Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 125, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL SOLUTIONS FOR FRACTIONAL VISCOELASTIC EQUATIONS WITH LOGARITHMIC NONLINEARITIES EUGENIO CABANILLAS LAPA In memory of my mother Celia Lapa C. Abstract. In this article we study a fractional viscoelastic equation of Kirch- hoff type with logarithmic nonlinearities. Under suitable conditions we prove the existence of global solutions and the exponential decay of the energy. 1. Introduction We consider the problem of finding u = u(x, t) weak solutions to the nonlinear heat equation of Kirchhoff type with variable exponent of nonlinearity, viscoelastic term and logarithmic source terms, involving the fractional Laplacian,( 1 + a|u|r(x)−2 ) ut +M(‖u‖2w0 )(−∆)su− ∫ t 0 g(t− τ)(−∆)su(τ)dτ = ( |u|ρ−2u+ |u|σ−2u ) log |u| =: f(u) in Ω×]0,∞[, u = 0 in (RN\Ω)× [0,∞[, u(x, 0) = u0(x) in Ω, (1.1) where Ω ⊆ RN is a smooth bounded domain, M(t) = tα−1 + 1, 0 ≤ t0, s ∈]0, 1[, 2 < N/s, α > 1, g : [0,∞[→]0,∞[ belongs to C1([0,∞[), g(0) > 0, l = 1 −∫∞ 0 g(τ)dτ > 0, g′(t) ≤ 0, ρ, σ > 2, and r is a continuous function. This type of problems without viscoelastic term (that is g = 0 ), with r(x) constant , M(t) = 1 and f a polynomial, have been considered by many authors with the standard Laplace operator (−4)s, s = 1, and can be seen as special case of doubly nonlinear parabolic type equations (ϕ(u))t −∆u = f(u), which appear in the mathematical modeling of various physical processes such as flows of incompressible turbulent fluids or gases in pipes, processes of filtration in porous media, glaciology, see [2, 3, 13, 35] and the further references therein. 2010 Mathematics Subject Classification. 35J30, 35J92,47H05, 47H30. Key words and phrases. Kirchhoff type problem; fractional Laplacian; Galerkin method; logarithmic nonlinearity. c©2020 Texas State University. Submitted January 19, 2020. Published December 20, 2020. 1 2 E. CABANILLAS L. EJDE-2020/125 The questions of solvability and the long time behavior of solutions to the doubly nonlinear nonlocal parabolic equation (ϕ(u))t − div σ(∇u) = ∫ t 0 g(t− τ)divσ(∇u(τ))dτ + f(x, t, u), were studied in [4, 19, 30, 31, 32, 34]. This equation arises from the study of heat conduction in materials with memory. On the other hand, many fractional and non- local operators are actively studied in recent years. This type of operators arises in a quite natural way in many interesting applications, such as, finance, physics, game theory, Lévy stable diffusion processes, crystal dislocation; see [5, 22, 36] and their references. The first result concerning fractional Kirchhoff problems was obtained by Fiscella and Valdinoci [18]. Pan et al [26] investigated for the first time the existence of global weak solutions for degenerate Kirchhoff-type diffusion problems involving