: Doan, Thai Son Doan; Huong, Phan Thi; Kloeden, Peter E. title: theta-scheme for solving Caputo fractional differential equations date: 2025 words: 4829 flesch: 77 summary: Write Y (n) k = Y (n)(kTn ) with Y (n) 0 = y0. , n− 1, which gives Y (n) 1 = Y (n) 0 + f(t0, Y (n) 0 ) T n , Y (n) 2 = Y (n) 0 + f(t0, Y (n) 0 ) T n + f(t1, Y (n) 1 ) T n = Y (n) 1 + f(t1, Y (n) 1 ) T n , and so on, culminating in Y (n) k+1 = Y (n) k + f(tk, Y (n) k ) T n . keywords: scheme cache: ejde-850.pdf plain text: ejde-850.txt item: #575 of 601 id: ejde-856 author: Agarwal, Ravi ; Baltaeva, Umida; Hubert, Florence; Khasanov, Boburjon title: Existence and uniqueness of  the solution to initial and inverse problems for integro-differential heat equations with fractional load date: 2024 words: 7840 flesch: 75 summary: Fractional diffusion equations are extensions of the basic equations of mathe- matical physics Find a solution u(x, t) in the domain (x, t) ∈ Rn T of the loaded heat equation ut − a(t)∆u = λD−α 0t u(x ′, t) + ∫ t 0 k(x′, τ)u(x, t− τ)dτ, (x, t) ∈ Rn T , (2.1) that satisfies the condition u(x, t) ∣∣ t=0 = φ(x), x ∈ Rn, (2.2) where D−α 0t is the Riemann-Liouville fractional integral operator of order α defined by D−α 0t u(x ′, t) keywords: a(θ−1(τ; equation; fractional; inverse; problem; t 0; ∫ rn; ∫ θ(t; ∫ θ−1(τ cache: ejde-856.pdf plain text: ejde-856.txt item: #576 of 601 id: ejde-86 author: Hao, Jianghao; Yang, Jing title: Exponential stability for porous thermoelastic systems with Gurtin-Pipkin flux date: 2023 words: 6384 flesch: 76 summary: ρv − µuxx − bϕx − γvxx = ρf2, (3.6) ϕ− w = f3, (3.7) Introduction In this work, we consider the porous thermoelastic transmission system with Gurtin-Pipkin flux, ρutt − µuxx − bϕx − γuxxt = 0 in (0, 1)× R+, Jϕtt − δϕxx + bux + ξϕ+ βθx keywords: stability; system; ∫ ∞ cache: ejde-86.pdf plain text: ejde-86.txt item: #577 of 601 id: ejde-860 author: Figueiredo, Giovany M.; Kiametis, George title: Caffarelli-Kohn-Nirenberg type problems with Berestycki-Lions type nonlinearities date: 2024 words: 4667 flesch: 85 summary: Let u ∈ Erad, then for almost every x ∈ RN\{0}, then there exists C = C(a, b, p) > 0 such that |u(x)| ≤ C 1 |x| (N−p)−ap∗ p ∥u∥. Proof. Hence, there exists u ∈ Erad such that, up to a subsequence, un ⇀ u in Erad. keywords: lemma; |x|−bp∗ cache: ejde-860.pdf plain text: ejde-860.txt item: #578 of 601 id: ejde-87 author: Zhang, Bo; Liu, Xiangqing title: Localized nodal solutions for semiclassical quasilinear Choquard equations with subcritical growth date: 2022 words: 10373 flesch: 90 summary: 2−p 2 ≤ c (∫ RN (|∇uk|p−2∇uk − |∇ul|p−2∇ul,∇uk −∇ul) dx )p/2 → 0, as k, l→∞, ∫ RN E(εx)|uk − ul|p dx ≤ c (∫ RN (|uk|p−2uk − |ul|p−2ul)(uk − ul) dx )p/2 × (∫ RN (|uk|p + |ul|p) dx ) 2−p 2 ≤ c (∫ RN (|uk|p−2uk − |ul|p−2ul)(uk − ul) dx )p/2 → 0, as k, l→∞ and ∫ RN exp{(m− p) dist(εx,M)}|(uk − ul)|m dx ≤ c (∫ RN (kε(x, uk)− kε(x, ul))(uk − ul) dx )m/2 × (∫ RN εm−p