ejde: A Pathfinder

This is a computer-generated pathfinder created against the Distant Reader study called ejde.

Each Distant Reader study carrel is composed of many individual items. Each item is bibliographically described with author, title, date, summary, and keyword values. Below is a list of the items' most signficant keywords as well as lists of the items themselves. Purpusing the content of this pathfinder provides the student, researcher, or scholar with one way to get their heads around the scope of the carrel. The keywords include:

Solutions; Lemma; System; Equations; Problem; Theorem; Equation; Function; Fractional; Differential; Lim; Proof; Model; Case; Wave; Schrödinger; Boundary; Sup; Stability; 1,p; Linear; I=1; Time; Operator; Periodic; Stochastic; Existence; G(x; U(x; Point; Energy; Matrix; Prey; P−1; Space; Log; U(t; Waves; Inequality; Spaces; Controllability; Proposition; Chemotaxis; Domain; Flow; Method; Poisson; Eigenvalue; F(z; |x−; Stokes; Set; Operators; Tmax; Div; Estimates; Non; A(x; Regularity; Bifurcation; Parameters; Eigenvalues; P(x; Groups; Kam; H1(ω; Kirchhoff; Inequalities; Λ,µ; Mean; 1,p(rn; 2−α; Beam; A(t; Soliton; |x|t; Γ(1; Global; M(c; Value; Ai(x; Eigenfunctions; Error; R2−n; 2∗α; |x|; Τ(t; E(t; H1(rn; Inverse; S(r; Scheme; Infection; H11; Nakra; Y|µ|y|α; 2∗∗; J−1; Resp; Y(r; 1,η; N−2α; M−1)/m; Q(x; P−q; C(α; Plasma; Functional; Φ(x; Landesman; 1,m; |∇u|2; Λ)h1(qt; Dr′; Varying; Er(t; N(a; D1,p(rn; P∗(α; R2n+1; A+1; P(t; T)3/2; G(t; Λ1(m; Λn−d; Lipschitz; Bégout; Θn−1; Aαβij; Kλ(x)−; Σ(rn; Β(log; |x|2; A)β; Β(0; Observability; Bridge; S−τ1(s; B21; |x′|2; M1,a; Pl)φ(tl; Klein; 2−β; Q(c; Ecm; R1−1; Cosα; Decomposition; 0,n; Θ−τω; Cosh; Sα(x; |x|α

Depending on how this pathfinder was created, many of the bibliographic sections will include elaborations on the meaning(s) of the given keywords. These elaborations were generated by feeding the items' summaries to a large langauge model and asking the model to address the question, "What is X?", where "X" is the keyword. The result will be a few sentences of elaboration. Be forewarned. The elaborations are often plausible, but they should not be take as truth. Instead, they should be taken as points for consideration.

• (omega, c)-periodic solutions for non-instantaneous impulsive systems with unbounded time-varying coefficients by Liu, Kui; Feckan, Michal; O'Regan, Donal; Wang, Jinrong (2022) - = 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
• Existence and multiplicity of solutions to quasilinear Dirac-Poisson systems by Yang, Minbo; Zhou, Fan (2025) - (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
• Existence of undercompressive travelling waves of a non-local generalised Korteweg-de Vries-Burgers equation by Achleitner, Franz ; Cuesta, Carlota M.; Diez-Izagirre, Xuban (2025) - 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
• Asymptotic profile of least energy solutions to the nonlinear Schrodinger-Bopp-Podolsky system by Ramos, Gustavo de Paula (2025) - 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
• Variational and numerical aspects of a system of ODEs with concave-convex nonlinearities by Agudelo, Oscar; Holubova, Gabriela; Kudlac, Martin (2025) - 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
• Positive solution to quasilinear Schrodinger equations via Orlicz space framework by Sun, Rui; Liu, Duchao (2022) - [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
• Existence, uniqueness and multiplicity of nontrivial solutions for biharmonic equations by Feng, Meiqiang; Lu, Yichen (2025) - 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
• Existence of three positive solutions for a p-sublinear problem involving a Schrodinger p-Laplacian type operator by Herron, Sigifredo; Lopera, Emer ; Sanchez, Diana (2025) - 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
• Asymptotic behavior of blowup solutions for Henon type parabolic equations with exponential nonlinearity by Chang, Caihong; Zhang, Zhengce (2023) - 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
• Concentrating normalized solutions for 2D nonlocal Schrodinger equations with critical exponential growth by Shen, Liejun; Squassin, Marco (2025) - 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
• New type of multi-bump solutions for Schrodinger-Poisson systems by Wang, Tao; Tian, Xiaoyu; He, Wenling (2025) - 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+
• Existence of at least four solutions for Schrodinger equations with magnetic potential involving and sign-changing weight function by de Paiva, Francisco Odair; Lima, Sandra Machado de Souza; Miyagaki, Olimpio Hiroshi (2023) - [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
• Complete classification of self-similar solutions for singular polytropic filtration equations by Zheng, Yanzhi; Yin, Jingxue; Ji, Shanming (2025) - ≤ −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
• Periodic solutions of stochastic Volterra equations by Chen, Feng (2022) - 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
• Localized nodal solutions for semiclassical nonlinear Kirchhoff equations by Wang, Lixia (2022) - 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
• Nehari manifold for degenerate logistic parabolic equations by Fernandes, Juliana; Maia, Liliane (2025) - 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
• Stationary and oscillatory dynamics of Nicholson's blowflies equation with Allee effect by Guo, Cuiping; Guo, Shangjiang (2022) - 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
• Existence and multiplicity of solutions to triharmonic problems by Wei, Qifan; Zhang, Xuemei (2025) - 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
• Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions by Zhao, Qiulan; Cheng, Hongbiao; Li, Xinyue; Li, Chuanzhong (2022) - (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
• Inverse scattering method for an integrable system of derivative nonlinear Schrodinger equations by Unlu, Mehmet (2025) - = 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
• Mixed local and nonlocal Schrodinger-Poisson type system involving variable exponents by Lin, Xiaolu; Zheng, Shenzhou (2022) - = 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
• Existence of positive periodic solutions for a nonlinear system of second-order ordinary differential equations by Han, Xiao; He, Yujing; Wei, Hui (2022) - 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
• Thermoelastic plates with type I heat conduction with second gradient by Munoz Rivera, Jaime; Ochoa Ochoa, Elena; Quintanilla, Ramon (2025) - 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
• Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2 by Xie, Zheng; Chen, Jing (2023) - • 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
• Regular solutions to elliptic equations by Castro, Alfonso; Jacobsen, Jon (2023) - 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
• Resonant solutions for elliptic systems with Neumann boundary conditions by Delgado, Briceyda B.; Pardo, Rosa (2023) - 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
• Yamabe boundary problem with scalar-flat manifolds target by Ghimenti, Marco G.; Micheletti, Anna Maria (2023) - 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
• Construction of single-peak solutions for Grushin equations via reduction method by Wei, Yawei; Zhou, Xiaodong (2025) - = ( 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
• Bifurcation from infinity with oscillatory nonlinearity for Neumann problems by Chhetri, Maya; Mavinga, Nsoki; Pardo, Rosa (2022) - 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
• Infinitely many solutions and asymptotics for resonant oscillatory problems by Korman, Philip; Schmidt, Dieter S. (2022) - 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
• Connected components of positive solutions of biharmonic equations with the clamped plate conditions in two dimensions by Ma, Ruyun; Zhao, Zhongzi; Yan, Dongliang (2021) - 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
• Radial and non-radial solutions for a nonlinear Schrodinger equation with a constraint by Yang, Jiaxuan; Li, Yongqing; Wang, Zhi-Qiang (2021) - 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
• On solutions arising from radial spatial dynamics of some semilinear elliptic equations by Valdebenito, Dario A. (2022) - 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
• Solutions to viscous Burgers equations with time dependent source term by Engu, Satyanarayana; Sahoo, Manas R.; Berke, Venkatramana P. (2021) - ∫ 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
• An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property by Sourdis, Christos (2021) - 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
• Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities by Antontsev, Antontsev; Ferreira, Jorge; Piskin, Erhan (2021) - = ∫ 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
• Ground state and multiple solutions for critical fractional Schrodinger-Poisson equations with perturbation terms by Liu, Lintao; Teng, Kaimin (2021) - (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
• Existence and concentration results for fractional Schrodinger-Poisson system via penalization method by Yang, Zhipeng; Zhang, Wei; Zhao, Fukun (2021) - 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
• Exact forms of entire solutions for Fermat type partial differential equations in C^2 by Chen, Yu Xian; Xu, Hong Yan (2021) - = 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
• Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N by Pu, Hongling; Li, Shiqi; Liang, Sihua; Repovs, Dusan D. (2021) - 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
• Periodic traveling waves and asymptotic spreading of a monostable reaction-diffusion equations with nonlocal effects by Han, Bang-Sheng; Kong , De-Yu Kong; Shi, Qihong; Wang, Fan (2021) - 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
• Existence of sign-changing solutions for radially symmetric p-Laplacian equations with various potentials by Wang, Wei-Chuan (2021) - [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
• Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent by Lin, Xiaolu; Zheng, Shenzhou (2021) - 〈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
• Monotone solutions of first order nonlinear differential systems by Wang, Lianwen; Mubarak, Abdulrahman (2021) - 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
• Non-radial normalized solutions for a nonlinear Schrodinger equation by Tong, Zhi-Juan; Chen, Jianqing; Wang, Zhi-Qiang (2023) - 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
• Normalized solutions for Sobolev critical Schrodinger-Bopp-Podolsky systems by Li, Yuxin; Chang, Xiaojun; Feng, Zhaosheng (2023) - 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
• Ground state solutions for quasilinear equations of Kirchhoff type by Zhao, Junfang; Liu, Xiangqing (2020) - 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
• Time periodic solutions for the non-isentropic compressible quantum hydrodynamic equations with viscosity in R^3 by Li, Min (2020) - − ‖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
• Short term unpredictability of high Reynolds number turbulence - rough dependence on initial data by Feng, Zaichun; Li, Y. Charles (2020) - 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
• Prescribed energy saddle-point solutions of nonlinear indefinite problems by Il'yasov, Yavdat; da Silva, Edcarlos Domingos; da Silva, Maxwell Lizete (2023) - 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
• Periodic solutions for evolution equations by Bostan, Mihai (2002) - |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
• Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations by Soriano Hernandez, Lorena; Siciliano, Gaetano (2023) - 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
• Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments by Chhetri, Maya; Girg, Petr; Hollifield, Elliott (2020) - 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
• Qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction by Iagar, Razvan Gabriel; Munoz, Ana I.; Sanchez, Ariel (2023) - 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
• Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations by Ahmad, Bashir; Alsaedi, Ahmed; Berbiche, Mohamed; Kirane, Mokhtar (2020) - = 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
• Continuous dependence of recurrent solutions for stochastic differential equations by Qiu, Haijing; Wang, Yan (2020) - [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
• Existence and multiplicity of homoclinic solutions for a difference equation by Heidarkhani, Shapour; Gharehgazlouei, Fariba; Imbesi, Maurizio (2020) - = 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
• Global low-energy weak solutions for compressible magneto-micropolar fluids with discontinuous initial data in R^3 by Wu, Wanping; Zhang, Yinghui (2023) - = 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
• Localized nodal solutions for semiclassical Choquard equations with critical growth by Zhang, Bo; Zhang, Wei (2024) - 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
• Crossed differential systems of equations and Clunie lemma by Gao, Yingchun; Liu, Kai; Qi, Xiaoguang (2024) - 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
• Periodic solutions in distribution for stochastic lattice differential equations by Gao, Yue; Yang, Xue (2024) - 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
• Existence of maximal and minimal weak solutions and finite difference approximations for elliptic systems with nonlinear boundary conditions by Bandyopadhyay, Shalmali; Lewis, Thomas; Mavinga, Nsoki (2025) - ∈ ∂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
• Existence of semi-nodal solutions for elliptic systems related to Gross-Pitaevskii equations by Silva, Joao Pablo Pinheiro da; Silva, Edcarlos Domingos da (2024) - = (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
• Forms of entire solutions of partial differential difference equations with constant coefficients by Ding, Xin; Zheng, Xiu Min Zheng (2024) - 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
• Semigroup theory and asymptotic profiles of solutions for a higher-order Fisher-KPP problem in R^N by Díaz Palencia, José Luis (2023) - − 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
• Solvability of an attraction-repulsion chemotaxis Navier-Stokes system with arbitrary porous medium diffusion by Karuppusamy, Yadhavan; Lingeshwaran, Shangerganesh; Jeyaraj, Manimaran (2024) - ∥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
• Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2 by Xu, Hong Yan; Haldar, Goutam (2023) - 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
• Entire solutions for non-linear differential-difference equations by Banagere Erajikkappa, Manjunath; Waghamore, Harina P. (2025) - 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
• Nonlinear non-autonomous Boussinesq equations by Ludu, Andrei; Khanal, Harihar; Carstea, Adrian Stefan (2024) - 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
• Normalized solutions of fractional Kirchhoff equations: the defocusing case by Zhang, Tianqing; Guo, Zhenyu (2025) - 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