fractional p-Laplacian, by combining the Galerkin method with potential well theory, for the special function M(t) = tλ−1(t ≥ 0). Mingqi et al [24] proved the existence and blow-up of solutions for a similar equation with more general con- ditions on M which cover the degenerate case. Recently, logarithmic nonlinearity appears frequently in partial differential equations which describes important phys- ical phenomena, see [12, 14, 20, 23, 37] and the references therein. Ding and Zhou [14] studied the semilinear parabolic problem of Kirchhoff type with logarithmic nonlinearity, ut −M([u]2s)LKu = |u|p−2u log |u|. They obtained results of global solutions and of finite time blow-up of solutions, when the initial energy is subcritical and critical, by using the potential well method. In the works mentioned above, there are only a few about global existence and exponential decay rate for doubly nonlinear parabolic equations involving variable exponent, viscoelastic term in the fractional setting, and logarithmic nonlinear terms. Motivated by this, we study global solutions for (1.1) by using Galerkin’s method and similar arguments as those in Tartar [33]. Also, we give the exponential decay rate of the energy via the energy perturbation method. It is worth mentioning that we do not use the logarithmic Sobolev inequality to obtain our results. The article is organized as follows. In Section 2, we give the preliminaries for our research. In Section 3, by using the Galerkin approximation method we obtain a global solution, and finally, we obtain the exponential decay under certain class of initial data. 2. Preliminaries In this section, we present some material and assumptions needed in the rest of this paper. We denote Q = R2N \ (CΩ× CΩ), CΩ := RN \ Ω, and W = { u : RN → R : u|Ω ∈ L2(Ω), ∫∫ Q |u(x)− u(y)|2 |x− y|N+2s dx dy <∞ } , where u|Ω represents the restriction to Ω of function u(x). Also, we define the linear subspace of W , W0 = { u ∈W : u = 0 a.e. in RN \ Ω } . The linear space W is endowed with the norm ‖u‖W := ‖u‖L2(Ω) + (∫∫ Q |u(x)− u(y)|2 |x− y|N+2s dx dy )1/2 . EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 3 It is easily seen that ‖ · ‖W is a norm on W and C∞0 (Ω) ⊆W0. The functional ‖u‖W0 = (∫∫ Q |u(x)− u(y)|2 |x− y|N+2s dx dy )1/2 , is a equivalent norm on W0 = {u ∈W : u(x) = 0 a.e. in RN \ Ω} which is a closed linear subspace of W . Furthermore (W0, ‖ · ‖W0 ) is a Hilbert space with inner product 〈u, v〉W0 = ∫∫ Q (u(x)− u(y))(v(x)− v(y)) |x− y|N+2s dxdy. Now we review the main embedding results for the space W0. Lemma 2.1 ([27, 28, 29]). The embedding W0 ↪→ Lr(Ω) is continuous for any r ∈ [1, 2∗s], and compact for any r ∈ [1, 2∗s[. Lemma 2.2 ([25, Lemma 2.1]). Let N ≥ 1, 0 < s < 1, p > 1, q ≥ 1, τ > 0 and 0 < θ < 1 be such that 1 τ = θ( 1 p − s N ) + 1−θ q . Then ‖u‖Lτ (RN ) ≤ ‖u‖θW