exp{(m− p) dist(εx,M)}(|uk|m + |ul|m) dx ) 2−m 2 ≤ c ( ∫ RN (kε(x, uk)− kε(x, ul))(uk − ul) dx )m/2 → 0, as k, l→∞. So {un} is a Cauchy sequence in Xε. � 10 B. ZHANG, X. LIU EJDE-2022/11 3. keywords: 1,p(rn; choquard; dx dy; equations; kε(x; lemma; y|α; |x−; ∫ rn cache: ejde-87.pdf plain text: ejde-87.txt item: #579 of 601 id: ejde-88 author: Mishra, Indira title: Homogenization of boundary optimal control problem date: 2022 words: 8744 flesch: 81 summary: Let Y , T and Y ∗ be as follows: Y = (0, 1)N is a reference cell, or more generally a set having the paving property with respect to a basis (b1, . . = 0 in Y ∗, A(x, y)[∇yµi(x, y) + ei] · ν = 0 on ∂Y ∗\∂Y, y 7→ µi(x, y) is Y -periodic. keywords: boundary; control; problems; ω×y cache: ejde-88.pdf plain text: ejde-88.txt item: #580 of 601 id: ejde-881 author: Zhang, Xiang; Kang, Ming; Geng, Fengjie title: Dynamic behavior of a stochastic predator-prey model with stage-structure and nonlinear perturbation date: 2025 words: 8631 flesch: 72 summary: = − lnx1 − lnx2. 6 X. ZHANG, M. KANG, F. GENG EJDE-2025/32 Combining (3.1) and (3.2), we have LV1 = −r x2 x1 + αy +m+ d1 + (σ11 + σ12x1) 2 2 −m x1 x2 + sx2 + βy (1 + ax2)(1 + by) + d2 + (σ21 + σ22x2) 2 2 ≤ −rx2 x1 − mx1 x2 + αy +m+ d1 + σ2 11 2 + σ11σ12x1 + σ2 12 2 x2 1 + sx2 + β b + d2 + σ2 21 + σ2 22x 2 2 ≤ ( −2 √ rm+m+ d1 + β b + d2 + σ2 11 2 + σ2 21 ) + αy + σ11σ12x1 + σ2 12 2 x2 1 + sx2 + σ2 22x 2 2. (3.3) We define V2(x1) = u1(x1 + u2) v v , where u1 and u2 are positive constants which will be determined later, v ∈ (0, 1) is adequately small. +mx1( 1 √ x2 − 1 x2 )− σ2 12x 5 2 1 4 + σ2 12x 2 1 + pα 2 x2 1 − σ2 22x 5 2 2 4 + σ2 22x 2 2 + sx2 + qβ 2 x2 2 − σ2 32y 5 2 4 + αy + pα 2 y + qβ 2 y + δy + σ2 32y 2 +m 4 X. ZHANG, M. KANG, F. GENG EJDE-2025/32 + d1 + d2 + d3 + β b + σ2 11 + σ2 21 + σ2 31 ≤ rx2 4 + mx1 4 − σ2 12x 5 2 1 4 + σ2 12x 2 1 + pα 2 x2 1 − σ2 22x 5 2 2 4 + σ2 22x 2 2 + sx2 + qβ 2 x2 2 − σ2 32y 5 2 4 + αy + pα 2 y + qβ 2 y + δy + σ2 32y 2 +m+ d1 + d2 + d3 + β b + σ2 11 + σ2 21 + σ2 31 = ( −σ2 12x 5 2 1 4 + σ2 12x 2 1 + pα 2 x2 1 + mx1 4 ) keywords: + σ2; predator; prey; σ11σ12d2 cache: ejde-881.pdf plain text: ejde-881.txt item: #581 of 601 id: ejde-885 author: Gerberry, David; Joshi, Hem; Peloquin, Mac; Vargas, Sonia title: Respiratory Illness clinical trial modeling date: 2024 words: 5895 flesch: 53 summary: Of course, it is important to note that vaccine clinical trials do not occur in a vacuum but rather involve individuals (both in the control and treatment arms of the clinical trial) interacting with the general population under the current condi- tions of the epidemic. General simulation linking epidemiological dynamics to those of vaccine clinical trials in adults and children. keywords: adults; children; days; kids; population; trial; vaccine; wave cache: ejde-885.pdf plain text: ejde-885.txt item: #582 of 601 id: ejde-886 author: Kimura, Yasunori; Sasaki, Kazuya; Torii, Kakeru title: Convergence theorems of implicit type iterations in geodesic spaces with negative curvature date: 2024 words: 6193 flesch: 88 summary: For x, y ∈ X and l ≥ 0, a mapping c : Then, we denote the image of the geodesic with endpoints x, y ∈ X by [x, y], which is well defined. keywords: cosh; d(xn cache: ejde-886.pdf plain text: ejde-886.txt item: #583 of 601 id: ejde-887 author: Ludu, Andrei; Khanal, Harihar; Carstea, Adrian Stefan title: Nonlinear non-autonomous Boussinesq equations date: 2024 words: 10173 flesch: 61 summary: Boussinesq non-autonomous nonlinear system We consider a non-autonomous and nonlinear Boussinesq-type of differential system in the form qzt + (qu+ αzu)x + β 3 (qu)xxx = 0, qut + zx + αuux = 0, (2.1) for the solutions z(x, t), u(x, t) where (x, t) ∈ (−L,L)×[0,∞) and the space domain can be arbitrary extended L to∞. Subscripts x, t represent differentiation. Boussinesq non-autonomous nonlinear system 2.1. keywords: amplitude; boussinesq; case; coefficient; equations; nonlinear; numerical; soliton; solutions; system; variable; waves cache: ejde-887.pdf plain text: ejde-887.txt item: #584 of 601 id: ejde-888 author: Sharma, Anshul; Mishra, Suyash Narayan; Shukla, Anurag title: Asymptotic stability for Hilfer-like nabla nonlinear fractional difference equations date: 2024 words: 5554 flesch: 74 summary: We examined the existence and uniqueness theorem, asymptotic sta- bility of fractional nonlinear difference equations. Hilfer-like nabla operator; asymptotic stability; fractional difference equations; Lyapunov direct method. keywords: difference; fractional; stability; η(ω cache: ejde-888.pdf plain text: ejde-888.txt item: #585 of 601 id: ejde-889 author: Tarfulea, Nicoleta E. title: On drug therapy for an HIV infection age model with cellular and immune delays date: 2024 words: 4875 flesch: 68 summary: There have been a variety of modifications of HIV mathematical models that have resulted from incorporating drug therapies. Without treatment with HIV medicines, HIV infection advances in stages, getting worse over time. keywords: cell; hiv; infection; model; virus cache: ejde-889.pdf plain text: ejde-889.txt item: #586 of 601 id: ejde-890 author: Verma, Vijai Shanker; Kunwar, Laxman Bahadur title: Impact of vaccination and sterilization on the transmission dynamics of rabies date: 2024 words: 8317 flesch: 66 summary: Based on the parameter val- ues listed in the Table 2, we have used model (1.1)-(1.2) to simulate the data and we predicted the trend of exposed to rabies human population in Nepal. Thus, with the current control and prevention measures, dog and human rabies will persist endemically, which is also justified in Figure 3. 