• A priori estimates for the linearized relativistic Euler equations with a physical vacuum boundary and an ideal gas equation of state by Luczak, Brian B. (2025) - ∂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 −
• Long-time behavior of solutions to the 2D magnetic B\'enard problem in porous media on unbounded domains by Son, Dang Thanh (2025) - 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
• Normalized solutions for fractional Schrodinger-Choquard systems with Sobolev critical coupled nonlinearity by Chen, Zilin; Yang, Yang (2025) - 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
• Minimizers for fractional Schrodinger equations with inhomogeneous perturbation by Zhang, Lei; Liu, Lintao; Chen, Haibo (2025) - 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
• Existence of global solutions and blow-up for p-Laplacian parabolic equations with logarithmic nonlinearity on metric graphs by Wang, Ru; Chang, Xiaojun (2022) - 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
• Multiple solutions for parametric weighted (p,q)-equations by Zhang, Xiaohui; Xu, Xian (2025) - 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
• Structure and stability of global attractors for a Cahn-Hilliard tumor growth model with chemotaxis by Yayla, Sema (2025) - − 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
• Controllability under positivity constraints for non-linear and non-local parabolic PDEs by Nunez-Chavez, Miguel R. (2025) - − 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
• Blow-up prevention and rate of convergence of solutions for N-dimensional parabolic-parabolic systems with consumption of chemoattractant by Zheng, Jiashan; Wang, Yuying (2025) - + ∫ 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
• Existence of positive solutions for fractional Laplacian systems with critical growth by Correia, Jeziel N.; Oliveira, Claudionei P. (2022) - × 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−
• Gradient regularity for non-autonomous functionals with Dini or non-Dini continuous coefficients by Baroni, Paolo; Coscia, Alessandra (2022) - 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
• Well-posedness of stochastic time fractional 2D-Stokes models with finite and infinite delay by Xu, Jiaohui; Caraballo, Tomas (2022) - + ∫ 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
• Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions by Mavinga, Nsoki; Morris, Quinn A.; Robinson, Stephen B. (2023) - 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
• Generalized quasilinear equations with critical growth and nonlinear boundary conditions by Maia, Liliane de A.; Oliveira Junior, Jose Carlos; Ruviaro, Ricardo (2022) - 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
• Existence of multiple positive solutions for fractional Laplace problems with critical growth by Zhang, Yajing; Li, Qiaoqin; Pang, Lu (2021) - − 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
• Optimization problems and mathematical analysis of optimal values in Orlicz spaces by Donyari, Zahra; Zivari-Rezapour, Mohsen; Emamizadeh, Behrouz (2021) - 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
• Asymptotic behavior of linearized Boltzmann equations for soft potentials with cut-off by Wu, Yakui; Sun, Jiawei (2021) - 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
• Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator by Alvarez-Caudevilla, Pablo; Colorado, Eduardo; Ortega, Alejandro (2021) - 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
• Attractors for dissipative lattice differential equations with local and nonlocal nonlinearities by Pereira, Jardel Morais (2021) - 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
• Cubic differential systems with invariant straight lines of total multiplicity seven and four real distinct infinite singularities by Bujac, Cristina; Schlomiuk, Dana; Vulpe, Nicolae (2021) - = 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
• Extending Putzer's representation to all analytic matrix functions via omega matrix calculus by Neto, Antonio Francisco (2021) - 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
• Existence results for nonlinear Schrodinger equations involving the fractional (p,q)-Laplacian and critical nonlinearities by Lv, Huilin; Zheng, Shenzhou; Feng, Zhaosheng (2021) - 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
• Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities by Yang, Jie; Chen, Haibo; Feng, Zhaosheng (2020) - = 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
• Blow-up criteria and instability of standing waves for the fractional Schrodinger Poisson equation by Mo, Yichun; Zhu, Min; Feng, Binhua (2023) - 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
• Multiple positive solutions for biharmonic equation of Kirchhoff type involving concave-convex nonlinearities by Meng, Fengjuan; Zhang, Fubao; Zhang, Yuanyuan (2020) - 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
• Positive solutions for asymptotically 3-linear quasilinear Schrodinger equations by Li, Guofa; Cheng, Bitao; Huang, Yisheng (2020) - (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
• Palais-Smale approaches to semilinear elliptic equations in unbounded domains by Wang, Hwai-chiuan (2004) - 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
• Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems by Squassina, Marco (2006) - 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; ≤ ∫; ≥ ∫
• KdV type asymptotics for solutions to higher-order nonlinear Schrodinger equations by Naumkin, Pavel I.; Sanchez-Suarez, Isahi (2020) - 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
• Existence and concentration of positive ground states for Schrodinger-Poisson equations with competing potential functions by Wang, Wenbo; Li, Quanqing (2020) - 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
• Existence of solution for a segmentation approach to the impedance tomography problem by Mendoza, Renier; Keeling, Stephen (2020) - 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
• Time discretization of an abstract problem from linearized equations of a coupled sound and heat flow by Kurima, Shunsuke (2020) - 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∑
• Continuity of attractors for C^1 perturbations of a smooth domain by Barbosa, Pricila S.; Pereira, Antonio L. (2020) - − 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
• Nehari manifold approach for fractional p(.)-Laplacian system involving concave-convex nonlinearities by Biswas, Reshmi; Tiwari, Sweta (2020) - (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→∞; − ∫
• Asymptotically linear and superlinear elliptic equations with gradient terms by Wei, Yuanhong; Tian, Jian (2020) - 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
• Pyramidal traveling fronts in the Belousov-Zhabotinskii reaction-diffusion systems in R^3 by Ma, Luyi; Niu, Hong-Tao; Wang, Zhi-Cheng (2020) - = (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
• Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth by Pardo, Rosa; Sanjuan, Arturo (2020) - (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
• Concentration of nodal solutions for semiclassical quadratic Choquard equations by Yang, Lu; Liu, Xiangqing; Zhou, Jianwen (2023) - 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
• Existence of KAM tori for presymplectic vector fields by Bauer, Sean; Petrov, Nikola P. (2020) - = 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
• Stability of ground states of nonlinear Schrodinger systems by Cely, Liliana (2023) - = 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
• Asymptotic behavior of solutions to nonclassical diffusion equations with degenerate memory and a time-dependent perturbed parameter by Zhang, Jiangwei; Xie, Zhe; Xie, Yongqin (2024) - 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; ∫ ∞
• Existence of global solutions for cross-diffusion models in a fractional setting by Perez-Lopez, Jhean E.; Rueda-Gomez, Diego A.; Villamizar-Roa, Elder J. (2023) - 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
• A KAM theorem for degenerate infinite-dimensional reversible systems by Lou, Zhaowei; Wu, Youchao (2024) - 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
• Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion by Li, Yuequn; Liu, Hui; Guo, Fei (2024) - (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
• Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity by Severo, Uberlandio B.; Ribeiro, Bruno H. C.; Germano, Diogo de S. (2024) - 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
• Existence of positive solutions for systems of quasilinear Schrodinger equations by Baig, Ayesha; Li , Zhouxin (2025) - (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
• Ground state solutions for the nonlinear Schrödinger-Bopp-Podolsky systems with nonperiodic potentials by Jiang, Qiaoyun; Li, Lin; Chen, Shangjie; Siciliano, Gaetano (2024) - (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
• Global gradient estimates  for  shear thinning-type Stokes system on the non-smooth domains by Cho, Namkyeong (2024) - 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
• Existence for a nonlocal Penrose-Fife type phase field system with inertial term by Kurima , Shunsuke (2023) - 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
• Space-time decay rates of a two-phase flow model with magnetic field in R^3 by Ye, Qin; Zhang, Yinghui (2023) - 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
• Instability of energy solutions, travelling waves, and scaling invariance for a fourth-order p-Laplacian operator with superlinear reaction by Diaz Palencia, Jose Luis (2024) - 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
• Existence and uniqueness of global strong solutions for 3D fractional compressible systems by Liu, Mengqian; Niu, Lei; Wu, Zhigang (2025) - 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
• Caffarelli-Kohn-Nirenberg type problems with Berestycki-Lions type nonlinearities by Figueiredo, Giovany M.; Kiametis, George (2024) - 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∗
• Localized nodal solutions for semiclassical quasilinear Choquard equations with subcritical growth by Zhang, Bo; Liu, Xiangqing (2022) - 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
• Infinitely many sign-changing solutions for an asymptotically linear and nonlocal schrodinger equation by Qiu, Ruowen; You, Renqing; Zhao, Fukun (2025) - 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

