Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 53, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.53 OPTIMAL ENERGY DECAY RATES FOR VISCOELASTIC WAVE EQUATIONS WITH NONLINEARITY OF VARIABLE EXPONENT MUHAMMAD I. MUSTAFA Abstract. In this article, we consider the viscoelastic wave equation utt −∆u + ∫ t 0 g(t− s)∆u(s)ds + a|ut|m(·)−2ut = 0 with a nonlinear feedback having a variable exponent m(x). We investigate the interaction between the two types of damping and establish an optimal decay result under very general assumptions on the relaxation function g. We construct explicit formulae which provide faster energy decay rates than the ones already existing in the literature. 1. Introduction In this article we consider the problem utt −∆u+ ∫ t 0 g(t− s)∆u(s)ds+ a|ut|m(·)−2ut = 0, in Ω× (0, T ) u = 0, on ∂Ω× (0, T ) u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, (1.1) a wave equation subject to the effect of an internal frictional damping and a vis- coelastic damping. Here Ω is a bounded domain of Rn (n ≥ 1) with a smooth boundary ∂Ω, a is a positive constant, g is a non-increasing positive function and m ∈ C(Ω) is satisfying 1 < m1 ≤ m(x) ≤ m2 < 2∗, (1.2) with m1 = inf x∈Ω m(x), m2 = sup x∈Ω m(x), 2∗ = { 2n n−2 , if n > 2 ∞, if n = 1, 2, and satisfying the log-Hölder continuity condition |m(x)−m(y)| ≤ − A log |x− y| , (1.3) for a.e. x, y ∈ Ω with |x− y| < δ, 0 < δ < 1, A > 0. 2020 Mathematics Subject Classification. 35B40, 74D99, 93D15, 93D20. Key words and phrases. Viscoelasticity; frictional damping; variable exponent; energy decay. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 22, 2023. Published August 28, 2023. 1 2 M. I. MUSTAFA EJDE-2023/53 In the absence of the viscoelastic term, when m is a constant satisfying 1 < m < 2∗, there have been many results about the existence and energy decay rates of the solutions, we refer the readers to [8, 15, 17, 29, 33] and the references therein. In recent years, more attention has been paid to the study of mathematical nonlinear models of hyperbolic, parabolic and elliptic equations with variable exponents of nonlinearity. Some models from physical phenomena like flows of electro-rheological fluids or fluids with temperature-dependent viscosity, filtration processes in a porous media, nonlinear viscoelasticity, and image processing, give rise to such problems. More details on the subject can be found in [4, 5]. Regarding hyperbolic problems with nonlinearities of variable-exponent type, only few works have appeared. The issues of existence and blow up of solutions were treated in [2, 3, 13, 21, 23, 32]. For the stability, Messaoudi and Talahmeh [22] looked at utt −∆u+ α|ut|m(x)−1ut = 0, (1.4) with α ≡ 1 and 2 ≤ m(x) < 2∗, and proved decay estimates for the solution