3.2. keywords: dogs; equilibrium; human; model; population; rabies cache: ejde-890.pdf plain text: ejde-890.txt item: #587 of 601 id: ejde-897 author: Huang, Rui; Ji, Shanming; Ma, Yansheng title: Cauchy problem for the Lane-Emden heat flow with sign-changing initial data date: 2024 words: 5350 flesch: 75 summary: Comparison principle of the heat equation implies that u(x, t) ≥ u(x) for t ∈ (0, Tmax), which means that Ω̂+ u0 ⊂ Ω+ u(x,t) and Ω− u(x,t) ⊂ v(x)dx ≥ (1− α1−p) (∫ Ω̂+ u0 u+(x, t) · v(x)dx )p(∫ Ω̂+ u0 v(x)dx )1−p = (1− α1−p) · (∫ Ω̂+ u0 v(x)dx )1−p zp(t), t ∈ (τ, Tmax), (2.10) which implies that z(t) blows up in finite time since p > 1 and z(τ) > 0. □ Proof of Theorem 1.1. keywords: u(x; ω̂+ cache: ejde-897.pdf plain text: ejde-897.txt item: #588 of 601 id: ejde-902 author: Salwahan, Shraddha; Abbas, Syed; Tridane, Abdessamad title: Optimal switching of vaccination for an infectious disease model date: 2024 words: 4791 flesch: 59 summary: Finally, some numerical simulations are performed to compare continuous vaccination programs and optimal switching of vaccination control. Hence, the system switches between two subsystems according to the presence of vaccination control. keywords: control; disease; model; switching; system; vaccination cache: ejde-902.pdf plain text: ejde-902.txt item: #589 of 601 id: ejde-911 author: Bai, Ruobing; Saanouni, Tarek title: Non global solutions for non-radial inhomogeneous nonlinear Schrodinger equations date: 2025 words: 9720 flesch: 87 summary: |−τ |u|p ) |x|−τ |u|p dx ) + 8 p (−τ − N − α 2 ) ∫ RN ( Jα ∗ | · = ( ( A B )1− B 2 (M[φ])p−1(M[u0]) −A/2 ) 2 B−2 ( 1− 2 B ) = B − 2 A ( (M[u0]) −A/2(M[φ])p−1 ) 2 B−2 = B − 2 A ( M[u0] )−αc ( M[φ] )2/sc . (3.67) Relations (3.66) and (3.67) imply that E [u0] < F (x1). keywords: schrödinger; |u|p; |−τ cache: ejde-911.pdf plain text: ejde-911.txt item: #590 of 601 id: ejde-92 author: Sabina de Lis, Jose C. title: Remarks on the second Neumann eigenvalue date: 2022 words: 5072 flesch: 78 summary: In the first one, X = Ω is a bounded set of Rn, endowed with the measure dµ = m(x)dx where m ∈ L1(Ω), m(x) > 0 if 1 < p < N, > 1 if p = N, = 1 if p > N. We define λ̂(m) = inf u∈M0\{0} ∫ Ω |∇u|p dx∫ Ω |u|pmdx , (3.1) with M0 = {u ∈ W 1,p(Ω) : ∫ Ω |u|p−2umdx = 0}. keywords: eigenvalue cache: ejde-92.pdf plain text: ejde-92.txt item: #591 of 601 id: ejde-931 author: Qiu, Ruowen; You, Renqing; Zhao, Fukun title: Infinitely many sign-changing solutions for an asymptotically linear and nonlocal schrodinger equation date: 2025 words: 7550 flesch: 81 summary: Indeed, if 0 ∈ I(W ∩ ∂M), then there exists u ∈ W ∩ ∂M such that ∫ RN F (x, u+) dx =∫ RN F (x, u−) dx. Now, we choose {yn} ⊂ RN such that RN ⊂ ⋃∞ i=1 Br(yi) and each x ∈ RN is covered by at most 2N balls. keywords: f(x; lemma cache: ejde-931.pdf plain text: ejde-931.txt item: #592 of 601 id: ejde-932 author: Chen, Kai; Wang, Jinrong title: Ulam type stability for nonlinear Hahn difference equations with delay date: 2024 words: 5132 flesch: 82 summary: Secondly, we examine the equation D2 q,ωx(s) = F (s, x(s),Dq,ωx(s), x(Θ(s))), s ∈ I1, x(s) = y(s), Dq,ωx(s) = Dq,ωy(s), s ∈ I2, (1.2) where F : I1 × R3 → R is continuous at s = ω0. ,D n−1 