s,p(RN )‖u‖ 1−θ Lq(RN ) , ∀u ∈ C1 0 (RN ). Now, we recall some background concerning the generalized Lebesgue-Sobolev spaces. We refer the reader to [15, 16, 17] for details. Set C+(Ω) = {p(x) : p(x) ∈ C(Ω), p(x) > 1, for all x ∈ Ω}. For p ∈ C+(Ω) we define p+ = max{p(x) : x ∈ Ω}, p− = min{p(x);x ∈ Ω}, and the space Lp(x)(Ω) = { u : u is a measurable real-valued function, ∫ Ω |u(x)|p(x)dx <∞ } , with ‖u‖p(x) ≡ ‖u‖Lp(x)(Ω) = inf { λ > 0 : ∫ Ω ∣∣u(x) λ ∣∣p(x) dx ≤ 1 } which is a Banach space [21]. We also define the space W 1,p(x)(Ω) = {u ∈ Lp(x)(Ω) : |∇u| ∈ Lp(x)(Ω)}, equipped with the norm ‖u‖W 1,p(x)(Ω) = ‖u(x)‖p(x) + ‖∇u(x)‖p(x). We denote by W 1,p(x) 0 (Ω) the closure of C∞0 (Ω) in W 1,p(x)(Ω). Of course the norm ‖u‖ = ‖∇u‖p(x) is an equivalent norm in W 1,p(x) 0 (Ω). Proposition 2.3 ([16]). (i) The conjugate space of Lp(x)(Ω) is Lp ′(x)(Ω), where 1 p(x) + 1 p′(x) = 1. For any u ∈ Lp(x)(Ω) and v ∈ Lp′(x)(Ω), we have∫ Ω |uv|dx ≤ ( 1 p− + 1 p′− ) ‖u‖p(x)‖v‖p′(x) ≤ 2‖u‖p(x)‖v‖p′(x). (ii) If p1(x), p2(x) ∈ C+(Ω) and p1(x) ≤ p2(x) for all x ∈ Ω, then Lp2(x)(Ω) ↪→ Lp1(x)(Ω) and the embedding is continuous. Proposition 2.4 ([16]). Set ρ(u) = ∫ Ω |∇u(x)|p(x) dx, then for u ∈W 1,p(x) 0 (Ω) and (uk) ⊂W 1,p(x) 0 (Ω), we have 4 E. CABANILLAS L. EJDE-2020/125 (1) ‖u‖ < 1 (resp. = 1;> 1) if and only if ρ(u) < 1 (resp. = 1;> 1); (2) for u 6= 0, ‖u‖ = λ if and only if ρ(u/λ) = 1; (3) if ‖u‖ > 1, then ‖u‖p− ≤ ρ(u) ≤ ‖u‖p+ ; (4) if ‖u‖ < 1, then ‖u‖p+ ≤ ρ(u) ≤ ‖u‖p− ; (5) ‖uk‖ → 0 (resp. →∞) if and only if ρ(uk)→ 0 (resp. →∞). For x ∈ Ω, let us define p∗(x) = { Np(x) N−p(x) if p(x) < N, ∞ if p(x) ≥ N. Proposition 2.5 ([17]). If q ∈ C+(Ω) and q(x) ≤ p∗(x) (q(x) < p∗(x)) for x ∈ Ω, then there is a continuous (compact) embedding W 1,p(x)(Ω) ↪→ Lq(x)(Ω). Lemma 2.6. Let 2 < r < ρ < 2∗s.For each ε > 0, there exists a positive constant Cε such that ‖v‖ρρ ≤ ε‖v‖2W0 + Cε‖v‖krr , for all v ∈W0 ∩ Lr(Ω) where k = 2ρ(1− θ) r(2− ρθ) , θ = (1 r − 1 ρ )( s N − 1 2 + 1 r )−1 . The above lemma immediately follows from Lemma 2.2 and Young’s inequality. Lemma 2.7 ([21, Theorem 1, pag 23 ]). Suppose that r ∈ L∞+ (Ω), r− ≥ 2, w ∈ Lr(x)(Ω×]0, T [) and ∂ ∂t (|w|r(x)−2w) ∈ Lr ′(x)(Ω×]0, T [). Then, for any s, τ ∈ [0, T ] with s < τ we have formula of integration by parts,∫ τ s ∫ Ω w ( 1 r(x)− 1 |w|r(x)−2w ) dx dt = ∫ Ω 1 r(x) |w(τ)|r(x) dx− ∫ Ω 1 r(x) |w(s)|r(x) dx. 3. Existence of global solutions and exponential decay In this section, we focus our attention on global solutions and exponential decay for problem (1.1). Definition 3.1. Let T > 0. A weak solution of (1.1) is a function u ∈ L∞(0, T ;W0), with ut ∈ L2(0, T ;L2(Ω)) and (|u|r(x)/2)t ∈ L2(Ω×]0, T [) such that∫ T 0 ∫ Ω ( 1 + a|u|r(x)−2 ) utw dxdt+M(‖u‖2w0 ) ∫ T 0 〈u,w〉W0 dt − ∫ T 0 ∫ t 0 g(t− τ)〈u(τ), w〉W0dτ dt = ∫ T 0 ∫ Ω f(u)w dxdt, for all w ∈ L2(0, T ;W0) and u(x, 0) = u0(x) ∈ W0, where for s ∈ R, f(s) =( |s|ρ−2s+ |s|σ−2s ) log |s|. Theorem 3.2 (Local solution). Assume u0 ∈ W0 \ {0}, 2 < r− < 2∗s, r+, ρ, σ ∈ ]2, 2∗s[ , then problem (1.1) has a unique weak solution u for T small enough. EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 5 Proof. We prove the existence of weak solutions by using the Faedo-Galerkin method with ideas from [7]. We choose a sequence {wν}ν∈N ⊆ C∞0 (Ω) such that C∞0 (Ω) ⊆ ∪∞ν=1Vm C1(Ω) and {wν} is a standard orthonormal basis with respect to the Hilbert space L2(Ω) and an orthogonal basis in W0, where Vm = spam{w1, w2, ....wm}. Now, we con- struct approximate solutions um (m = 1, 2, . . . ), of the problem (1.1), in the form um(x, t) = m∑ i=1 gjm(t)wj(x), where the coefficient functions gjm satisfy the system of ordinary differential equa- tions ∫ Ω ( 1 + a|um(t)|r(x)−2 ) umt(t)wj dx+M(‖um(t)‖2w0 )〈um(t), wj〉W0 − ∫ t 0 g(t− τ)〈um(τ), wj〉W0 dτdt = ∫ Ω f(um)wj dx, j = 1, 2, . . .m; um(x, 0) = u0 m(x)→ u0(x) in W0. (3.1) Let us show that the system (3.1) is locally solvable. It is clear that (3.1) can be rewritten in the form d dt Φ(gm(t)) = −M ( ‖ m∑ i=1 gjm(t)wj(x)‖2W0 ) Bgm(t) + ∫ t 0 g(t− τ)Bgm(τ)dτ + F (gm(t)), (3.2) where gm(t) = (gm1(t), gm2(t), . . . , gmm(t))T , B = [〈wi, wj〉]1≤i,j≤m, Φ(η) = (Φ1(η),Φ2(η), . . . ,Φm(η))T with η = (η1, η2, . . . , ηm) ∈ Rm, Φi(η) = ∫ Ω { m∑ j=1 ηjwj + a r(x)− 1 ∣∣∣ m∑ k=1 ηkwk ∣∣∣r(x)−2 m∑ k=1 ηkwk } wi dx i = 1, 2, . . . ,m; F (η) = (∫ Ω f ( m∑ k=1 ηjwj ) w1 dx, . . . , ∫ Ω f ( m∑ k=1 ηjwj ) wm dx )T . This system is equivalent to Φ(gm(t)) = Φ(gm(0)) + ∫ t 0 [ −M (∥∥ m∑ i=1 gjm(t)wj(x) ∥∥2 W0 ) Bgm(t) + ∫ ξ 0 g(ξ − τ)Bgm(τ)dτ + F (gm(ξ)) ] dξ. and the fact that the map s 7→ f(s) is increasing for large s, we obtain (Φ(ζ)− Φ(η), ζ − η)Rm ≥ Cm|ζ − η|2Rm (3.3) 6 E. CABANILLAS L. EJDE-2020/125 for ζ, η ∈ Rm, where Cm is a constant such that, for any gm in Rm,∫ Ω |um|2 dx ≥ Cm|gm|2Rm . So, by the elementary inequality s log s ≥ s − 1,∀s > 0, we deduce that Φ is monotone coercive. Also it is obviously continuous. So, by the Brouwer theorem Φ is onto. In view of (3.3), Φ−1 is locally Lipchitz continuous. Consider the map L : C(0, T,Rm)→ C(0, T,Rm), defined by L(gm)(t) = Φ−1 ( Φ(gm(0)) + ∫ t 0 [ −M ( ‖ m∑ i=1 gjm(t)wj(x)‖2W0 ) Bgm(t) + ∫ ξ 0 g(ξ − τ)Bgm(τ)dτ + F (gm(ξ)) ] dξ ) , t ∈ [0, T ]. It is not hard to prove that L is completely continuous and that there exist (sufficient small) Tm > 0 and (sufficient large) R > 0 such that L(BR) ⊆ BR, where BR is the ball in C(0, Tm,Rm) with center the origin and radius R. Consequently, by Schauder’s theorem, the operator L has a fixed point in C(0, Tm,Rm). This fixed point is a solution of (3.2). So, we can obtain an approximate solution um(t) of (3.1) in Vm over [0, Tm[ and it can be extended to the whole interval [0, T ], for all T > 0, as a consequence of the a