under suitable assumptions on the variable exponent m. Mustafa et al. [25, 28] obtained decay rate estimates for (1.4) in bounded and unbounded domains with 1 < m(x) < 2∗ and a nonconstant time-dependent coefficient α(t). On the other hand, when the unique damping mechanism is given by the memory term, many stability results have been established with different types of relaxation function g. When g decays exponentially (resp. polynomially), we refer to [16, 24, 31] for subsequent results showing that the energy decays at the same rate of g. For more general types of g, Messaoudi [19] used the condition g′(t) ≤ −ξ(t)g(t) (1.5) where ξ is a non-increasing differentiable function, and established a more general decay result. Also, a condition of the form g′(t) ≤ −H ( g(t) ) (1.6) where H is a convex function satisfying some smoothness properties, was introduced by Alabau-Boussouira and Cannarsa [1] to obtain decay results in terms of H. Mustafa [25] studied viscoelastic wave equations with relaxation functions of more general type than the ones in (1.5) and (1.6), namely g′(t) ≤ −ξ(t)H (g(t)) (1.7) where H is increasing and convex without any additional constraints, and estab- lished energy decay results that address both the optimality and generality. The interaction between viscoelastic and frictional dampings was given a great deal of attention. We first refer to the work of Fabrizio and Polidoro [10] who studied the following equation utt −∆u+ ∫ t 0 g(t− τ)∆u(τ)dτ + h(ut) = 0, (1.8) with linear function h, and showed that the the viscoelasticity with poorly behaving relaxation kernel destroys the exponential decay rates generated by linear frictional dissipation. Cavalcanti and Oquendo [9] treated (1.8) and established exponential (resp. polynomial) stability for g decaying exponentially (resp. polynomially) and h has linear (resp. polynomial) growth near 0. Recently, Mustafa [27] obtained energy decay rates for (1.8) with general h and general g satisfying (1.7). EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 3 System (1.1), with constant m satisfying 1 < m < 2∗ and g satisfying (1.5), was investigated by Messaoudi [20] and stability results depending on m and ξ were obtained. Later on, Belhannache et al. [7] extended the result of [20] to the case when g satisfies (1.7). For variable exponent m(x), we refer to Gao and Gao [12] and Park and Kang [30] who studied (1.1), with nonlinear source term, and proved existence and blow up results. Hassan et al. [14] treated (1.1), used condition (1.7) and provided general energy estimates, but their results lack optimality in some cases. Our aim in this work is to investigate (1.1), with m(x) satisfying (1.2) and (1.3) and g satisfying (1.7). We study both cases when m1 ≥ 2 and m1 < 2 and establish explicit formulae depending on both m and g which combine the generality and optimality and provide faster energy decay rates than the ones obtained in [14]. 