q,ω x(s), x(Θ(s))), s ∈ I1, x(s) = y(s), Dj q,ωx(s) = keywords: equation; stability; ulam cache: ejde-932.pdf plain text: ejde-932.txt item: #593 of 601 id: ejde-942 author: Zhang, Sen; Zu, Jian; Zhang, Jingqi title: Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems date: 2024 words: 7207 flesch: 62 summary: It is not difficult to know that θ′ = √ Λk − ρ0 − (ηu(x)− ρ0) sin 2 θ√ Λk − ρ0 ≤ √ Λk − ρ0. |Λj − Λk| − |λk − Λk| ≥ |j2 + ρ0 − k2 − ρ1| − |k2 + ρ1 − k2 − ρ0| ≥ |j2 − k2| − 2|ρ1 − ρ0| ≥ 1. keywords: boundary; eigenpairs; eigenvalue; liouville; method; problems; sturm cache: ejde-942.pdf plain text: ejde-942.txt item: #594 of 601 id: ejde-955 author: Ma, Wenhui; Ma, Qiaozhen title: Random attractors and their stability for nonclassical diffusion equations driven by additive white noise with delay and intensity date: 2025 words: 8303 flesch: 77 summary: Then for every τ ∈ R, ω ∈ Ω, ϵ ∈ (0, 1] and D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D, the solution of problem (3.4)-(3.5) satisfies ∥ d dt v(t, τ − t, θ−τω, ψ)∥2H1(Rn) ≤ Q1(∥v(t, τ − t, θ−τω, ψ)∥2H1(Rn) + ∥g(t)∥2 + ϵ|y(θt−τω)|2 + ∥u(t− ρ, τ − t, θ−τω, ϕ)∥2H1(Rn) + 1), (4.14) where Q1 > 0 is a constant independent of τ, ω,D and ψ ∈ D(τ − t, θ−tω). Proof. Then for every τ ∈ R, ω ∈ Ω, s ∈ [−ρ, 0], D = {D(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D and ψ ∈ D(τ − t, θ−tω), the solution of problem (3.4)-(3.5) satisfies lim k,t→+∞ ∫ Oc k ∥v(τ + s, τ − t, θ−τω, ψ)∥2H1(Rn)dx = 0. (4.22) Proof. keywords: h1(rn; ∈ r cache: ejde-955.pdf plain text: ejde-955.txt item: #595 of 601 id: ejde-96 author: Melzi, Imane; Atik, Youcef title: A nonlinear mathematical model for two-phase flow in nanoporous media date: 2022 words: 12468 flesch: 87 summary: For 1 ≤ p < ∞ and B a Banach space, we denote Lp(I;B) the Bochner space (of classes with respect to equiva- lence a.e.) of Bochner integrable functions u : I −→ B satisfying ∫ T 0 ‖u(t)‖pB dt < +∞. , N, (4.2) ∫ T 0 (φ∂−αt sα, ψ) dt+ ∫ ΩT λw(sα)K(∇pα)∇pα · ∇ψ dx dt + ∫ ΩT Λε(s α)p′c(s α)K(∇pα)∇sα · keywords: dx dt; sα(x; − ∫; ∫ ω cache: ejde-96.pdf plain text: ejde-96.txt item: #596 of 601 id: ejde-963 author: Bunoiu, Renata; Ramdani, Karim; Timofte, Claudia title: Asymptotic analysis of  sign-changing transmission problems with rapidly oscillating interface date: 2024 words: 7873 flesch: 75 summary: [7] A. Bonnet-Ben Dhia, L. Chesnel, P. Ciarlet Jr.; T -coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM Math. Positive and negative materials; transmission problem; asymptotic analysis; oscillating interface; imperfect interfaces; flux jump. keywords: problem cache: ejde-963.pdf plain text: ejde-963.txt item: #597 of 601 id: ejde-97 author: Li, Chunyang; Dong, Xiu; Wang, Jinliang title: Stability analysis of an age-structured viral infection model with latency date: 2022 words: 10642 flesch: 85 summary: + ∫ t 0 ξ(a)Ω(a)ê(t− a) da+ ∫ ∞ t ξ(a) Ω(a) Ω(a− t) e0(a− t) da. + ∫ t 0 