• Final evolutions for Lotka-Volterra systems in R^3 having a Darboux invariant by Llibre, Jaume; Zhao, Yulin (2025) - 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
• Turrittin's normal forms for linear systems of meromorphic ODEs over the real field by Barkatou, Moulay; Carnicero, Félix Álvaro; Sanz, Fernando (2023) - 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
• Global well-posedness to a multidimensional parabolic-elliptic-elliptic attraction-repulsion chemotaxis system by Liu, Ling (2025) - 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
• Single-component regularity criterion and inviscid limit for axially symmetric MHD-Boussinesq systems by Xing, Zhaojun (2025) - ∥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
• Traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with nonlocal nonlinearities by Yang, Zhi-Jiao; Zhang, Guo-Bao; He, Juan (2025) - [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
• Dynamics of stochastic Lotka-Volterra predator-prey models driven by three independent Brownian motions by Li, Shangzhi; Guo, Shangjiang (2022) - 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
• Asymptotic stability for thermodiffusion Timoshenko systems of type III by Qin, Jiali; Hao, Jianghao (2025) - 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
• Long-time dynamics and upper-semicontinuity of attractors for a porous-elastic system with nonlinear localized damping by Santos, Mauro L.; Freitas, Mirelson M.; Caljaro, Ronal Q. (2025) - [ 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; − ∫
• Local solutions for a Brinkman equation coupled with heat-convective and concentration-diffusive equations and a volumetric mass source by Hong, Hakho (2025) - 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
• Stability of bistable traveling wavefronts for a nonlocal dispersal epidemic system by Hao, Yu-Cai; Zhang, Guo-Bao (2022) - 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
• Global asymptotic stability in quadratic systems by Llibre, Jaume; Valls, Claudia (2025) - 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
• Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force by Qin, Liuna; Xiao, Changguo; Zhang, Yinghui (2022) - 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
• Wave-breaking for two-component Fornberg-Whitham systems with dissipation by Zhu, Xi; Zhu, Min; Wang, Ying; Wang, Ke (2025) - 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
• Existence of solutions for a non-isothermal Navier-Stokes-Allen-Cahn system with thermo-induced coefficients by Lopes, Juliana Honda; Planas, Gabriela (2022) - 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
• Global well-posedness of 3D inhomogeneous incompressible nematic liquid crystal systems in critical Besov spaces with initial density perturbed around the equilibrium by Wu, Xinrui; Liang, Xingyu (2025) - 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 +
• Extending Lagrangian transformations to nonconvex scalar conservation laws by Dutta, Prerona (2022) - � 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
• Dynamics of a May-Leonard asymmetric system of ordinary differential equations by Dias, Fabio Scalco; Oliveira, Regilene; Valls, Claudia (2025) - 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
• Periodic solution and stationary distribution of a stochastic epidemic modelwith two different epidemics and different epidemiological frameworks by Mishra, Shivam Kumar; Abbas, Syed; Nieto, Juan Jose (2025) - − 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
• Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis by Zhao, Zhihong; Hu, Huanqin (2023) - (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
• Multiple positive solutions for nonhomogeneous Schrodinger-Poisson systems with Berestycki-Lions type conditions by Huang, Lan-Xin; Wu, Xing-Ping; Tang, Chun-Lei (2021) - 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
• A parabolic system with strong absorption modeling dry-land vegetation by Diaz, Jesus Ildefonso; Hilhorst, Danielle; Kyriazopoulos, Paris (2021) - 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
• Modeling porcine pseudorabies with age structure by Long, Yuhua; Chen, Yining (2021) - 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
• Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq by Llibre, Jaume; Oliveira, Regilene; Rodrigues, Camila A. B. (2021) - = 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
• Existence of bounded global solutions for fully parabolic attraction-repulsion by Chiyo, Yutaro; Mizukami, Masaaki (2021) - 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
• Dynamics of flocking models with two species by Zhao, Qingjian; Shi, Shaoyun; Li, Wenlei (2021) - ∈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
• Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent by Su, Yu; Chen, Haibo; Liu, Senli; Fang, Xianwen (2020) - = 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
• Analysis and simulations of the HANDY model with social mobility, renewables and nonrenewables by Shillor, Meir; Kadhim, Thanaa Ali (2023) - − 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
• Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential by Chen, Qing; Wu, Guochun; Zhang, Yinghui; Zou,  Lan (2020) - 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
• Asymptotic stability of a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps by Zhang, Mengqing; Tian, Jing; Zou, Keyue (2023) - 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
• Piecewise linear differential systems with an algebraic line of separation by Gasull, Armengol; Torregrosa,  Joan; Zhang, Xiang (2020) - − y = 0. (3.4) − y = 0, (3.5) that passes trough the origin. Keywords: curve; cycles; differential; limit; linear; piecewise; system
• Crossing limit cycles for a class of piecewise linear differential centers separated by a conic by Jimenez, Johana; Llibre, Jaume; Medrado, Joao C. (2020) - − 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
• Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity by Su, Si; Zhang, Guo-Bao (2020) - ‖(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
• Phase portraits of Bernoulli quadratic polynomial differential systems by Llibre, Jaume; Pereira, Weber F.; Pessoa, Claudio (2020) - 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
• Global dynamics of the May-Leonard system with a Darboux invariant by Oliveira, Regilene; Valls, Claudia (2020) - 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
• Controllability and stabilization of a nonlinear hierarchical age-structured competing system by He, Ze-Rong; Zhou, Nan (2020) - ∂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; − ∫
• Stability and sensitivity analysis of the epidemiological model Be-CoDiS predicting the spread of human diseases between countries by Ivorra, Benjamin; Ngom, Diene; Ramos, Angel M. (2020) - [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
• Singularities of transition processes in dynamical systems: Qualitative theory of critical delays by Gorban, Alexander N. (2004) - 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
• Asymptotic behavior of stochastic three-species predator-prey systems with white and Levy noise by Zhao, Yihan; Xia, Yuanpei; Yang, Zhichun (2020) - = (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
• An algorithm for constructing Lyapunov functions by Hafstein, Sigurdur Freyr Hafstein (2007) - ∈ 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
• Period functions and critical periods of piecewise linear system by Wang, Shaoqing; Yang, Jiazhong (2020) - 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
• Existence and stability of traveling waves for a competitive-cooperative recursion system by Bao, Xiongxiong; Li, Ting (2020) - 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
• Stabilization of coupled thermoelastic Kirchhoff plate and wave equations by Mansouri, Sabeur; Tebou, Louis (2020) - 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
• Global analysis on a continuous planar piecewise linear differential system with three zones by Jia, Man; Su, Youfeng; Chen, Hebai (2023) - = {(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
• Asymptotic stabilization for Bresse transmission systems with fractional damping by Hao, Jianghao; Wang, Dingkun (2023) - 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‖+
• Internal stabilization of interconnected heat-wave equations by Yu, Xiu-Fang; Wang, Jun-Min; Zhang, Han-Wen (2023) - 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
• Dynamics of traveling waves for predator-prey systems with Allee effect and time delay by Hua, Yang; Lin, Xiaojie; Liu, Jiang; Lu, Haixia (2024) - 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
• Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity by Gnanasekaran, Shanmugasundaram; Nithyadevi, Nagarajan (2024) - ′ ≤ 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
• Output tracking for a 1-D wave equations with spatially varying coefficients and subject to unknown disturbances by Jia, Yan-Na; Jin, Can; Yu, Xiu-Fang (2024) - = −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
• Exponential stability for porous thermoelastic systems with Gurtin-Pipkin flux by Hao, Jianghao; Yang, Jing (2023) - 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; ∫ ∞
• Optimal switching of vaccination for an infectious disease model by Salwahan, Shraddha; Abbas, Syed; Tridane, Abdessamad (2024) - 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
• Strong solutions to density-dependent incompressible smectic-A liquid crystal equations by Zhang, Xue; Zhao, Xiaopeng (2025) - ∥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

• Practical stability of stochastic differential delay equations driven by G-Brownian motion with general decay rate by Caraballo, Tomas; Ezzine, Faten; Hammami, Mohamed Ali (2024) - 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
• Boundary-domain integral equations for Dirichlet diffusion problems with non-smooth coefficient by Fresneda Portillo, Carlos; Woldemicheal, Zenebe W. (2022) - 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
• Lower and upper solutions for delay evolution equations with nonlocal and impulsive conditions by Zhang, Xuping (2002) - 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
• Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term by Vidhyaa, Kumar S.; Thandapani, Ethiraju; Alzabut, Jehad; Ozbekler, Abdullah (2023) - [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
• Optimal control and  approximate controllability for second-order integro-differential equations with state-dependent delay and non-instantaneous impulses by Bensalem, Abdelhamid; Salim, Abdelkrim; Benchohra, Mouffak; N'Guerekata, Gaston M. (2025) - 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
• Oscillation of modified Euler type half-linear differential equations via averaging technique by Hasil, Petr; Sisolakova, Jirina; Vesely, Michal (2022) - = 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
• Solution estimates and stability tests for nonlinear delay integro-differential equations by Pinelas, Sandra; Tunc, Osman (2022) - , 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
• Topological properties of the solution set and T-controllability for second-order neutral evolution equations with delay by Wang, Tiantian; Fan, Hongxia (2025) - = 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
• Evolution psi-Hilfer fractional differential equations in Banach spaces by Liang, Jin; Mu, Yunyi; Xiao, Ti-Jun (2025) - 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
• Optimal control for solutions to Sobolev stochastic equations by Bychkov, Evgeniy; Sviridyuk, Georgy; Bogomolov, Alexey (2021) - 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
• Propagating interface in reaction-diffusion equations with distributed delay by Wang, Haoyu; Tian, Ge (2021) - 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
• Solving singular evolution problems in sub-Riemannian groups via deterministic games by Ochoa, Pablo; Ruiz, Julio Alejo (2021) - 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
• Existence of solutions to infinite systems of nonlinear integral equations on the real half-axis by Chlebowicz, Agnieszka (2021) - → 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
• Stochastic Newtonian equations with mean boundary conditions by Guo, Ying-Jia; Jiang, Xiao-Meng (2021) - = ∫ 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
• Existence and stabilization for impulsive differential equations of second order with multiple delays by Pinelas, Sandra; Tunç, Osman; Korkmaz, Erdal; Tunç, Cemil (2024) - = { 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
• Oscillation criteria of fourth-order nonlinear semi-noncanonical neutral differential equations via a canonical transform by Purushothaman, Ganesh; Suresh, Kannan; Tunc, Ercan; Thandapani, Ethiraju (2023) - [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
• Existence of rational solutions for q-difference Painleve equations by Yan Xu, Hong; Tu, Jin (2020) - [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
• Abstract degenerate Volterra inclusions in locally convex spaces by Kostic, Marko (2023) - 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
• Stabilization of the wave equation with variable coefficients and a dynamical boundary control by Zhang, Zhifei (2020) - 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
• Nonlocal problems for hyperbolic equations from the viewpoint of strongly regular boundary conditions by Pulkina, Ludmila S. (2020) - 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
• Oscillatory behavior of solutions to third-order nonlinear differential equations with a superlinear neutral term by Tunc, Ercan; Grace, Said R. (2020) - Oscillation of solutions; asymptotic behavior; neutral differential equation. [5] P. Das; Oscillation criteria for odd order neutral equations, J. Math. Keywords: differential; equations; order
• Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science by Floridia, Giuseppe (2020) - 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
• Inviscid limit of linearly damped and forced nonlinear Schrodinger equations by Gialelis, Nikolaos (2020) - [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
• Monotone iterative method for retarded evolution equations involving nonlocal and impulsive conditions by Zhang, Xuping; Chen, Pengyu; Li, Yongxiang (2020) - 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
• Propagation of coupled porosity and fluid-concentration waves in isotropic porous media by Fama, Alessio; Restuccia, Liliana (2020) - = 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
• Existence and stability for fractional order pantograph equations with nonlocal conditions by Ahmad, Israr; Nieto, Juan Jose; Rahman, Ghaus ur; Shah, Kamal (2020) - − δ1 − µ1)| , K2 |λ2 − δ2 − µ2)| ) . − δ1 − µ1| ‖x1 − x̄1‖ , ‖H2x2 −H2x̄2‖ ≤ K2 |λ2 − (δ2 + µ2)| ‖x2 − x̄2‖ . Keywords: differential; equations; fractional; |λ1
• Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces by Melo, Wilberclay G.; Rocha, Nata F.; Costa, Natielle dos Santos (2023) - Assume that f ∈ X sa,σ(R3). [−1, 0)× (1,+∞) ) ∪ ( [0,+∞)×{0}× [1,+∞) ) and u0 ∈ X sa,σ(R3). Keywords: equations; x sa
• Entire solutions to Fermat-type difference and partial differential-difference equations in C^n: System of Fermat-type difference equations in $ \mathbb{C}^n $ by Xu, Hong Yan; Haldar, Goutam (2024) - − ( 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
• Asymptotic behavior of stochastic functional differential evolution equation by Clark, Jason; Misiats, Oleksandr; Mogylova, Viktoriia; Stanzhytskyi, Oleksandr (2023) - 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
• Stabilization of semilinear wave equations with time-dependent variable coefficients and memory by Li, Sheng-Jie; Chai, Shugen (2022) - 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
• Solutions of linear and non-linear partial differential equations by means of tensor product theory of Banach space by Alshanti, Waseem Ghazi (2024) - 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
• Decay rates for two Cauchy thermoelastic laminated Timoshenko problems of type III with interfacial sli by Guesmia, Aissa (2022) - − 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
• Null-controllability for 1-D degenerate quasilinear parabolic equations by de Carvalho, Pitágoras; Demarque, Reginaldo; Límaco, Juan; Viana, Luiz (2025) - 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
• Well-posedness of solutions for the 2D stochastic quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces by Khaider, Hassan; Azanzal, Achraf ; Raji, Abderrahmane (2024) - 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
• Asymptotic behavior of solutions to 3D Kelvin-Voigt-Brinkman-Forchheimer equations with unbounded delays by huy, Le Thi (2022) - − 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
• Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations by Purushothaman, Ganesh ; Suresh, Kannan ; Thandapani, Ethiraju ; Tunc, Ercan (2024) - 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