2. Preliminaries In this section, we present some preliminary facts about Lebesgue and Sobolev spaces with variable exponents (see [11, 17]) and introduce our assumptions. Let p : Ω → [1,∞) be a measurable function, where Ω is a domain of Rn. We define the Lebesgue space with a variable exponent p(·) by Lp(·)(Ω) = { u : Ω→ R,measurable in Ω and %p(·)(u) = ∫ Ω |u(x)|p(x) dx <∞}. Equipped with the Luxembourg-type norm ‖u‖p(·) := inf{λ > 0 : %p(·) (u λ ) ≤ 1}, Lp(·)(Ω) is a Banach space. If 1 < p1 ≤ p(x) ≤ p2 < ∞ holds, then, for any u ∈ Lp(·)(Ω), min{‖u‖p1p(·), ‖u‖ p2 p(·)} ≤ %p(·)(u) ≤ max{‖u‖p1p(·), ‖u‖ p2 p(·)}. We, next, define the variable-exponent Sobolev space W 1,p(·)(Ω) = {u ∈ Lp(·)(Ω) : ∇u exists and |∇u| ∈ Lp(·)(Ω)}. This space is a Banach space with respect to the norm ‖u‖W 1,p(·)(Ω) = ‖u‖p(·) + ‖∇u‖p(·). Furthermore, we set W 1,p(·) 0 (Ω) to be the closure of C∞0 (Ω) in W 1,p(·)(Ω). Here we note that the space W 1,p(·) 0 (Ω) is usually defined in a different way for the variable exponent case. However, both definitions are equivalent under (1.3). Hölder’s Inequality: Let p, q, s ≥ 1 be measurable functions defined on Ω such that 1 s(y) = 1 p(y) + 1 q(y) , for a.e. y ∈ Ω. If f ∈ Lp(·)(Ω) and g ∈ Lq(·)(Ω), then fg ∈ Ls(·)(Ω) and ‖fg‖s(·) ≤ 2 ‖f‖p(·)‖g‖q(·). Poincaré’s Inequality: Let Ω be a bounded domain of Rn and p(·) satisfies (1.3), then ‖u‖p(·) ≤ C‖∇u‖p(·), for all u ∈W 1,p(·) 0 (Ω), where the positive constant C depends on p1, p2 and Ω only. In particular, the space W 1,p(·) 0 (Ω) has an equivalent norm given by ‖u‖ W 1,p(·) 0 (Ω) = ‖∇u‖p(·). 4 M. I. MUSTAFA EJDE-2023/53 Embedding Property: Let Ω be a bounded domain in Rn with a smooth boundary ∂Ω. Assume that p, q ∈ C(Ω) such that 1 < p1 ≤ p(x) ≤ p2 < +∞, 1 < q1 ≤ q(x) ≤ q2 < +∞, for all x ∈ Ω, and q(x) < p∗(x) in Ω with p∗(x) = { np(x) n−p(x) , if p2 < n +∞, if p2 ≥ n, then there is a continuous and compact embedding W 1,p(·)(Ω) ↪→ Lq(·)(Ω). On the relaxation function g and the variable exponent m(x), we consider the following assumption (A1) m ∈ C(Ω) is satisfying (1.2) and (1.3) and g : [0,∞) → (0,∞) is a C1 function satisfying 1− ∫ +∞ 0 g(s)ds = l > 0, (2.1) g′(t) ≤ −ξ(t)H1(g(t)), ∀t ≥ 0, (2.2) where ξ : [0,∞) → (0,∞) is a non-increasing differentiable function and H1 : (0,∞)→ (0,∞) is a C1 function which is linear or strictly increasing and strictly convex 2 function on (0, r1], with H1(0) = H ′1(0) = 0. Remarks. (1) The function H2(t) = t m1 2m1−2 is strictly increasing and strictly con- vex when 1 < m1 < 2. We will be using H(t) = min{H1(t), H2(t)} and r ≤ r1 is small enough so that