ξ(a)Ω(a)ê(t− a)da. (4.5) keywords: infection; t t; t ∗v; β1 t; β2 t; ∫ ∞ cache: ejde-97.pdf plain text: ejde-97.txt item: #598 of 601 id: ejde-974 author: Xie, Junhui; Li, Pengfei title: Existence of solutions to fractional p-Laplacian problems with Robin boundary conditions date: 2025 words: 7060 flesch: 86 summary: Hence g′(t) = −tr−p−1(r − q)(r − p) ∫ Ω |u|r |x|α dx < 0. On the contrary, ifN0 λ ̸= ∅, then there exists u ∈ N0 λ, this implies ⟨Ψ′(u), u⟩ = 0, we can deduce that (p− q)∥u∥p Xs,p β ≤ (p− q)∥u∥p Xs,p β + (p− q) ∫ Rn\Ω β(x)|u|pdx = (r − q) ∫ Ω |u|r |x|α dx, (3.3) EJDE-2025/13 SOLUTIONS TO FRACTIONAL P-LAPLACIAN PROBLEMS 7 and (r − p)∥u∥p Xs,p β ≤ (r − p)∥u∥p Xs,p β + (r − p) ∫ Rn\Ω β(x)|u|pdx = (r − q)λ ∫ Ω |u|qdx. (3.4) By (2.2), we obtain (r − q) ∫ Ω |u|r |x|α dx ≤ (r − q)S−r/p α Ĉ−1∥u∥rXs,p β . keywords: |x|α cache: ejde-974.pdf plain text: ejde-974.txt item: #599 of 601 id: ejde-984 author: Zhang, Tianqing; Guo, Zhenyu title: Normalized solutions of fractional Kirchhoff equations: the defocusing case date: 2025 words: 8347 flesch: 82 summary: Setting u ∈ Pc,µ, by fractional Gagliardo-Nirenberg inequality and µ < 0 we obtain that a|(−∆)s/2u|22 ≤ δs,p|u|pp ≤ δs,pC(s, p) p| −∆s/2u|pδs,p2 |u|p(1−δs,p) 2 . (3.26) From the definition of Pc,µ, we obtain u ∈ Sc, namely, |u|2 = c. Let N ≥ 2, then Hs r (RN ) is compactly embedding into Lp(RN ) for p ∈ (2, 2∗s). keywords: fractional; pδs; solutions cache: ejde-984.pdf plain text: ejde-984.txt item: #600 of 601 id: ejde-988 author: Zhang, Xue; Zhao, Xiaopeng title: Strong solutions to density-dependent incompressible smectic-A liquid crystal equations date: 2025 words: 7035 flesch: 80 summary: ∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥∇∆φt∥4L2 + ∥∇φt∥4L2 + ∥∆2φ∥4L2 + ∥ut∥4L2 + 1), (3.39) J24 ≤ C∥∇ut∥L2∥∇φt∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇φt∥2L2 , (3.40) J25 ≤ C∥∆φt∥L2∥∇φ∥2L6∥ut∥L6 ≤ C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥H1 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇∆φt∥L2∥∇φt∥L2∥∇φ∥4H1 + C∥∇∆φt∥1/2L2 ∥∇φt∥1/2L2 ∥∇φ∥2H1∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥ut∥6L2 + ∥∇∆φt∥6L2 + ∥∇φt∥6L2 + ∥∇φ∥6H1 + 1), (3.41) J26 ≤ C∥∇φt∥L2∥∇φ∥L6∥∆φ∥L6∥ut∥L6 ≤ C∥∇φt∥L2∥∇φ∥2H2∥ut∥H1 ≤ µ4 28 ∥∇ut∥2L2 + C∥∇φt∥2L2∥∇φ∥4H2 + C∥∇φt∥L2∥∇φ∥2H2∥ut∥L2 ≤ µ4 28 ∥∇ut∥2L2 + C(∥∇∆φ∥6L2 + ∥∇φ∥6L2 + ∥∇φt∥6L2 + ∥ut∥6L2 + 1). Also, ∫ Ω |u||∇φ||φ|dx ≤ ∥u∥L3∥∇φ∥L2∥φ∥L6 ≤ C∥u∥H1∥φ∥2H1 ≤ C(∥u∥3H1 + ∥φ∥3H1). keywords: inequality; liquid; system cache: ejde-988.pdf plain text: ejde-988.txt item: #601 of 601 id: ejde-996 author: Belin, Théo; Lafitte, Pauline title: Quantitative estimates of L^p maximal regularity for nonautonomous operators and global existence for quasilinear equations date: 2025 words: 17719 flesch: 73 summary: Also for t ∈ Denote for ϵ > 0, Eϵ := {t ∈ I : ρ(t) > ϵ}. keywords: bounded; constant; continuity; mrp(i; nonautonomous; operators; regularity; theorem cache: ejde-996.pdf plain text: ejde-996.txt