• Multiplicity results for Schrodinger type fractional p-Laplacian boundary value problems by Lopera, Emer; Recova, Leandro; Rumbos, Adolfo (2024) - 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
• Uniqueness for optimal control problems of two-dimensional second grade fluids by Almeida, Adilson; Chemetov, Nikolai V.; Cipriano, Fernanda (2022) - 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
• Exact boundary controllability for wave equations with fixed and moving boundaries in two-dimensional convex-complemented domains by Nunes, Ruikson; Nunez-Chavez, Miguel R. (2025) - 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
• Exponential stability of a damped beam-string-beam transmission problem by Barraza Martinez, Bienvenido; Hernandez Monzon, Jairo; Vergara Rolong, Gustavo (2022) - − 〈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
• Modeling of groundwater flow in porous medium layered over inclined impermeable beds by Girg, Petr; Kotrla, Lukas (2025) - 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
• Existence of solutions to nonlocal elliptic problems with singular and combined nonlinearities by Tordecilla, Jesus Alberto Leon (2022) - 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
• Existence of solutions for a problem with multiple singular weighted p-Laplacians and vanishing potentials by Alves, Maria Jose; Assuncao, Ronaldo B. (2022) - 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
• Asymptotic behavior of cooperative systems involving p-Laplacian operators by Alvarez-Caudevilla, Pablo (2022) - 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; − ∫
• 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 by Diaz, Jesus Ildefonso; Shaposhnikova, Tatiana A.; Zubova, Maria N. (2022) - = −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
• Combined effects of critical Hardy-Sobolev exponent and singular nonlinearities in nonlocal problems with variable weights by Almutairi, Sarah; Saoudi, Kamel (2025) - □ 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
• Monotonicity properties of the eigenvalues of nonlocal fractional operators and their applications by Molica Bisci, Giovanni; Servadei, Raffaella; Zhang, Binlin (2022) - 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
• Existence and regularity of solutions for elliptic systems with mixed boundary conditions by Boussetouan, Imane; Amrouche, Cherif (2025) - (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
• Non-steady magneto-hydrodynamics-heat system with joule and buoyancy effects under mixed boundary conditions by Kim, Tujin (2025) - (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
• Space-time analyticity of weak solutions to semilinear parabolic systems with variable coefficients by Baustian, Falko; Takac, Peter (2021) - ∈ 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
• Eigenvalues and bifurcation for Neumann problems with indefinite weights by Calanchi, Marta; Ruf, Bernhard (2021) - 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
• Limit for the p-laplacian equation with dynamical boundary conditions by Ozturk, Eylem; Rossi, Julio D. (2021) - ∀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
• An asymmetric problem at resonance with a one-sided Ahmad-Lazer-Paul condition by Recova, Leandro L.; Rumbos, Adolfo J. (2021) - 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
• Existence and global behavior of weak solutions to a doubly nonlinear evolution by Giacomoni, Jacques; Gouasmia, Abdelhamid; Mokrane, Abdelhafid (2021) - = ( ‖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
• Existence and multiplicity for radially symmetric solutions to Hamilton-Jacobi-Bellman equations by Li, Xiaoyan; Yang, Bian-Xia (2021) - 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
• Turing instability analysis of a singular cross-diffusion problem by Galiano, Gonzalo; Gonzalez-Tabernero, Victor (2021) - 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
• T-coercivity for the asymptotic analysis of scalar problems with sign-changing coefficients in thin periodic domains by Bunoiu, Renata; Karim, Karim; Timofte, Claudia (2021) - 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
• Periodic orbits of the spatial anisotropic Kepler problem with anisotropic perturbations by Li, Mengyuan; Liu, Qihuai (2021) - × ( (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
• Well-posedness and energy decay of a transmission problem of Kirchhoff type wave equations with damping and delay terms by Liu, Zhiqing; Gao, Cunchen; Fang, Zhong Bo (2021) - 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
• Asymptotic behavior of solutions to porous medium equations with boundary degeneracy by hao, Xutong; Zhou, Mingjun; Jing, Xinxin (2021) - = ∫ 1 0 u(x, t)ηδ(x)dx, t ≥ 0. = ∫ 1 0 u(x, t)η(x)dx, t ≥ 0. Keywords: problem
• Asymptotic analysis of perturbed Robin problems in a planar domain by Musolino, Paolo; Dutko, Martin; Mishuris, Gennady (2023) - 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
• Passing to the limit on small parameters for generalized viscous Cahn-Hilliard type equations with nonlinear source by Le Trong Thanh, Bui; Ngoc Quoc Thuong, Nguyen (2020) - + 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
• An optimal transport problem with storage fees by Bansil, Mohit; Kitagawa, Jun (2023) - 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; − ∫
• Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term by Sremr, Jiri (2023) - [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
• Homogeneous Boltzmann equation in quantum relativistic kinetic theory by Escobedo, Miguel; Mischler, Stephane; Valle, Manuel A. (2003) - = ∫ ∞ 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; ∫ ∞; ∫ ∫
• Dynamics of a diffusive competitive model on a periodically evolving domain by Zhu, Jiazhen; Zhou, Jiazheng; Lin, Zhigui (2020) - 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
• Dirichlet problem for second-order abstract differential equations by Dore, Giovanni (2020) - = 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
• Discussion of a uniqueness result in "Equilibrium Configurations for a Floating Drop" by Treinen, Raymond (2023) - 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
• Strange non-local operators homogenizing the Poisson equation with dynamical unilateral boundary conditions: asymmetric particles of critical size by Diaz, Jesus Ildefonso; Shaposhnikova, Tatiana A.; Podolskiy, Alexander V. (2024) - 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
• Optimal mass of structure with motion described by Sturm-Liouville operator: design and predesign by Belinskiy, Boris P.; Smith, Tanner A. (2024) - 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
• Dirichlet problems with anisotropic principal part involving unbounded coefficients by Motreanu, Dumitru; Tornatore, Elisabetta (2024) - . , 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
• Signorini's problem for the Bresse beam model with localized Kelvin-Voigt dissipation by Munoz Rivera, Jaime E.; Baldez, Carlos A. da Costa; Cordeiro, Sebastiao M. S. (2024) - − 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
• Viscosity solutions to the infinity Laplacian equation with lower terms by Li, Cuicui; Liu, Fang (2023) - 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
• Duality arguments for well-posedness of history-dependent variational inequalities by Hu, Rong; Sofonea, Mircea (2022) - 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
• Optimal control problem for Stokes systems: asymptotic analysis via unfolding method in a perforated domain by Garg, Swati; Sardar, Bidhan Chandra Sardar (2023) - [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
• Mathematical analysis of a Dupuit-Richards model by Al Nazer, Safaa; Rosier, Carole; Tsegmid, Munkhgerel (2022) - 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
• Existence and uniqueness of  the solution to initial and inverse problems for integro-differential heat equations with fractional load by Agarwal, Ravi ; Baltaeva, Umida; Hubert, Florence; Khasanov, Boburjon (2024) - 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
• Asymptotic analysis of  sign-changing transmission problems with rapidly oscillating interface by Bunoiu, Renata; Ramdani, Karim; Timofte, Claudia (2024) - [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

• Stability of Leray weak solutions to 3D Navier-Stokes equations by Zhang, Zujin; Yuan, Weijun; Yao, Zhengan (2025) - [(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
• New approach to the Lagrange-Burmann theorem via omega calculus and applications by Neto, Antonio Francisco (2025) - = 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
• Global bifurcation for semilinear eigenvalue problems involving nonlocal terms by Liu, Qingbo; Zhao, Lan (2025) - In this situation, ψ′(|α|2/2) = λ1 − λ1 = 0. Case 2: λ = λ1. Keywords: eigenvalue; theorem
• Friedrichs extension of singular symmetric differential operators by Bao, Qinglan; Wei, Guangsheng; Zettl, Anton (2023) - 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
• Spectral theory of C-symmetric non-selfadjoint differential operators of order 2n by Behncke, Horst; Hinton, Don (2023) - 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
• the mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane) by Mawhin, Jean (2021) - 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
• Hilbert Space Methods for Partial Differential Equations by Showalter, Ralph E. (1994) - 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
• Hille-Nehari type non-oscillation criteria for half-linear dynamic equations with mixed derivatives on a time scale by Ishibashi, Kazuki (2021) - 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
• Generalizations of the drift Laplace equation in the Heisenberg group and Grushin-type spaces by Bieske, Thomas; Blackwell, Keller (2021) - 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
• Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure by Kijowski, Antoni (2020) - 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
• Linearization via the Lie derivative by Chicone, Carmen; Swanson, Richard (2000) - = 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
• Sub-super solution method for nonlocal systems involving the p(x)-Laplacian operator by dos Santos, Gelson C. G.; Figueiredo, Giovany M.; Tavares,  Leandro S. (2020) - 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
• Mathematical methods for the randomized non-autonomous Bertalanffy model by Calatayud, Julia; Caraballo, Tomas; Cortes, Juan Carlos; Jornet, Marc (2020) - [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
• The contraction mapping principle and some applications by Brooks, Robert M.; Schmitt, Klaus (2009) - 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
• General p-curl systems and duality mappings on Sobolev spaces for Maxwell equations by Adhikari, Dhruba R.; Stachura, Eric (2020) - (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
• Periodic unfolding method for domains with very small inclusions by Avila, Jake; Cabarrubias, Bituin (2023) - | 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
• Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence by Boccardo, Lucio; Diaz, Jesus Ildefonso; Gomez-Castro, David (2024) - 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
• Behavior near the extinction time for systems of differential equations with sublinear dissipation terms by Hoang, Luan (2025) - 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
• Nodal sets and continuity of eigenfunctions of Krein-Feller operators by Ngai , Sze-Man; Zhang, Meng-Ke; Zhao, Wen-Quan (2025) - 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
• Quantitative estimates of L^p maximal regularity for nonautonomous operators and global existence for quasilinear equations by Belin, Théo; Lafitte, Pauline (2025) - Also for t ∈ Denote for ϵ > 0, Eϵ := {t ∈ I : ρ(t) > ϵ}. Keywords: bounded; constant; continuity; mrp(i; nonautonomous; operators; regularity; theorem