either H(t) = H1(t) or H(t) = H2(t) on the interval (0, r]. (2) The well-known Jensen inequality will be of essential use in establishing our result. If Y is a convex function on [d1, d2], y : Ω → [d1, d2] and w are integrable functions on Ω, w(x) ≥ 0, and ∫ Ω w(x) dx = d3 > 0, then Jensen’s inequality states Y [ 1 d3 ∫ Ω y(x)w(x) dx ] ≤ 1 d3 ∫ Ω Y [y(x)]w(x) dx. (3) If H∗ be the convex conjugate of H in the sense of Young [6, p. 61-64], then H∗(s) = s(H ′)−1(s)−H[(H ′)−1(s)] and H∗ satisfies the generalized Young inequality AB ≤ H∗(A) +H(B). (2.3) (4) If H is a strictly increasing and strictly convex 2 function on (0, r], with H(0) = H ′(0) = 0, then it has an extension H which is strictly increasing and strictly convex 2 function on (0,∞). For instance, if H(r) = a,H ′(r) = b,H ′′(r) = c, we can define H, for t > r, by H(t) = c 2 t2 + (b− cr)t+ (a+ c 2 r2 − br). (2.4) At the end of this section, we state, without proof, the following existence and regularity result. Proposition 2.1 ([11, 30]). Let (u0, u1) ∈ H1 0 (Ω)×L2(Ω) be given. If (A1) holds, then problem (1.1) has a unique global (weak) solution u ∈ L∞((0, T );H1 0 (Ω)), ut ∈ L∞((0, T );L2(Ω)) ∩ Lm(·)((0, T )× Ω). EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 5 Moreover, if (u0, u1) ∈ ( H2(Ω) ∩H1 0 (Ω) ) ×H1 0 (Ω), then the solution satisfies u ∈ L∞((0, T );H2(Ω) ∩H1 0 (Ω)) ∩W 1,∞((0, T );H1 0 (Ω)) ∩W 2,∞((0, T );L2(Ω)). 3. Technical Lemmas We introduce the energy functional E(t) := 1 2 ∫ Ω ( u2 t + [1− ∫ t 0 g(s) ds]|∇u|2 ) dx+ 1 2 (g ◦ ∇u)(t), where (g ◦ v)(t) = ∫ Ω ∫ t 0 g(t− s)|v(t)− v(s)|2 ds dx. In this section, we establish several lemmas and construct a Lyapunov functional L equivalent to E. We will use c, in this paper, to denote a generic positive constant. Lemma 3.1. Let u be the solution of (1.1). Then the energy functional satisfies E′(t) = 1 2 (g′ ◦ ∇u)(t)− 1 2 g(t) ∫ Ω |∇u|2 dx− a ∫ Ω |ut|m(x) dx ≤ 0. (3.1) Proof. By multiplying equation (1.1) by ut and integrating over Ω, using integration by parts, hypothesis (A) and some manipulations, we obtain (3.1). � We consider the following partition of Ω, Ω∗ = {x ∈ Ω : m(x) < 2} and Ω∗∗ = {x ∈ Ω : m(x) ≥ 2}. Lemma 3.2. The functional K1 defined by K1(t) := ∫ Ω uut dx (3.2) satisfies, along the solution of (1.1), the estimate K ′1(t) ≤ − l 4 ∫ Ω |∇u|2 dx+ ∫ Ω u2 t dx+ Cα l (h ◦ ∇u)(t) + c ∫ Ω∗ |ut|2m(x)−2 dx + c ∫ Ω |ut|m(x) dx (3.3) for any 0 < α < 1, where Cα = ∫ ∞ 0 g2(s) αg(s)− g′(s) ds and h(t) = αg(t)− g′(t). (3.4) Proof. Direct computations, using (1.1), (2.1), and Young’s inequality, yield K ′1(t) = ∫ Ω u2 t dx+ ∫ Ω u∆u dx− ∫ Ω u ∫ t 0 g(t− s)∆u(s) ds dx− a ∫ Ω u|ut|m(x)−2ut dx = ∫ Ω u2 t dx− ( 1− ∫ t 0 g(s)ds )∫ Ω |∇u|2 dx + ∫ Ω ∇u · ∫ t 0 g(t− s)(∇u(s)−∇u(t)) ds dx− a ∫ Ω |ut|m(x)−2utu dx ≤ ∫ Ω u2 t