priori estimates that shall be proven in the next step. First estimate. Multiplying (3.1) by gjm(t) and adding in j = 1, . . . ,m, we have∫ Ω ( 1 + a|um(t)|r(x)−2 ) umt(t)um(t) dx+M(‖um(t)‖2w0 )〈um(t), um(t)〉W0 − ∫ t 0 g(t− τ)〈um(τ), um(t)〉W0 dτdt = ∫ Ω (|um(t)|ρ + |um(t)|σ) log |um(t)| dx (3.4) which implies, integrating with respect to the time variable from 0 to t on both sides, using Lemma 2.7 that Sm(t) = Sm(0) + ∫ t 0 dλ ∫ λ 0 g(λ− τ)〈um(τ), um(λ)〉W0 dτ + ∫ t 0 ∫ Ω (|um(τ)|ρ + |um(τ)|σ) log |um(τ)| dx dτ, (3.5) where Sm(t) = ∫ Ω |um(t)|2 dx+ a ∫ Ω 1 r(x) |um(t)|r(x) dx + ∫ t 0 ( ‖um(τ)‖2αW0 + ‖um(τ)‖2W0 ) dτ. EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 7 Let us introduce the function Θ(λ) = ∫ λ 0 g(λ−τ)‖um(τ)‖W0 . Estimating the second term on right-hand side of (3.5) we have∫ t 0 dλ ∫ λ 0 g(λ− τ)〈um(τ), um(λ)〉W0 dτ ≤ 1 2 ∫ t 0 ( ‖um(τ)‖2αW0 + ‖um(τ)‖2W0 ) dτ + 1 2 ∫ t 0 Θ2(λ) dλ. (3.6) But, using Young Inequality and noting that ∫∞ 0 g(τ)dτ < 1, we obtain ∫ t 0 Θ2(λ)dλ ≤ ∫ ∞ 0 g(τ)dτ ∫ t 0 ( ‖um(τ)‖2αW0 + ‖um(τ)‖2W0 ) dτ. (3.7) Let %ρ := 2∗s − ρ, %σ := 2∗s − σ. Since log(|u|%) ≤ |u|% it follows that∫ Ω |um(t)|ρ log |um(t)| dx = 1 %ρ ∫ Ω |um(t)|ρ log(|um(t)|%ρ) dx ≤ 1 %ρ ∫ Ω |um(t)|ρ+%ρ dx (3.8) Plugging (3.6)-(3.8) into (3.5), it follows that Sm(t) ≤ Sm(0) + 1 2 ( 1 + ∫ ∞ 0 g(τ)dτ )∫ t 0 ( ‖um(τ)‖2αW0 + ‖um(τ)‖2W0 ) dτ + ∫ t 0 ( 1 %ρ ‖um(τ)‖%ρ+ρ %ρ+ρ + 1 %σ ‖um(τ)‖%σ+σ %σ+σ ) dτ. (3.9) To estimate the last term in (3.9) we use Lemma 2.6,∫ t 0 ( 1 %ρ ‖um(τ)‖%ρ+ρ %ρ+ρ + 1 %σ ‖um(τ)‖%σ+σ %σ+σ ) dτ ≤ ε ∫ t 0 ( ‖um(τ)‖2αW0 + ‖um(τ)‖2W0 ) dτ + C0 ∫ t 0 ( Sk1m (λ) + Sk2m (λ) ) dλ, (3.10) where k1 = 2(%ρ + ρ)(1− θ) r−[2− (ρ+ %ρ)θ] > 1, k2 = 2(%σ + σ)(1− θ) r−[2− (%σ + σ)θ] > 1. Taking ε suitably small in (3.10), it follows from (3.5)-(3.10) that Sm(t) ≤ Ĉ0 + Ĉ1 ∫ t 0 ( Sk1m (λ) + Sk2m (λ) ) dλ. (3.11) Hence, by employing [10, Theorem 1.2], there exists a constant T0 such that Sm(t) ≤ CT0 , ∀t ∈ [0, T0]. (3.12) 8 E. CABANILLAS L. EJDE-2020/125 Second estimate. Multiplying (3.1) by g′jm(t) and adding in j = 1, . . . ,m, it follows that d dt { 1 2α ‖um(t)‖2αW0 + 1 2 ( 1− ∫ t 0 g(τ)dτ ) ‖um(t)‖2W0 + 1 2 (g � u)(t) − 1 ρ ∫ Ω |um(t)|ρ log |um(t)| dx+ 1 ρ2 ‖um(t)‖ρρ − 1 σ ∫ Ω |um(t)|σ log |um(t)| dx+ 1 σ2 ‖um(t)‖σσ } + ‖umt(t)‖22 + a ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx = 1 2 (g′ � u)(t)− 1 2 g(t)‖um(t)‖2W0 . (3.13) where (g � u)(t) = ∫ t 0 g(t− τ)‖u(t)− u(τ)‖2W0 dτ. Integrating (3.13) on [0, t], t ≤ T0 we obtain∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx+ 1 2α ‖um(t)‖2αW0 + l 2 ‖um(t)‖2W0 ≤ 1 2α ‖um(0)‖2αW0 + 1 2 ‖um(0)‖|2W0 + 1 ρ ∫ Ω |um(t)|ρ log |um(t)| dx + 1 ρ2 ‖um(0)‖ρρ + 1 σ ∫ Ω |um(t)|σ log |um(t)| dx+ 1 σ2 ‖um(0)‖σσ − 1 ρ ∫ Ω |um(0)|ρ log |um(0)| dx− 1 σ ∫ Ω |um(0)|σ log |um(0)| dx. From the assumptions on u0, (3.8), Lemma 2.6 and the estimate (3.12), it