• Traveling waves with singularities in a damped hyperbolic MEMS type equation in the presence of negative powers nonlinearity by Ichida, Yu (2023) - 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
• Persistence properties of solutions for multi-component Novikov equations by Xin; Xinglong (2025) - 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
• Existence of solution to critical Kirchhoff-type equation with dipole-type potential by Wang, Sainan; Su, Yu (2022) - 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
• Mild solutions to Love-type equations on R^2 by Nam, Bui Duc; Nghia, Bui Dai; Tuan, Nguyen Anh (2025) - 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
• Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R^3 by Fu, Song-Ren; Ning, Zhen-Hu (2022) - 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
• Global solutions and blow-up for wave equations with variable coefficients and boundary supercritical source by Ha, Tae Gab (2025) - 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
• Complicated asymptotic behavior of solutions to doubly nonlinear diffusionequation in unbounded spaces by Lu, Can; Wang, Liangwei; Yin, Jingxue; Zhou, Meiling (2025) - 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
• Determining the background driving process of the Ornstein-Uhlenbeck model by Mariani, Maria C.; Asante, Peter K.; Kubin, William; Tweneboah, Osei K.; Beccar-Varela, Maria (2023) - [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
• Nonlocal advection diffusion equations and the two-slit experiment in quantum mechanics by Webb, Glenn (2023) - 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
• Solutions of Kirchhoff plate equations with internal damping and logarithmic nonlinearity by Pereira, Ducival; Cordeiro, Sebastiao; Raposo, Carlos; Maranhao, Celsa (2021) - 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
• Blow-up criteria and instability of standing waves for the inhomogeneous fractional Schrodinger equation by Feng, Binhua; He, Zhiqian; Liu, Jiayin (2021) - 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
• Lie symmetry analysis and conservation laws for the (2+1)-dimensional Mikhalev equation by Li, Xinyue; Zhang, Yongli; Zhang, Huiqun; Zhao, Qiulan (2021) - (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
• Stability and bifurcation in a delayed predator-prey model with Holling-type IV response function and age structure by Cai, Yuting; Wang, Chuncheng; Fan, Dejun (2021) - 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
• Entire solutions for the heat equation by Papanicolaou, Vassilis G.; Kallitsi, Eva; Smyrlis, George (2021) - (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
• Exact boundary controllability for the wave equation with moving boundary domains in a star-shaped hole by Nunes, Ruikson S. O. (2021) - 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
• Controllability for the wave equation with moving boundary by Jesus, Isaias P. de; Cabanillas Lapa, Eugenio; Limaco, Juan (2021) - 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
• Symmetry analysis for a second-order ordinary differential equation by Feng, Sebert (2021) - 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
• Coupled porosity-fluid concentration flux-temperature waves in isotropic porous media by Fama, Alessio; Restuccia, Liliana (2021) - 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
• Curvature blow-up for the periodic CH-mCH-Novikov equation by Zhu, Min; Wang, Ying; Chen, Lei (2021) - 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
• Stochastic attractor bifurcation for the two-dimensional Swift-Hohenberg equation with multiplicative noise by Xiao, Qingkun; Gao, Hongjun (2023) - 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
• Almost optimal local well-posedness for modified Boussinesq equations by Geba, Dan-Andrei; Lin, Bai (2020) - 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
• Spatial dynamics of a nonlocal bistable reaction diffusion equation by Han, Bang-Sheng; Chang, Meng-Xue; Yang, Yinghui (2020) - 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
• Energy decay for variable coefficient viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary by Hao, Jianghao; Lv, Mengxian (2020) - 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
• Variety of solutions and dynamical behavior for YTSF equations by Chen, Wei (2023) - [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
• Global well-posedness for Cauchy problems of Zakharov-Kuznetsov equations on cylindrical spaces by Osawa, Satoshi; Takaoka, Hideo (2024) - | |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
• Solving linear differential equations with mixed arguments by Karakostas, George L. (2024) - 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
• Asymptotic behavior of eigenvalues of fourth-order differential operators with spectral parameter in the boundary conditions by Polyakov, Dmitry M. (2024) - 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
• Construction of solutions to PDEs using holomorphic functions of several variables by Shpakivskyi, Vitalii (2024) - 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
• Ulam type stability for nonlinear Hahn difference equations with delay by Chen, Kai; Wang, Jinrong (2024) - 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

• Properties of the Dirichlet Green's function for linear diffusions on a half line by Conlon, Joseph G.; Dabkowski , Michael (2025) - = −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
• Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits by Huzak, Renato; Mardesic, Pavao; Resman, Maja; Zupanovic, Vesna (2025) - 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
• Multi-dimensional c-almost periodic type functions and applications by Kostic, Marko (2022) - 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
• Compact almost automorphic dynamics of linear non-autonomous differential equationswith exponential dichotomy and of delayed biological models by Chavez, Alan; Aragones, Nelson; Zavaleta, Ulices; Pinto, Manuel (2025) - 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
• A parabolic bipolynomial fractional Dirichlet-Laplace problem by Idczak, Dariusz (2022) - 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
• Numeric estimates of the principal eigenvalue of the p-Laplacian using interval arithmetic by Benedikt, Jiri; Pulpan, Jan (2023) - 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
• Discrete Aleksandrov solutions of the Monge-Ampere equation by Awanou, Gerard (2022) - 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
• Continuous imbedding in Musielak spaces with an application to anisotropic nonlinear Neumann problems by Youssfi, Ahmed; Khatri, Mohamed Mahmoud Ould (2021) - , 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
• Hadamard type inequalities via fractional calculus in the space of exp-convex functions and applications by Ma, Li Ma; Yang, Guangzhengao (2021) - 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
• Age-dependent branching processes and applications to the Luria-Delbruck experiment by Montgomery-Smith, Stephen J.; Oveys, Hesam (2021) - 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
• Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator by Idczak, Dariusz (2021) - ∈ 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
• Space-time behavior for radiative hydrodynamics model with or without heat conduction by Liu, Mengqian; Wu, Zhigang (2023) - − 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
• Properties of the resolvent of singular q-Dirac operators by  Allahverdiev, Bilender P.; Tuna, Huseyin (2020) - 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
• µ pseudo rotating-periodic solutions for differential equations by Li, Dandan; Du, Jiayin (2020) - 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
• Solutions to mean curvature equations in weighted standard static spacetimes by de Lima, Henrique F.; Ramalho, Andre F. A.; Velasquez, Marco Antonio L. (2020) - = 〈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
• Asymptotic behavior for a non-autonomous model of neural fields with variable external stimuli by da Silva, Severino Horacio (2020) - 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
• Nonlinear degenerate elliptic equations in weighted Sobolev spaces by Benali, Aharrouch; Jaouad, Bennouna (2020) - − Tk(u)| ≤ η}; and since {x ∈ Ω : |uε α 0 φ(0, s)− φ(k, s)ds = 1 α ∫ |u|Keywords: function
• Reduction principle for partial functional differential equation without compactness by El Attaouy, Meryem; Ezzinbi, Khalil; ˜N'Guerekata, Gaston Mandata (2023) - 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
• Eigenvalue problems for Kirchhoff-type equations in variable exponent Sobolev spaces by Aramaki, Junichi (2025) - 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

• Existence and uniqueness of generalized normal solutions to first order fractional differential equations and applications by Lan, Kunquan (2024) - 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
• Inequalities for fractional derivatives via the Marchaud derivative by Webb, Jeffrey R. L. (2025) - 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−
• Three-point integral boundary-value problems for piecewise fractional impulsivedifferential equations with p-Laplacian operator by Chen, Xiao; Zhou, Wenxue (2025) - 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
• Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency by Qu, Siqi; He, Xiaoming (2022) - 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
• Decay estimates and extinction properties of parabolic equations with classical and fractional time derivatives by Meng, Fanmeng; Zhou, Xian-Feng (2025) - 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
• Positive solutions for nonlinear fractional Laplacian problems by Hollifield, Elliott (2023) - 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
• Existence of solutions for critical fractional p-Laplacian equations with indefinite weights by Cui, Na; Sun, Hong-Rui (2021) - 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
• A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems by Webb, Jeffrey R. L. (2021) - 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
• Convergence of solutions of fractional differential equations to power-type functions by Dahan Kassim, Mohammed; Eddine Tatar, Nasser (2020) - 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
• Nonexistence results for fractional differential inequalities by Webb, Jeffrey (2024) - 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
• Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance by Sousa, Jose Vanterler da C.; Pigossi, Mariane; Nyamoradi, Nemat (2024) - 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
• Existence of pseudosolutions for dynamic fractional differential equations by Sikorska-Nowak, Aneta (2024) - 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
• Existence of positive S-asymptotically omega-periodic solutions of time-space fractional  nonlocal reaction-diffusion equations by Zhang, Xuping; Ding, Kaibo; Chen, Pengyu (2025) - [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
• Asymptotic stability for Hilfer-like nabla nonlinear fractional difference equations by Sharma, Anshul; Mishra, Suyash Narayan; Shukla, Anurag (2024) - 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

• Oscillation for second order nonlinear differential equations with a sub-linear neutral term by Wu, Yingzhu; Yu, Yuanhong; Xiao , Jinsen (2022) - 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
• Existence and controllability for neutral partial differential inclusions nondenselly defined on a half-line by Anh, Nguyen Thi Van; Yen, Bui Thi Hai (2023) - 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‖
• Delay-dependent stability conditions for delay differential equations with unbounded operators in Banach spaces by Gil, Michael (2024) - = 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

• Some applications of Lyapunov regularity by Barreira, Luis; Valls, Claudia (2022) - , 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
• Spherical compactifications of central force equations by Gingolld, Harry; Quaintance, Jocelyn (2025) - = 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
• Traveling wave solutions for three-species nonlocal competitive-cooperative systems by Wu, Hong-Jie; Han, Bang-Sheng; Mi, Shao-Yue; Shen, Liang-Bin (2023) - + [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
• Transition fronts of two species competition lattice systems in random media by Cao, Feng; Gao, Lu (2020) - 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
• Existence of weak solutions to superlinear elliptic systems without the Ambrosetti-Rabinowitz condition by Wang, Xiaohui; Zhao, Peihao (2020) - 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
• Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure by Huang, Chuangxia; Wang, Jiafu; Huang, Lihong (2020) - 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
• Massera type theorems for abstract non-autonomous evolution equations by Zheng, Lan-Ling; Ding, Hui-Sheng (2024) - 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

• Oscillatory behavior for nonlinear homogeneous neutral difference equations of second order with coefficient changing sign by Bhuyan, Ajit Kumar; Padhy, Laxmi Narayan; Rath, Radhanath (2020) - 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

• Mathematical models for the transmission of malaria with seasonality and ivermectin by Zhao, Zhihong; Shaochun, Shaochun; Lu, Yulan (2022) - 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
• Traveling waves of a diffusive modified Leslie-Gower model with chemotaxis by Wang, Shuna; Liu, Jiang; Fang, Jun; Lin, Xiaojie (2025) - 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
• p-Laplacian in phenomenological modeling of flow in porous media and CFD simulations by Girg, Petr; Kotrla, Lukas; Svandova, Anezka (2025) - 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
• Traveling wave solutions for an epidemic model by Zhang, Zhenbu (2025) - 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
• Modeling the control of COVID-19: impact of policy interventions and meteorological factors by Jia, Jiwei; Ding,  Jian; Liu, Siyu; Liao, Guidong; Li, Jingzhi; Duan, Ben; Wang, Guoqing; Zhang, Ran (2020) - 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
• Assessing the efficiency of different control strategies for the COVID-19 epidemic by Castilho, Cesar; Gondim, Joao A. M.; Marchesin, Marcelo; Sabeti, Mehran (2020) - 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
• Optimal control analysis applied to a two-patch model for Guinea worm disease by Mushayabasa, Steady; Losio, Anthony A. E.; Modnak, Chairat; Wang, Jin (2020) - 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
• Optimal control applied to a visceral leishmaniasis model by Pantha, Buddhi; Agusto, Folashade B.; Elmojtaba, Ibrahim M. (2020) - 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
• On drug therapy for an HIV infection age model with cellular and immune delays by Tarfulea, Nicoleta E. (2024) - 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
• Impact of vaccination and sterilization on the transmission dynamics of rabies by Verma, Vijai Shanker; Kunwar, Laxman Bahadur (2024) - 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