dx− l ∫ Ω |∇u|2 dx+ 1 l ∫ Ω (∫ t 0 g(t− s)|∇u(s)−∇u(t)|ds )2 dx 6 M. I. MUSTAFA EJDE-2023/53 + l 4 ∫ Ω |∇u|2 dx+ a ∫ Ω∗ |ut|m(x)−1|u| dx+ a ∫ Ω∗∗ |ut|m(x)−1|u| dx. Using Cauchy-Schwarz inequality, we have∫ Ω (∫ t 0 g(t− s)|∇u(s)−∇u(t)| ds )2 dx = ∫ Ω (∫ t 0 g(t− s)√ αg(t− s)− g′(t− s) √ αg(t− s)− g′(t− s)|∇u(s)−∇u(t)|ds )2 dx ≤ (∫ t 0 g2(s) αg(s)− g′(s) ds )∫ Ω ∫ t 0 [ αg(t− s)− g′(t− s) ] |∇u(s)−∇u(t)|2 ds dx ≤ Cα(h ◦ ∇u)(t). (3.5) Now, using Young’s and Poincaré’s inequalities, a ∫ Ω∗ |ut|m(x)−1|u| dx ≤ ∫ Ω∗ [ δ0|u|2 + a2 4δ0 |ut|2m(x)−2 ] dx ≤ cδ0‖∇u‖22 + a2 4δ0 ∫ Ω∗ |ut|2m(x)−2 dx. On the other hand, If meas (Ω∗∗) 6= 0 so m2 ≥ 2, we use Young’s inequality with p(x) = m(x) m(x)−1 and p′(x) = m(x) for x ∈ Ω∗∗ and the Sobolev embedding H1 0 (Ω) ↪→ Lm2(Ω) to obtain a ∫ Ω∗∗ |ut|m(x)−1|u| dx ≤ a ∫ Ω∗∗ [ ε|u|m(x) + Cε(x)|ut|m(x) ] dx ≤ aε ∫ Ω ( |u|2 + |u|m2 ) dx+ a ∫ Ω∗∗ Cε(x)|ut|m(x) dx ≤ cε ( ‖∇u‖22 + ‖∇u‖m2 2 ) + a ∫ Ω∗∗ Cε(x)|ut|m(x) dx ≤ cε ( 1 + ‖∇u‖m2−2 2 ) ‖∇u‖22 + a ∫ Ω∗∗ Cε(x)|ut|m(x) dx ≤ cε ( 1 + E(0) m2−2 2 ) ‖∇u‖22 + a ∫ Ω∗∗ Cε(x)|ut|m(x) dx. If we fix ε = l 4c ( 1+E(0) m2−2 2 ) , then Cε(x) = m(x)− 1 [m(x)] m(x) m(x)−1 ε 1 m(x)−1 is bounded since m(x) is bounded, and we obtain a ∫ Ω∗∗ |ut|m(x)−1|u| dx ≤ l 4 ‖∇u‖22 + c ∫ Ω |ut|m(x) dx. Combining all the above estimates, with δ0 small enough, gives (3.3). � Lemma 3.3. The functional K2 defined by K2(t) := − ∫ Ω ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx (3.6) EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 7 satisfies, for any 0 < δ < 1, the estimate K ′2(t) ≤ − (∫ 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. By using (1.1) and integrating by parts, we have K ′2(t) = ∫ Ω ∇u · ∫ t 0 g(t− s)(∇u(t)−∇u(s)) ds dx − ∫ Ω (∫ t 0 g(t− s)∇u(s)ds )(∫ t 0 g(t− s)(∇u(t)−∇u(s))ds ) dx − ∫ Ω ut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx− (∫ t 0 g(s)ds )∫ Ω u2 t dx + a ∫ Ω |ut|m(x)−2ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx = ( 1− ∫ t 0 g(s)ds )∫ Ω ∇u · ∫ t 0 g(t− s)(∇u(t)−∇u(s)) ds dx + ∫ Ω (∫ t 0 g(t− s)(∇u(t)−∇u(s))ds )2 dx − ∫ Ω ut ∫ t 0 g′(t− τ)(u(t)− u(τ))dτ dx− (∫ t 0 g(s) ds )∫ Ω u2 t dx + a ∫ Ω∗ |ut|m(x)−2ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx + a ∫ Ω∗∗ |ut|m(x)−2ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx Using Young’s and Poincaré’s inequalities and similar calculations as in (3.5), we obtain ( 1− ∫ t 0 g(s)ds )∫ Ω ∇u · ∫ t 0 g(t− s)(∇u(t)−∇u(s)) ds dx ≤ δ ∫ Ω |∇u|2 dx+ c δ Cα(h ◦ ∇u)(t), − ∫ Ω ut ∫ t 0 g′(t− s)(u(t)− u(s)) ds dx = ∫ Ω ut ∫ t 0 h(t− s)(u(t)− u(s)) ds dx− ∫ Ω ut ∫ t 0 αg(t− s)(u(t)− u(s)) ds dx ≤ δ ∫ Ω u2 t dx+ 1 2δ ∫ Ω (∫ t 0 √ h(t− s) √ h(t− s)|u(s)− u(t)|ds )2 dx + α2 2δ ∫ Ω (∫ t 0 g(t− s)|u(s)− u(t)|ds )2 dx ≤ δ ∫ Ω u2 t dx+ ( ∫ t 0 h(s) ds ) 2δ (h ◦ u)(t) + α2Cα 2δ (h ◦ u)(t) 8 