follows that ∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx + 1 2α ‖um(t)‖2αW0 + l 2 ‖um(t)‖2W0 ≤M1, (3.14) for some constant M1 > 0. By the above estimates (3.12) and (3.14), {um} have subsequences still denoted by {um} such that um → u weakly* in L∞(0, T0;W0), umt → ut weakly in L2(0, T0;L2(Ω)),( |um|r(x)/2 ) t → χ weakly in L2(0, T0;L2(Ω)). (3.15) Also, reasoning as in [12], taking into account the compact embedding of W0 into Lβ(Ω), β = ρ, σ, we have |um|ρ−2um log |um| → |u|ρ−2u log |u| weakly* in L∞(0, T0;L ρ ρ−1 (Ω)), |um|σ−2um log |um| → |u|σ−2u log |u| weakly* in L∞(0, T0;L σ σ−1 (Ω)) (3.16) Employing the same arguments as in [9] we can prove that χ = ( |u|r(x)/2 ) t , |um|r(x)/2umt → |u|r(x)/2ut weakly in L2(Ω×]0, T0[) (3.17) EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 9 Therefore, passing to the limit in (3.1) as m→ +∞, by (3.15)– (3.17), we can show that u satisfies the initial condition u(0) = u0 and∫ T 0 ∫ Ω ( 1 + a|u|r(x)−2 ) utw dxdt+M(‖u‖2w0 ) ∫ T 0 〈u,w〉W0 dt − ∫ T 0 ∫ t 0 g(t− τ)〈u(τ), w〉W0 dτdt = ∫ T 0 ∫ Ω f(u)w dxdt, for all w ∈ L2(0, T0;W0). The uniqueness property of solutions can be derived as in [13, Theorem 3, p. 1095], observing that ( u+ a r(x)−1 |u| r(x)−2u ) ∈ L2(Ω×]0, T0[), F (s) = f(s) log(|s|) is locally Lipschitz continuous and Au = M(‖u‖2w0 )(−∆)su is a monotone operator. We omit the details. � Next, we consider the existence of global solutions and their energy decay for problem (1.1). For this purpose we define the energy associated with problem (1.1) by E(t) = 1 2α ‖u(t)‖2αW0 + 1 2 ( 1− ∫ t 0 g(τ)dτ ) ‖u(t)‖2W0 + 1 2 (g � u)(t) − 1 ρ ∫ Ω |u(t)|ρ log |u(t)| dx+ 1 ρ2 ‖u(t)‖ρρ − 1 σ ∫ Ω |u(t)|σ log |u(t)| dx+ 1 σ2 ‖u(t)‖σσ. (3.18) Then, we easily can check that d dt E(t) = 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 ≤ 0 (3.19) for any regular solution. This remains valid for weak solutions by simple density argument. This shows that E(t) is a nonincreasing function. Before going on, we introduce the following notation B1 = sup u∈W0, u 6=0 ‖u‖ρ+%ρ√ l‖u‖2W0 , B2 = sup u∈W0, u 6=0 ‖u‖σ+%σ√ l‖u‖2W0 , γ1 = 1 ρ%ρ B ρ+%ρ 1 , γ2 = 1 σ%σ Bσ+%σ 2 . Define the function h(λ) = 1 4 λ2 − γ1λ ρ+%ρ − 3 2 γ2λ σ+%σ . (3.20) Then h′(λ) = 1 2 λ− (ρ+ %ρ)γ1λ ρ+%ρ−1 − 3 2 (σ + %σ)γ2λ σ+%σ−1. So, choosing λ ∈ R such that 0 ≤ λρ+%ρ−1 ≤ 1 4(ρ+ %ρ)γ1 and 0 ≤ λσ+%σ−1 ≤ 1 6(σ + %σ)γ2 10 E. CABANILLAS L. EJDE-2020/125 we obtain that these λs satisfy the inequality 1 2 λ ≥ (ρ+ %ρ)γ1λ ρ+%ρ−1 + 3 2 (σ + %σ)γ2λ σ+%σ−1 and h′(λ) ≥ 0 for 0 ≤ λ ≤ λ1 where λ1 = min {[ 1 4(ρ+ %ρ)γ1 ]1/(ρ+%ρ−1) , [ 1 6(σ + %σ)γ1 ]1/(σ+%σ−1) } . Thus h(0) = 0 and h(λ) ≥ 0 for all λ ∈ [0, λ1[ Therefore, from (3.20), we have 1 4 λ2 − γ1λ ρ+%ρ − 3 2 γ2λ σ+%σ ≥ 0, ∀λ ∈ [0, λ1[. (3.21) Now, if one considers l‖u(t)‖2W0 + (g � u)(t) < λ2 1, (3.22) from (3.21), we obtain 1 4 (√ l‖u(t)‖2W0 + (g � u)(t) )2 + 1 2 γ2 (√ l‖u(t)‖2W0 + (g � u)(t) )σ+%σ ≤ 1 2 (√ l‖u(t)‖2W0 + (g � u)(t) )2 − γ1 (√ l‖u(t)‖2W0 + (g � u)(t) )ρ+%ρ − γ2 (√ l‖u(t)‖2W0 + (g � u)(t) )σ+%σ which implies E(t) ≥ l 2 ‖u(t)‖2W0 + 1 2 (g � u)(t)− γ1 (√ l‖u(t)‖2W0 + (g � u)(t) )ρ+%ρ − γ2 (√ l‖u(t)‖2W0 + (g � u)(t) )σ+%σ ≥ 1 4 (√ l‖u(t)‖2W0 + (g � u)(t) )2 + 1 2 γ2 (√ l‖u(t)‖2W0 + (g � u)(t) )σ+%σ , t ≥ 0. (3.23) Now, we are ready to state our main result. Theorem 3.3. Assume that hypotheses of Theorem 3.2 are satisfied. Consider u0 ∈W0, satisfying 0 < l1/2‖u0‖W0 < λ1, (3.24)(4 l E1 )1/2 < λ1, (3.25) where E1 = 1 2α ‖u0‖2αW0 + 1 2 ‖u0‖2W0 − 1 ρ ∫ Ω |u0|ρ log |u0| dx+ 1 ρ2 ‖u0‖ρρ − 1 σ ∫ Ω |u0|σ log |u0| dx+ 1 σ2 ‖u0‖σσ. (3.26) Then the problem admits a global weak solution in time. In addition, if there exists a constant ξ0 > 0 such that g′(t) ≤ −ξ0g(t), then this solution satisfies E(t) ≤ L0e −γt, ∀t ≥ 0, (3.27) where L0 and γ are positive constants. EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 11 Proof. We will get global estimates for um(t) solution of the approximate system (3.1) under the conditions (3.24)–(3.25) for u0. For this, it suffices to show that Em(t) + ∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx, where Em(t) is defined in (3.18) with u(t) replaced by um(t), is bounded and independently of t. From (3.13) and the definition of energy, we have Em(t) + ∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx ≤ Em(0). (3.28) From the convergence u0m → u0 in W0 we see that Em(0) < l 4λ 2 1 for sufficiently large m. We claim that there exists an integer ν0 such that√ l‖um(t)‖2W0 + (g � um)(t) < λ1 ∀t ∈ [0, Tm[,m ≥ ν0. (3.29) Supposing that the claim is proved, h (√ l‖um(t)‖2W0 + (g � um)(t) ) ≥ 0 and from (3.23), (3.28)–(3.29) we obtain ‖um(t)‖2αW0 + ‖um(t)‖2W0 + ∫ t 0 ‖umt(t)‖22 + a ∫ t 0 ∫ Ω |um(t)|r(x)−2|umt(t)|2 dx ≤ C. where C is a constant independent of m. Thus, we obtain the global solution. Proof of Claim: Suppose (3.29) is not true. Thus, for each m > ν0, there exists t1 ∈ [0, Tm[ such that √ l‖um(t1)‖2W0 + (g � um)(t1) ≥ λ1. (3.30) Here, we observe that, from (3.24) and the convergence u0m → u0 in W0 there exists ν1 such that l1/2‖um(0)‖W0 < λ1 ∀m > ν1. Hence, by continuity there exists t∗ = inf{t ∈ [0, Tm[: √ l‖um(t)‖2W0 + (g � um)(t) ≥ λ1}, such that √ l‖um(t∗)‖2W0 + (g � um)(t∗) = λ1. (3.31) By (3.23), we see that Em(t∗) ≥ h (√ l‖um(t∗)‖2W0 + (g � um)(t∗) ) ≥ 1 4 (l‖u(t∗)‖2W0 + (g � u)(t∗)) = 1 4 λ2 1 (3.32) which contradicts Em(t) ≤ Em(0) < l 4λ 2 1. Therefore our claim is true. The above estimates permit us to pass to the limit in the approximate equation. To show the uniform decay of the solution we introduce the perturbed energy functional F (t) = E(t) + εΦ(t), (3.33) where ε is a positive constant which shall be determined later, and Φ(t) = ∫ Ω (|u|2 + a r(x) |u|r(x)) dx. (3.34) 12 