• Remarks on periodic Zakharov systems by Kishimoto, Nobu (2022) - 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
• Smoothing properties for a coupled Zakharov-Kuznetsov system by Levandosky, Julie L.; Vera, Octavio (2023) - 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
• Phase portraits of a family of Kolmogorov systems depending on six parameters by Diz-Pita, Erika; Libre, Jaume; Otero-Espinar, M. Victoria (2021) - 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
• Abel quadratic differential systems of second kind by Artes, Joan C.; Llibre, Jaume; Schlomiuk, Dana; Vulpe, Nicolae (2024) - 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 =
• L^2 Solutions for cubic NLS equation with higher order fractional elliptic/hyperbolic operators on R cross T and  R^2 by Corcho, Adan J.; Mallqui, Lindolfo P. (2025) - 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

• Existence and nonlinear stability of solitary wave solutions for coupled Schrodinger-KdV systems by Cui, Pengxue; Ji, Shuguan (2021) - ‖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
• Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source by Salako, Rachidi B.; Shen, Wenxian (2020) - = c in the interval (0,min{ √ a, √ λ+τa (1−τ)+ }). Parabolic chemotaxis system; logistic source; traveling wave solution; minimal wave speed. Keywords: wave
• Respiratory Illness clinical trial modeling by Gerberry, David; Joshi, Hem; Peloquin, Mac; Vargas, Sonia (2024) - 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

• Holder regularity of weak solutions to nonlocal p-Laplacian type Schrodinger  equations with A_1^p-Muckenhoupt potentials by Kim, Yong-Cheol (2025) - (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
• Lower bounds on the fundamental spectral gap with Robin boundary conditions by Ahrami, Mohammed; El Allali, Zakaria (2022) - 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
• Standing waves to Chern-Simons-Schrodinger systems with critical exponential growth by Wan, Youyan; Tan, Jinggang (2021) - 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
• Local well-posedness and standing waves with prescribed mass for Schrodinger-Poisson systems with a logarithmic potential in R^2 by Dou, Xuechao; Sun, Juntao (2023) - 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
• Positive solutions of Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity by Lan, Yongyi; Tang, Biyun; Hu, Xian (2020) - 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
• Non global solutions for non-radial inhomogeneous nonlinear Schrodinger equations by Bai, Ruobing; Saanouni, Tarek (2025) - |−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

• Fourth-order differential operators with interior degeneracy and generalized Wentzell boundary conditions by Camasta, Alessandro; Fragnelli, Genni (2022) - 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
• A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs by Feng, Xiaobing; Lewis, Thomas; Ward, Kellie (2022) - 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
• Initial-boundary layer associated with the 3-D Boussinesq system for Rayleigh-Benard convection by Fan, Xiaoting; Wang, Shu; Xu, Wen-Qing (2020) - = 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
• Supercooled Stefan problem with a Neumann type boundary condition by Briozzo, Adriana C. (2020) - 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
• Homogenization of boundary optimal control problem by Mishra, Indira (2022) - 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
• Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems by Zhang, Sen; Zu, Jian; Zhang, Jingqi (2024) - 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

• Upper semicontinuity of uniform attractors for singular perturbed second order nonautonomous delay lattice systems by Zhou, Yao; Liu, Hongliang (2025) - + ∫ 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
• Approximations of Euler-Peano scheme for reflected stochastic differential equations with non-Lipschitz coefficients by Zhang, Mingbo (2025) - = ∫ 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
• Zero-viscosity-capillarity limit for the contact discontinuity for the 1-D full compressible  Navier-Stokes-Korteweg equations by Chen, Jiaxue; Li, Yeping; Yin, Rong (2025) - 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
• Low regularity of non-L^2(R^n) local solutions to gMHD-alpha systems by Riva, Lorenzo; Pennington, Nathan (2020) - ︸ ︷︷ ︸ 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
• Convergence of delay equations driven by a Holder continuous function of order 1/3 by Besalu, Mireia; Binotto, Giulia; Rovira, Carles (2020) - 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

• Linear stability of the Couette flow for non-isentropic compressible fluids by Zhai, Xiaoping (2025) - 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
• Hyers-Ulam stability of linear quaternion-valued differential equations by Lv, Jiaojiao; Wang, Jinrong; Liu, Rui (2023) - [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
• Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity by Su, Si; Zhang, Guo-Bao (2020) - − 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
• Stability and rate of decay for solutions to stochastic differential equations with Markov switching by Lu, Shuaishuai; Yang, Xue (2024) - = 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

• Weighted (p,q)-equations with gradient dependent reaction by Jing, Zhao; Liu, Zhenhai; Papageorgiou, Nikolaos S. (2025) - 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
• Twin positive solutions for resonant singular (p,q)-equations by Onete, Florin I.; Papageorgiou, Nikolaos S.; Radulescu, Vicentiu D. (2021) - (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
• Dirichlet (p,q)-equations with gradient dependent and locally defined reaction by Liu, Zhenhai; Papageorgiou, Nikolaos S. (2021) - 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
• Existence of solutions for implicit obstacle problems involving nonhomogeneous partial differential operators and multivalued terms by Zeng, Shengda; Bai, Yunru; Gasinski, Leszek; Krech, Ireneusz (2021) - 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
• Positive solutions for singular (p,q)-Laplacian equations with negative perturbation by Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca (2023) - 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
• Multiple solutions for nonhomogeneous Schrodinger-Poisson system with p-Laplacian by Huang, Lanxin; Su, Jiabao (2023) - EJDE-2023/28 SCHRÖDINGER-POISSON SYSTEMS WITH p-LAPLACIAN 7∫ R3 |un − u|p dx ≤ C (∫ R3 (|un|p−2un − |u|p−2u)(un − u) dx )p/2 . (3.4) It follows from (3.3) and (3.4) that∫ R3 en + (|un|p−2un − |u|p−2u)(un − u) dx = o(1). Keywords: 1,p
• Positive solution for a nonlinear elliptic equation on symmetric domains by Faria, Luiz F. O.; Montenegro, Marcelo (2024) - 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
• A weighted (p,2)-equation with double resonance by Liu, Zhenhai; Papageorgiou, Nikolaos S. (2023) - 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
• Existence of global weak solutions for a p-Laplacian inequality with strong issipation in noncylindrical domains by Ferreira, Jorge; Piskin, Erhan; Shahrouzi, Mohammad; Cordeiro , Sebastiao; Raposo, Carlos Alberto (2022) - 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

• Existence and uniqueness results for fourth-order four-point BVP arising in bridge design in the presence of reverse ordered upper and lower solutions by Urus, Nazia; Verma, Amit K. (2023) - 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
• Linear type global centers of cubic Hamiltonian systems symmetric with respect to the x-axis by Barreira, Luis; Llibre, Jaume; Valls, Claudia (2020) - 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
• Global dynamics of a special class of planar sector-wise linear systems by Han, Qian-Qian; Huan, Song-Mei (2024) - = −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

• Nonstationary Lame system without definite sign energy by Millla Miranda, Manuel; Louredo, Aldo Trajano; Clark, Marcondes Rodrigues; Gouveia, Giovana Siracusa (2022) - (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
• Eigenvalue bounds for the clamped plate problem of L^2_xi operator by Zeng, Lingzhong; Zhou, Ziyi (2025) - 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
• Existence of a solution and its numerical approximation for a strongly nonlinear coupled system in anisotropic Orlicz-Sobolev spaces by Ortegon Gallego, Francisco; Ouyahya, Hakima; Rhoudaf, Mohamed (2022) - 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
• Global solutions to a quasilinear hyperbolic equation by Milla Miranda, Manuel; Medeiros, Luiz A.; Louredo, Aldo T. (2020) - = ∫ 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
• Exponential decay and blow-up for nonlinear heat equations with viscoelastic terms and Robin-Dirichlet conditions by Phuong, Le Thi; Thanh Long, Nguyen (2020) - . , uN ) + Fi(x, t), (1.1) where 0 < x < 1, t > 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
• Pyramidal traveling fronts of a time periodic diffusion equation with degenerate monostable nonlinearity by Bu, Zhen-Hui; Wang, Chen-Lu; Zhang, Xin-Tian (2023) - 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

• Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model by Ding, Hang; Zhou, Jun (2022) - − 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
• Cauchy problems for chemotaxis systems with chemo attractant and repellent by Lagha, Aesha; Hattori, Hattori (2022) - 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
• Almost automorphic solutions to non-autonomous dynamic equations with Stepanov-like almost automorphic forcing terms on time scales by Zheng, Feng-Xia; Li, Hong-Xu (2025) - ∈ 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
• Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy by Chen, Shaohua; Xu, Runzhang; Yang, Chao (2022) - (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
• Oscillation time and damping coefficients in a nonlinear pendulum by Arango, Jaime (2021) - 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

• Normal solvability and Fredholm properties for special classes of hypoelliptic operators by Tumanyan, Ani (2025) - 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 ∈
• Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators by Adhikari, Dhruba R.; Aryal, Ashok; Bhatt, Ghanshyam; Kunwar, Ishwari J.; Puri, Rajan; Ranabhat, Min (2022) - 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

• S-asymptotically omega-periodic mild solutions to fractional differential equations by Brindle, Darin; N'Guerekata, Gaston M. (2020) - = ∫ 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
• Pseudo almost periodicity for stochastic differential equations in infinite dimensions by Chen, Ye-Jun; Ding, Hui-Sheng (2023) - 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

• Existence of solutions to stochastic p(t,x)-Laplace equations and applications by Liang, Chen; Yan, Lixu; Fu, Yongqiang (2024) - 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
• P-mean (mu1,mu2)-pseudo almost periodic processes and application to integro-differential stochastic evolution equations by Ayachi, Moez; Abbas, Syed (2024) - = 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

• Existence and blow up in a system of wave equations with nonstandard nonlinearities by Messaoudi, Salim A.; Bouhoufani, Oulia; Hamchi, Ilhem; Alahyane, Mohamed (2021) - , 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

• Solutions to magnetic Schrodinger equations with arbitrary growth at infinity by Almeida, Wendy F.; Figueiredo, Giovany M. (2025) - 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
• Harnack's inequality for quasilinear elliptic equations with generalized Orlicz growth by Shan, Maria A.; Skrypnik, Igor I.; Voitovych, Mykhailo V. (2021) - 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
• Radial bounded solutions for modified Schrodinger equations by Mennuni, Federica; Salvatore, Addolorata (2024) - [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
• Multiple solutions for p(x)-Kirchhoff type problems with extended Robin boundary conditions by Baladi, Houssam; Aglzim, Abdellatif; Filali, Mohammed; Tsouli, Najib (2024) - +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
• Existence and nonexistence of positive solutions for fourth-order elliptic problems by Feng, Meiqiang; Chen, Haiping (2023) - 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

• Critical Fujita exponents for a class of quasilinear coupled parabolic  equations by Nie, Yuanyuan; Leng, Yan; Zhao, Xu; Zhou, Qian (2025) - 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|+
• Weakly monotone decreasing solutions to elliptic Schrodinger integral system by Chernysh, Edward (2021) - 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|
• Cauchy problem for the Lane-Emden heat flow with sign-changing initial data by Huang, Rui; Ji, Shanming; Ma, Yansheng (2024) - 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+

• Open mappings: The case for a new direction in fixed point theory by Burton, Theodore A.; Purnara, Ioannis K. (2022) - 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

• Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations by You, Song; Zhao, Peihao; Wang, Qingxuan (2021) - 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 ∗
• Space versus energy oscillations of Prufer phases for matrix Sturm-Liouville and Jacobi operators by Schulz-Baldes, Hermann; Urban, Liam (2020) - 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

• Stability analysis of the Peaceman-Rachford method for parabolic equations with nonlocal conditions by Sapagovas, Mifodijus; Novickij, Jurij; Ciupaila, Regimantas (2022) - 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