M. I. MUSTAFA EJDE-2023/53 ≤ δ ∫ Ω u2 t dx+ c δ (h ◦ ∇u)(t) + cCα δ (h ◦ ∇u)(t), and a ∫ Ω∗ |ut|m(x)−2ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx ≤ ∫ Ω∗ (∫ t 0 g(t− s)(u(t)− u(s)) ds )2 dx+ a2 4 ∫ Ω∗ |ut|2m(x)−2 dx ≤ cCα(h ◦ ∇u)(t) + a2 4 ∫ Ω∗ |ut|2m(x)−2 dx . On Ω∗∗, we use a similar argument as in Lemma 3.2 to obtain a ∫ Ω∗∗ |ut|m(x)−2ut ∫ t 0 g(t− s)(u(t)− u(s)) ds dx ≤ aδ ∫ Ω∗∗ ∣∣ ∫ t 0 g m(x)−1 m(x) (t− s)g 1 m(x) (t− s)(u(t)− u(s))ds ∣∣m(x) dx + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx ≤ aδ ∫ Ω∗∗ (∫ t 0 g(s)ds )m(x)−1(∫ t 0 g(t− s)|u(t)− u(s)|m(x)ds ) dx + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx ≤ cδ ∫ Ω ∫ t 0 g(t− s) ( |u(t)− u(s)|2 + |u(t)− u(s)|m2 ) ds dx + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx ≤ cδ ∫ t 0 g(t− s) ( ‖∇u(t)−∇u(s)‖22 + ‖∇u(t)−∇u(s)‖m2 2 ) ds + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx ≤ cδ ∫ t 0 g(t− s) ( 1 + ‖∇u(t)−∇u(s)‖m2−2 2 ) ‖∇u(t)−∇u(s)‖22 ds + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx ≤ cδ ( 1 + E(0) m2−2 2 ) (g ◦ ∇u)(t) + a ∫ Ω∗∗ Cδ(x)|ut|m(x) dx. Combining the above estimates, (3.7) is established. � Next, we use the functional K3(t) = ∫ Ω ∫ t 0 f(t− s)|∇u(s)|2 ds dx (3.8) where f(t) = ∫∞ t g(s)ds. EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 9 Lemma 3.4. The functional K3 satisfies, along the solution of (1.1), the estimate K ′3(t) ≤ −1 2 (g ◦ ∇u)(t) + 3(1− l) ∫ Ω |∇u(t)|2 dx. (3.9) Proof. By Young’s inequality and the fact that f ′(t) = −g(t), we see that K ′3(t) = f(0) ∫ Ω |∇u(t)|2 dx− ∫ Ω ∫ t 0 g(t− s)|∇u(s)|2 ds dx = − ∫ Ω ∫ t 0 g(t− s)|∇u(s)−∇u(t)|2 ds dx − 2 ∫ Ω ∇u(t) · ∫ t 0 g(t− s)(∇u(s)−∇u(t)) ds dx+ f(t) ∫ Ω |∇u(t)|2 dx. But − 2 ∫ Ω ∇u(t) · ∫ t 0 g(t− s)(∇u(s)−∇u(t)) ds dx ≤ 2(1− l) ∫ Ω |∇u(t)|2 dx+ ∫ t 0 g(s)ds 2(1− l) ∫ Ω ∫ t 0 g(t− s)|∇u(s)−∇u(t)|2 ds dx. Then, as f(t) ≤ f(0) = (1− l) and ∫ t 0 g(s)ds ≤ (1− l), we obtain (3.9). � Lemma 3.5. The functional L defined by L(t) := NE(t) +N1K1(t) +N2K2(t) for suitable choice of N,N1, N2 > 0 and for all t ≥ t1, satisfies L′(t) ≤ −4(1−l) ∫ Ω |∇u|2 dx− ∫ Ω u2 t dx+ 1 4 (g◦∇u)(t)+c ∫ Ω∗ |ut|2m(x)−2 dx (3.10) and L(t) ∼ E(t) (3.11) which means that, for some constants a1, a2 > 0, a1E(t) ≤ L(t) ≤ a2E(t). Proof. Let g1 = ∫ t1 0 g(s)ds > 0 for some fixed t1 > 0. By combining (3.1), (3.3), (3.7), recalling that g′ = (αg − h), and taking δ = 1/(8cN2), we obtain that for all t ≥ t1, L′(t) ≤ − ( l 4 N1 − 1 8c ) ∫ Ω |∇u|2 dx− ( g1N2 − 1 8c −N1 ) ∫ Ω u2 t dx + (α 2 N + 1 8 ) (g ◦ ∇u)(t)− (1 2 N − 8c2N2 2 − Cα[ 1 l N1 + 8c2N2 2 ] ) (h ◦ ∇u)(t) − ∫ Ω (aN − cN1 − aCδ(x)N2) |ut|m(x) dx+ ( cN1 + a2 4 N2 ) ∫ Ω∗ |ut|2m(x)−2 dx Now we choose N1 large enough so that l 4 N1 − 1 8c > 4(1− l), 10 M. I. MUSTAFA EJDE-2023/53 and N2 large enough so that g1N2 − 1 8c −N1 > 1. As δ now is