E. CABANILLAS L. EJDE-2020/125 It is straightforward to see that F (t) and E(t) are equivalent in the sense that there exist two positive constants β1 and β2 depending on ε such that for t ≥ 0, β1E(t) ≤ F (t) ≤ β2E(t). (3.35) By taking the time derivative of the function F defined in (3.33), using (3.19), and performing several integration by parts, we obtain d dt F (t) = 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) On the other hand, from (3.23) and the monotonicity of E(t) we have ‖u(t)‖W0 ≤ 4E(0)/ √ l, which implies∣∣ ∫ Ω uf(u) log |u| ∣∣ ≤ 1 %ρ ‖u(t)‖%ρ+ρ %ρ+ρ + 1 %σ ‖u(t)‖%σ+σ %σ+σ ≤ C %ρ+ρ ∗ρ %ρ ‖u(t)‖%ρ+ρ W0 + C%σ+σ ∗σ %σ ‖u(t)‖%σ+σ W0 ≤ [C%ρ+ρ ∗ρ %ρl ( 4√ l E(0) )%ρ+ρ−2 + C%σ+σ ∗σ %σl ( 4√ l E(0) )%σ+σ−2] l‖u(t)‖2W0 ≡ θl‖u(t)‖2W0 . (3.37) where 1 c∗δ = inf u∈W0\0 ‖u‖W0 ‖u‖δ , with δ = ρ, σ. From Young’s inequality and the fact that ∫ t 0 g(τ) dτ ≤ ∫∞ 0 g(τ) dτ = 1− l, it follows that∫ t 0 g(t− τ)〈u(τ), u(t)〉W0 dτ ≤ 1 2 ‖u(t)‖2W0 + 1 2 {∫ t 0 g(t− τ) (‖u(τ)− u(t)‖W0 + ‖u(t)‖W0 ) dτ }2 ≤ 1 2 ‖u(t)‖2W0 + 1 2 (1 + η) (∫ t 0 g(t− τ)‖u(t)‖W0 dτ )2 + 1 2 (1 + 1 η ) (∫ t 0 g(t− τ)‖u(τ)− u(t)‖W0 dτ )2 ≤ 1 2 ‖u(t)‖2W0 + 1 2 (1 + η)(1− l)2‖u(t)‖2W0 + 1 2 (1 + 1 η )(1− l)(g � u)(t). (3.38) for any η > 0. Now, letting η = l 1−l > 0, (3.38) yields∫ t 0 g(t− τ)〈u(τ), u(t)〉W0 dτ ≤ 2− l 2 ‖u(t)‖2W0 + 1− l 2l (g � u)(t). (3.39) EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 13 Substituting (3.39) into (3.36), we obtain d dt F (t) ≤ −1 2 ( ξ0 − ε 1− l l ) (g � u)(t)− ε‖u(t)‖2αW0 − εl 2 ‖u(t)‖2W0 + ε ∫ Ω uf(u) log |u(t)| dx. (3.40) Using the definition of E(t) and (3.37), for any positive constant M , we have d dt F (t) ≤ −MεE(t) + ε (M 2α − 1 ) ‖u(t)‖2αW0 + ε {[1 2 + cρ∗ρ ρ2 ( 4E(0)√ l )ρ−2 + cσ∗σ σ2 ( 4E(0)√ l )σ−2 ] M + (M + 2)θl − l } ‖u(t)‖2W0 + 1 2 [ ε( 1− l l + M 2 )− ξ0 ] (g � u)(t). (3.41) At this point, we choose 1 > M > 0 and E(0) small sufficiently such that M 2α−1 < 0 and [1 2 + cρ∗ρ ρ2 ( 4E(0)√ l )ρ−2 + cσ∗σ σ2 ( 4E(0)√ l )σ−2 ] M + (M + 2)θl − l < 0. After M is fixed, we choose ε small enough such that ε (1− l l + M 2 ) − ξ0 < 0. Inequality (3.41) becomes d dtF (t) ≤ −MεE(t). Then by (3.35), we have d dt F (t) ≤ −Mβ2εF (t). So F (t) ≤ Ce−Kt where K = Mβ2ε > 0. Consequently, by using (3.35) once again, we conclude the result. 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[37] Zhou, J.; Ground state solution for a fourth-order elliptic equation with logarithmic nonlin- earity modeling epitaxial growth. Comput. Math. Appl., 78 (6 (2019)), 1878–1886. EJDE-2020/125 FRACTIONAL VISCOELASTIC EQUATIONS 15 Eugenio Cabanillas Lapa Instituto de Investigación, Facultad de Ciencias Matemáticas UNMSM, Lima-Perú Email address: cleugenio@yahoo.com 1. Introduction 2. Preliminaries 3. Existence of global solutions and exponential decay First estimate Second estimate Acknowledgements References