• Dynamical behavior in a reaction-diffusion system with prey-taxis by Song, Yingwei; Zhang, Tie; Li, Jinpeng (2022) - [(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
• Diffusive predator-prey models with fear effect in spatially heterogeneous environment by Li, Shanbing; Xiao, Yanni; Dong, Yaying (2021) - − 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
• Enrichment paradox and applications by Feng, Zaichun (Z.C.); Li, Y. Charles (2023) - 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
• Dynamics and pattern formation in diffusive predator-prey models with predator-taxis by Sun, Zhongyuan; Wang, Jinfeng (2020) - 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
• Dynamic behavior of a stochastic predator-prey model with stage-structure and nonlinear perturbation by Zhang, Xiang; Kang, Ming; Geng, Fengjie (2025) - = − 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

• Boundary regularity for strongly degenerate operators of Grushin type by Di Fazio, Giuseppe; Fanciullo, Maria Stella; Zamboni, Piero (2022) - 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
• A nonexistence result for p-Laplacian systems in a ball by Abebe, Abraham; Chhetri, Maya (2023) - 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
• Nonexistence results for hyperbolic type inequalities involving the Grushin operator in exterior domains by Jleli, Mohamed; Samet, Bessem (2021) - −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
• Traveling waves for unbalanced bistable equations with density dependent diffusion by Drabek, Pavel; Zahradnikova, Michaela (2021) - 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
• Blow-up for parabolic equations in nonlinear divergence form with time-ependent coefficients by Shen, Xuhui; Ding, Juntang (2022) - 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

• Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model by Takac, Peter (2023) - = 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

• Non-perturbative positivity and weak Holder continuity of Lyapunov exponent of analytic quasi-periodic Jacobi cocycles defined on a high dimension torus by Tao, Kai (2020) - 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
• Global solutions for fractional viscoelastic equations with logarithmic nonlinearities by Cabanillas Lapa, Eugenio (2020) - = 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

• Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential by Zhang, Guoping; Aburamyah, Ghder (2024) - 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

• Existence and stability of forced waves for p-Laplace equations in a shifting habitat by Xu, Tianyuan; Liu, Gege; Yin, Jingxue (2025) - ∫∫ 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
• Regular traveling waves for a reaction-diffusion equation with two nonlocal delays by Zhao, Haiqin; Wu, Shi-Liang (2022) - 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

• Asymptotic behavior of Kirchhoff type plate equations with nonlocal weak damping, anti-damping and subcritical nonlinearity by Xu, Ling (2025) - + ∫ 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
• Orlicz-Sobolev inequalities and the Dirichlet problem for infinitely degenerate elliptic operators by Hafeez, Usman; Lavier, Theo; Williams, Lucas; Korobenko, Lyudmila (2021) - 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

• Local and global solvability of fractional porous medium equations in critical Besov-Morrey spaces by El Idrissi, Ahmed; Srhiri, Halima; El Boukari, Brahim; El Ghordaf, Jalila (2025) - 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
• Hardy operators and commutators on generalized central function spaces by Nghia, Le Trung (2025) - 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
• Well-posedness of generalized magnetohydrodynamic equations in variable Lebesgue spaces by Sun, Jinyi; Mai, Yuanwei; Yang, Minghua (2025) - 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
• Global well-posedness for Klein-Gordon-Hartree and fractional Hartree equations on modulation spaces by Bhimani, Divyang G. (2021) - = (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
• Boundedness on generalized Morrey spaces for the Schrodinger operator with potential in a reverse Holder class by Wang, Guiyun; Zheng, Shenzhou (2023) - 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

• Null controllability of coupled systems of degenerate parabolic integro-differential equations by Allal, Brahim; Fragnelli, Genni; Salhi, Jawad (2023) - = 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

• An elliptic equation involving the square root of the Laplacian without asymptotic limits by Chen, Yutong; Su, Jiabao; Sun, Mingzheng; Tian, Rushun (2021) - − ( ‖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

• Global boundedness in an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion by Wu, Chun (2025) - 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

• Singular Monge-Ampere equations over convex domains by Li, Mengni (2021) - 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

• Estimates of characteristics of a micropolar flow passing through an axially symmetric cell by Koroleva, Yulia O.; Yu, Daria (2021) - 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

• Local min-orthogonal principle and its applications for solving multiple solution problems by Li, Meiqin; Ji, Bingbing; Zhou, Jianxin (2023) - → 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
• Semi-Lagrangian forward methods for some time-dependent nonlinear partial differential equations by Guo, Daniel X. (2022) - 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
• Rothe's method for solving semi-linear differential equations with deviating arguments by Devi, Darshana; Chutia, Duranta; Haloi, Rajib (2020) - = 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

• Poisson measures on semi-direct products of infinite-dimensional Hilbert spaces by Penney, Richard C.; Urban, Roman (2022) - 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

• Remarks on the second Neumann eigenvalue by Sabina de Lis, Jose C. (2022) - 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

• Lower order for meromorphic solutions to linear delay-differential equations by Bellaama, Rachid; Belaidi, Benharrat (2021) - (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
• Positive and nodal solutions for nonlinear nonhomogeneous parametric Neumann problems by Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca (2020) - 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

• Ground state solutions for fractional p-Kirchhoff equation by Wang, Lixiong; Chen, Haibo; Yang, Liu (2022) - 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

• Existence of solutions to steady Navier-Stokes equations via a minimax approach by Fereidooni, Amin; Moameni, Abbas; Grewal, Anant (2023) - = ∆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

• Caratheodory periodic perturbations of degenerate systems by Calamai, Alessandro; Spadini, Marco (2024) - 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

• H-convergence for equations depending on monotone operators in Carnot groups by Maione, Alberto (2021) - 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
• G-convergence of elliptic operators in non divergence form in R^n by D'Onofrio, Luigi (2023) - 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

• Boundedness and asymptotic stability in a chemotaxis model with indirect signal production and logistic source by Ye, Xiaobing; Wang, Liangchen (2022) - 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

• Uniform regularity of fully compressible Hall-MHD systems by Fan, Jishan; Zhou, Yong (2021) - [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
• Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms by Li, Fengbai; Wang, Weike; Wang, Yutong (2020) - 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

• Bilinear estimates posed in finite domains in 2D and 3D by Chen, Zhuoru; Wang, Taige; Xie, Xiangfei (2025) - ∇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
• L^q-regularity estimates for double obstacle problems  with quasilinear operators and Schrodinger-type lower order terms by Lee, Mikyoung; Ok, Jihoon (2025) - 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

• Existence and non-existence of solutions for Hardy parabolic equations with singular initial data by Aparcana, Aldryn; Carhuas-Torre, Brandon; Castillo, Ricardo; Loayza, Miguel (2025) - 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

• Weak solution by the sub-supersolution method for a nonlocal system involving Lebesgue generalized spaces by Razani, Abdolrahman; Figueiredo, Giovany M. (2022) - = 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

• Logarithmically improved regularity criteria for the Navier-Stokes equations in homogeneous Besov spaces by Dao, Nguyen Anh; Diaz, Jesus Ildefonso (2021) - 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
• Maximal regularity for fractional difference equations of order 2 by Zhang, Jichao; Bu, Shangquan (2024) - − 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

• Complete classification of bifurcation curves for a multiparameter diffusive logistic problem with generalized Holling type-IV functional response by Ciou, Jyun-Yuan; Tzung-Shin, Tzung-Shin (2021) - 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

• Estimation of plate parameters from vertical displacement data using a family of plate models by White, Luther W.; Malysheva, Tetyana; Karlstrom, Leif (2023) - 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

• Principal eigenvalues for the fractional p-Laplacian with unbounded sign-changing weights by Asso, Oumarou; Cuesta, Mabel; Doumate, Jonas Tele; Leadi, Liamidi (2023) - 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

• Multiple solutions for p(x)-Kirchhoff type problems with Robin boundary conditions by Afrouzi, Ghasem A.; Chung, Nguyen Thanh; Naghizadeh, Zohreh (2022) - 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
• Evolution equations on time-dependent Lebesgue spaces with variable exponents by Simsen, Jacson (2023) - 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

• Resolvent kernel on H-type groups and a Green kernel for fractional powers of its sub-Laplacian by Mouhcine, Zakariyae (2022) - = (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

• A KAM theorem for higher dimensional reversible nonlinear Schrodinger equations by Lou, Zhaowei; Sun, Yingnan (2022) - 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

• Multiple solutions for semilinear Robin problems with superlinear reaction and no symmetries by Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca (2021) - 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+

• Kirchhoff-type problems with critical Sobolev exponent in a hyperbolic space by Carriao, Paulo Cesar; Costa, Augusto Cesar dos Reis; Miyagaki, Olimpio Hiroshi; Vicente, Andre (2022) - 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

• Hardy and Caffarelli-Kohn-Nirenberg inequalities with nonradial weights by Tuan Duy, Nguyen; Long Phi, Le; Thanh Son, Nguyen (2020) - 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∗
• Cylindrical Hardy inequalities on half-spaces by Tuan Duy, Nguyen; Nguyen, Huy Bac (2020) - [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|

• Stationary quantum Zakharov systems involving a higher competing perturbation by Yao, Shuai; Sun,  Juntao; Wu, Tsung-Fang (2020) - 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
• Multiplicity of positive solutions for a gradient type cooperative/competitive elliptic system by Silva, Kaye; Moreno Sousa, Steffanio (2020) - 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
• Polyharmonic systems involving critical nonlinearities with sign-changing weight functions by Rani, Anu; Goyal, Sarika (2020) - = (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

• Local bifurcation structure and stability of the mean curvature equation in the static spacetime by Gao, Siyu; Liu, Qingbo; Sun, Yingxin (2024) - 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

• A global compactness result for quasilinear elliptic problems with critical Sobolev nonlinearities and Hardy potentials on R^N by Jin, Lingyu; Wei, Suting (2024) - (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

• Global solution for coupled parabolic systems with degenerate coefficients and time-weighted sources by Castillo, Ricardo; Guzman, Omar; Loayza, Miguel; Zegarra, Maria (2024) - = ∫ 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

• Bresse systems with localized Kelvin-Voigt dissipation by Aguilera Contreras, Gabriel; Munoz-Rivera, Jaime E. (2021) - 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

• Free boundary value problem for compressible magnetohydrodynamic equations by Kong, Huihui; Lian, Ruxu (2020) - 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

• Complete noncompact and stochastically complete m-quasi Yamabe gradient solitons by Molica Bisci, Giovanni; De Lima, Henrique; Leite, Ary V. F. ARY V.F.; Velasquez, Marco A. L. (2025) - = 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

• Multiplicity results of nonlocal singular PDEs with critical Sobolev-Hardy exponent by Daoues, Adel; Hammami, Amani; Saoudi, Kamel (2023) - ‖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

• Gevrey regularity of the solutions of inhomogeneous nonlinear partial differential equations by Remy, Pascal (2023) - 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
• Representation of solutions of a second order delay differential equation by Qiu, Kee; Wang, Jinrong (2020) - ∫ 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

• Global unique solution for 3D incompressible inhomogeneous magneto-micropolar equations with discontinuous density by Song, Xiao; Wang, Chenhua; Wang, Xiaojie; Xu, Fuyi (2025) - [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

• Normalized ground state solutions for Kirchhoff equation with subcritical or critical perturbation by Fan, Zian (2025) - 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

• Existence of solutions for a n-dimensional systems of nonlocal boundary value problems by Karakostas, George L. (2025) - + ∫ 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

• Stability of anisotropic parabolic equations without boundary conditions by Zhan, Huashui; Feng, Zhaosheng (2020) - 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

• On the L^2-orthogonality of Steklov eigenfunctions by Cho, Manki; Rivas, Mauricio A. (2022) - 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

• Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs by Lewis, Thomas; Rapp, Aaron; Zhang, Yi (2022) - 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

• Existence and nonexistence for singular sublinear problems on exterior domains by Ali, Mageed; Iaia, Joseph A. Iaia (2021) - 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
• Infinitely many solutions for a singular semilinear problem on exterior domains by Ali, Mageed; Iaia, Joseph (2021) - 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

• Henon equation with nolinearities involving Sobolev critical growth in H^1 by Barboza, Eudes M.; Miyagaki, Olimpio H.; Pereira, Fabio R.; Santana, Claudia R. (2021) - 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

• Lower bounds at infinity for solutions to second order elliptic equations by Nguyen, Tu (2023) - 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|

• Improved oscillation criteria for first-order delay differential equations with variable delay by Dix, Julio G. (2021) - = ∫ 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
• Exponential stability of solutions to nonlinear time-varying delay systems of neutral type equations with periodic coefficients by Matveeva, Inessa I. (2020) - 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

• Optimal energy decay rates for viscoelastic wave equations with nonlinearity of variable exponent by Mustafa, Muhammad I. (2023) - ≤ − (∫ 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
• Decay of energy for viscoelastic wave equations with Balakrishnan-Taylor damping and memories by Wang, Fei; Hao, Jianghao (2020) - = 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; ∫ ∞

• Random attractors and their stability for nonclassical diffusion equations driven by additive white noise with delay and intensity by Ma, Wenhui; Ma, Qiaozhen (2025) - 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

• Inverse nodal problems for Dirac operators and their numerical approximations by Song, Fei; Wang, Yuping; Akbarpoor, Shahrbanoo (2023) - 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

• Existence and forms of entire solutions to system  of non-linear partial differential equations by Banerjee, Abhijit; Sarkar, Jhuma (2024) - 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

• A second order convergent difference scheme for the initial-boundary value problem of Rosenau-Burgers equation by Dong, Sitong; Zhang, Xin; Jin, Yuanfeng (2024) - − 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
• theta-scheme for solving Caputo fractional differential equations by Doan, Thai Son Doan; Huong, Phan Thi; Kloeden, Peter E. (2025) - 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

• Stability analysis of an age-structured viral infection model with latency by Li, Chunyang; Dong, Xiu; Wang, Jinliang (2022) - + ∫ 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; ∫ ∞

• k-Hessian curvature type equations in space forms by Zhou, Jundong (2022) - (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

• Blow-up solutions for damped Rao-Nakra beams with source terms by Freitas, Mirelson M.; Santos, Mauro L.; Raposo, Carlos A.; Ramos, Anderson A.; Ferreira, Jorge (2025) - [16], ρ1h1utt − E1h1uxx − τ = 0, (1.4) ρ3h3vtt − E1I1ϕ1,xx − h1 2 τ +G1h1(wx + ϕ1) Keywords: nakra; rao

• Fractional Kirchhoff Hardy problems with weighted Choquard and singular nonlinearity by Goyal, Sarika; Sharma, Tarun (2022) - 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

• Existence of nontrivial solutions for biharmonic equations with critical growth by He, Juhua; Wu, Ke; Zhou, Fen (2025) - 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∗∗

• Neumann-Robin problem for p(x)-Laplacian equations in a domain with the boundary edge by Borsuk, Mikhail (2025) - (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

• Classification of boundary-equilibria for two-dimensional continuous piecewise linear systems with two intersecting switching lines by Yang, Xin; Zhou, Jueliang (2025) - −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 ∈

• Shadowing properties of evolution equations with exponential trichotomy on Banach spaces by Tu, Kun; Ding, Hui-Sheng (2025) - + ∫ 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; ∫ +

• Double phase equations with an indefinite concave term by Liu, Zhenhai; Papageorgiou, Nikolaos S. (2022) - 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

• Higher differentiability for solutions to nonhomogeneous obstacle problems with 1 by Wang, Zhenqiang (2022) - [ 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

• Asymptotic behavior of solutions to coupled porous medium systems with boundary degeneracy by Zhao, Xutong; Zhou, Mingjun; Zhou, Qian (2022) - 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

• Heat kernel estimates for fourth-order non-uniformly elliptic operators with non-strongly convex symbols by Barbatis, Gerassimos; Branikas, Panagiotis (2022) - 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

• Kirchhoff systems involving fractional p-Laplacian and singular nonlinearity by Kratou, Mouna (2022) - 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

• Stochastic Burgers equations with fractional derivative driven by fractional noise by Duan, Yubo; Jiang, Yiming; Tian, Yang; Wei, Yawei (2023) - 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; ∫ ∞

• Initial-boundary value problem of plasma-charge model in the half space by Wu , Jingpeng; Zhu, Min (2025) - 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

• Inverse volatility problem for currency options by Knowles, Ian; Tamang, Sundar (2023) - − 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

• De Bruijn identities in different Markovian channels by Emamirad, Hassan; Rougirel, Arnaud (2023) - 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

• A third look at the first result of Landesman-Lazer type by Amster, Pablo (2021) - 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

• Localized nodal solutions for parameter-dependent quasilinear Schrodinger equations by He, Rui; Liu, Xiangqing (2021) - 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

• Existence of nontrivial solutions for Schrodinger-Kirchhoff equations with indefinite potentials by Jiang, Shuai; Yin, Li-Feng (2023) - = ∫ 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

• Impulsive regular q-Dirac systems by Allahverdiev, Bilender P.; Tuna, Huseyin; Isayev, Hamlet A (2023) - 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

• Energy-dependent Hamiltonian in a nuclear optical model by Huu-Tai, Pierre Chau; Ducomet, Bernard (2021) - 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′

• Asymptotic formulas for q-regularly varying solutions of half-linear q-difference equations by Katarina S. Djordjevic, Katarina S. (2021) - 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

• Blow up and asymptotic behavior of solutions for a p(x)-Laplacian equation with delay term and variable exponents by Antontsev, Stanislav; Ferreira, Jorge; Piskin, Erhan; Yuksekkaya, Hazal; Shahrouzi, Mohammad (2021) - = (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

• Periodic solutions for conformable type non-instantaneous impulsive differential equations by Ding, Yuanlin; Wang, Jinrong (2021) - = 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

• p-Laplacian equation with finitely many critical nonlinearities by Xia, Pengcheng; Su, Yu (2021) - 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

• Singular p-biharmonic problems involving the Hardy-Sobolev exponent by Drissi, Amor; Ghanmi, Abdeljabbar; Repovs, Dusan D. (2023) - 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

• Lifespan of solutions of a fractional evolution equation with higher order diffusion on the Heisenberg group by Alsaedi, Ahmed; Ahmad, Bashir; Kirane, Mokhtar; Nabti, Aberrazak (2020) - 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

• Convergence of approximate solutions to nonlinear Caputo nabla fractional difference equations with boundary conditions by Liu, Xiang; Jia, Baoguo; Gensler,  Scott; Erbe, Lynn; Peterson, Allan (2020) - = 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

• Orlicz estimates for general parabolic obstacle problems with p(t,x)-growth in Reifenberg domains by Tian, Hong; Zheng,  Shenzhou (2020) - 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

• Global classical solutions to equatorial shallow-water equations by Fang, Yue; Li, Kaiqiang; Xu, Xin (2023) - + 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

• Periodicity of non-homogeneous trajectories for non-instantaneous impulsive heat equations by Peng, Peng; Wang,  Jinrong; O'Regan, Donal (2020) - [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

• Maximum and antimaximum principles for the p-Laplacian with weighted Steklov boundary conditions by Cuesta, Mabel; Leadi, Liamidi; Nshimirimana, Pascaline (2020) - = 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

• Null controllability from the exterior of fractional parabolic-elliptic coupled systems by Louis-Rose, Carole (2020) - 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

• Explicit solutions of Jensen's auxiliary equations via extremal Lipschitz extensions by Charro, Fernando (2020) - 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

• Finite time extinction for a damped nonlinear Schrodinger equation in the whole space by Begout, Pascal (2020) - 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

• Global weak solutions to degenerate coupled transport processes in partially saturated deformable elastic-inelastic porous media by Benes, Michal (2020) - 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

• Holder continuity for vector-valued minimizers of quadratic functionals by Danecek, Josef; Viszus, Eugen (2020) - Case Aαβij = Aαβij (u). Case Aαβij = Aαβij (x, u). Keywords: aαβij

• Fractional-power approach for the study of elliptic second-order boundary-value problems with variable-operator coefficients in an unbounded domain by Boutaous, Fatiha (2020) - = − 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)−; ∫ +

• Local existence and blow-up criterion for the two and three dimensional ideal magnetic Benard problem by Manna, Utpal; Ashirbad Panda, Akash (2020) - 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

• Growth properties of solutions of complex differential equations with entire coefficients of finite (alpha,beta,gamma)-order by Belaidi, Benharrat; Biswas, Tanmay (2023) - [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

• Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential by Deng, Jin; Xia,  Aliang; Yang, Jianfu (2020) - = 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

• Stability for conformable impulsive differential equations by Ding, Yuanlin; Feckan, Michal; Wang, Jinrong (2020) - = 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

• Maximal regularity for non-autonomous Cauchy problems in weighted spaces by Mahdi, Achache; Hossni, Tebbani (2020) - (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

• Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations by da Silva Almeida Junior, Dilberto; Araujo Ramos, Anderson de Jesus; Pantoja Fortes, Joao Carlos; de Lima Santos, Mauro (2020) - = ∫ 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

• Uniform attractors of non-autonomous suspension bridge equations with memory by Wang, Lulu; Ma, Qiaozhen (2024) - = 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

• Existence of periodic solutions and stability for a nonlinear system of neutral differential equations by Li, Yang; Chen, Guiling (2024) - ∫ 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

• Form of solutions to quadratic trinomial partial differential equations with two complex variables by Tu, Jin; Wei, Huizhen (2024) - 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 +

• Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space by Jia, Xiaobiao; Ma, Shanshan (2023) - − 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

• Existence of two infinite families of solutions for singular superlinear equations on exterior domains by Iaia, Joseph (2024) - (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

• Properties of the solutions to periodic conformable non-autonomous non-instantaneous impulsive differential equations by Ding, Yuanlin; Liu, Kui (2024) - = 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

• Existence of high energy solutions for superlinear coupled Klein-Gordons and Born-Infeld equations by Wang, Lixia; Zhao, Pingping; Zhang, Dong (2024) - 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

• Mild solutions to fourth-order parabolic equations modeling thin film growth with time fractional derivative by Liu, Qiang; Zhu, Wanyu; Ye, Hailong (2024) - 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

• Normalized ground state of a mixed dispersion nonlinear Schrodinger equation with combined power-type nonlinearities by Ma, Zhouji; Chang, Xiaojun; Feng, Zhaosheng (2024) - = 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

• Solvability of transmission problems with generalized diffusion equation in L^p-spaces by Thorel, Alexandre (2024) - 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 +

• Existence and boundedness of solutions for a parabolic-parabolic predator-prey model by Zhao, Fengxiang; Tang, Haotian; Zheng, Jiashan; Li, Kaiqiang (2025) - 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

• Curved-pipe flow with boundary conditions involving Bernoulli pressure by Doresic, Tvrtko; Pazanin, Igor (2024) - − 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

• Oseledets decomposition on sub semiflow by Kryspin, Marek (2024) - 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

• Discrete Stein-Weiss inequalities by Li, Chunhong; Zhou, Tiantian (2025) - 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

• Dynamics of a non-autonomous stochastic weakly damped plate model with critical exponent by Wen, Lan; Yang, Lu (2022) - (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

• Convergence theorems of implicit type iterations in geodesic spaces with negative curvature by Kimura, Yasunori; Sasaki, Kazuya; Torii, Kakeru (2024) - 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

• A nonlinear mathematical model for two-phase flow in nanoporous media by Melzi, Imane; Atik, Youcef (2022) - 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

• Existence of solutions to fractional p-Laplacian problems with Robin boundary conditions by Xie, Junhui; Li, Pengfei (2025) - 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

Epilogue

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Created: 2025-12-25