fixed and Cδ(x) is bounded, we have − (aN − cN1 − aCδ(x)N2) ≤ − (aN − cN1 − cN2) . Next, as αg2(s) αg(s)−g′(s) < g(s), it is easy to show, using the Lebesgue dominated convergence theorem, that αCα = ∫ ∞ 0 αg2(s) αg(s)− g′(s) ds −→ 0 as α −→ 0. Hence, there is 0 < α0 < 1 such that if α < α0, then αCα < 1 16[ 1 lN1 + 8c2N2 2 ] . Let us choose N large enough and choose α satisfying aN − cN1 − cN2 > 0, 1 4 N − 8c2N2 2 > 0, α = 1 4N < α0 which implies 1 2 N − 8c2N2 2 − Cα[ 1 2l N1 + 8c2N2 2 ] > 0. So, we arrive at L′(t) ≤ −4(1− l) ∫ Ω |∇u|2 dx− ∫ Ω u2 t dx+ 1 4 (g ◦ ∇u)(t) + c ∫ Ω∗ |ut|2m(x)−2 dx. On the other hand, we find that |L(t)−NE(t)| ≤ N1|K1(t)|+N2|K2(t)| ≤ N1 ∫ Ω |uut| dx+N2 ∫ Ω ∣∣ut ∫ t 0 g(t− s)(u(t)− u(s))ds ∣∣ dx ≤ N1 2 ∫ Ω u2 dx+ N1 +N2 2 ∫ Ω u2 t dx+ N2 2 ∫ Ω ∣∣ ∫ t 0 g(t− s)(u(t)− u(s))ds ∣∣2 dx ≤ c [ ∫ Ω |∇u|2 dx+ ∫ Ω u2 t dx+ (g ◦ ∇u)(t) ] ≤ cE(t). Therefore, we can choose N even larger (if needed) so that (3.11) is satisfied. � 4. Main result Theorem 4.1. Assume that (A1) holds. Then there exist positive constants ε0 ≤ r, k1 ≤ 1, and k2 such that the energy functional satisfies E(t) ≤ k2H −1 0 ( k1 ∫ t 0 ξ(s)ds ) (4.1) EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 11 where H0(t) =  ∫ 1 t 1 sH′1(ε0s) ds, if m1 ≥ 2∫ 1 t 1 sH′2(ε0s) ds, if 1 < m1 < 2 and H1 is linear∫ 1 t 1 sH′(ε0s) ds, if 4 3 < m1 < 2 and H1 is nonlinear∫ 1 t 1 sH′(ε0s m1 2m1−2 ) ds, if 1 < m1 ≤ 4 3 and H1 is nonlinear. Here, H0 is strictly decreasing and convex on (0, r], with limt→0H0(t) = +∞. Proof. We start by estimating the last term in (3.10). When m1 ≥ 2, we have that meas (Ω∗) = 0 =⇒ ∫ Ω∗ |ut|2m(x)−2 dx = 0. But, if 1 < m1 < 2, with Ω1 = {x ∈ Ω∗ : |ut| ≤ 1}, Ω2 = Ω∗\Ω1 and 2m(x)−2 m(x) ≥ 2m1−2 m1 , we use Jensen’s inequality to obtain∫ Ω∗ |ut|2m(x)−2 dx = ∫ Ω1 |ut|2m(x)−2 dx+ ∫ Ω2 |ut|2m(x)−2 dx = ∫ Ω1 [ |ut|m(x) ] 2m(x)−2 m(x) dx+ ∫ Ω2 |ut|m(x)+m(x)−2 dx ≤ ∫ Ω1 [ |ut|m(x) ] 2m1−2 m1 dx+ ∫ Ω2 |ut|m(x) dx ≤ [ ∫ Ω1 |ut|m(x) dx ] 2m1−2 m1 + ∫ Ω2 |ut|m(x) dx ≤ c[−E′(t)] 2m1−2 m1 − cE′(t) = cH−1 2 (−E′(t))− cE′(t) (4.2) and use Young’s inequality to obtain E(t) 2−m1 2m1−2 [−E′(t)] 2m1−2 m1 E(t) 2−m1 2m1−2 ≤ εE(t) m1 2m1−2 − CεE′(t) E(t) 2−m1 2m1−2 = εE(t)− CεE ′(t) E(t) 2−m1 2m1−2 . With ε = b0 2c and b0 = min{2(1− l), 1/2}, this yields c ∫ Ω∗ |ut|2m(x)−2 dx ≤ b0 2 E(t)− cE′(t) E(t) 2−m1 2m1−2 − cE′(t). (4.3) Next, we prove that ∫ ∞ 0 E(s)ds <∞, if m1 > 4 3 ,∫ ∞ 0 E(s) m1 2m1−2 ds <∞, if 1 < m1 ≤ 4 3 (4.4) 12 M. I. MUSTAFA EJDE-2023/53 For this purpose, we use Lemmas 3.4 and 3.5 to deduce that L(t) = L(t) + K3(t) is nonnegative and it satisfies, for all t ≥ t1, L′(t) ≤ − [ (1− l) ∫ Ω |∇u|2 dx+ ∫ Ω u2 t dx+ 1 4 (g ◦ ∇u) ] + c ∫ Ω∗ |ut|2m(x)−2 dx ≤ −b0E(t) + c ∫ Ω∗ |ut|2m(x)−2 dx. (4.5) If m1 ≥ 2 then meas(Ω∗) = 0, so L′(t) ≤ −b0E(t) implies b0 ∫ t 0 E(s)ds ≤ − ∫ t 0 L′(s)ds ≤ L(0)− L(t) ≤ L(0). If 1 < m1 < 2, then, using (4.3), (4.5) becomes L′(t) ≤ −b0 2 E(t)− cE′(t) E(t) 2−m1 2m1−2 − cE′(t). Here, when 4/3 < m1 < 2, we notice that∫ t 0 −cE′(s) E(t) 2−m1 2m1−2 ds = c(2m1 − 2) 3m1 − 4 [ E(0) 3m1−4 2m1−2 − E(t) 3m1−4 2m1−2 ] ≤ c(2m1 − 2) 3m1 − 4 E(0) 3m1−4 2m1−2 = d0 implies b0 2 ∫ t 0 E(s)ds ≤ − ∫ t 0 [ L′(s) + cE′(t) + cE′(s) E(t) 2−m1 2m1−2 ] ds ≤ L(0) + cE(0) + d0 which gives (4.4)1. Otherwise, E(t) 2−m1 2m1−2 [L′(t) + cE′(t)] ≤ −b0 2 E(t) m1 2m1−2 − cE′(t). This means, as E(t) is decreasing, that L0(t) = E(t) 2−m1 2m1−2 [L(t) + cE(t)] + cE(t) is nonnegative and L′0(t) ≤ −b0 2 E(t) m1 2m1−2 , for all t ≥ t1 which now gives (4.4)2. To this end, we multiply (3.10) by ξ(t) and use (4.2), the fact that ξ is non- increasing and the functional F := ξL + cE satisfies F ∼ E and deduce, for some constant m > 0 and for all t ≥ t1, F ′(t) ≤ { −mξ(t)E(t) + cξ(t)(g ◦ ∇u)(t), if m1 ≥ 2 −mξ(t)E(t) + cξ(t)(g ◦ ∇u)(t) + cξ(t)H−1(−E′(t)), if 1 < m1 < 2. (4.6) Case 1: H1 is nonlinear on [0, r] and 1 < m1 ≤ 4/3. First, we define I(t) := ∫ t 0 ∫ Ω |∇u(t)−∇u(t− s)|2 dx ds EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 13 and consider H to be an extension of H such that H is a strictly increasing and strictly convex 2 function on (0,∞) [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) ) . (4.7) Inserting (4.7) into (4.6)2, defining F0(t) := H ′( ε0[ E(t) E(0) ] m1 2m1−2 ) F (t), with ε0 < r, and using the fact that E′ ≤ 0, H ′ > 0, and H ′′ > 0, give F ′0(t) ≤ H ′ ( ε0 [E(t) E(0) ] m1 2m1−2 ) F ′(t) ≤ −mξ(t)E(t)H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) + cξ(t)I(t)H −1 (−2E′(t) I(t)ξ(t) ) H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) + cξ(t)H−1(−E′(t))H ′ ( ε0 [E(t) E(0) ] m1 2m1−2 ) . (4.8) If A = H ′( ε0[E(t) E(0) ] m1 2m1−2 ) and one time B = H −1(−2E′(t) I(t)ξ(t) ) and another time B = H −1 (−E′(t)) are used in the generalized Young inequality (2.3), we obtain F ′0(t) ≤ −mξ(t)E(t)H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) + cε0ξ(t)I(t) [E(t) E(0) ] m1 2m1−2H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) − cE′(t) + cε0ξ(t) [E(t) E(0) ] m1 2m1−2H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) − cξ(t)E′(t). Using that E(t) 2−m1 2m1−2 I(t) is uniformly bounded by some constant C because of (4.4)2 and H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) = H ′ ( ε0[E(t) E(0) ] m1 2m1−2 ) , the choice of ε0, yields F ′0(t) ≤ −mξ(t)E(t)H ′ ( ε0 [E(t) E(0) ] m1 2m1−2 ) + cε0ξ(t) E(t) E(0) H ′ ( ε0 [E(t) E(0) ] m1 2m1−2 ) − cE′(t). 14 M. I. MUSTAFA EJDE-2023/53 Consequently, with F1 = F0 + cE, which for some α1, α2 > 0 satisfies α1F1(t) ≤ E(t) ≤ α2F1(t), (4.9) and with a suitable choice of ε0, we obtain, for some constant k > 0 and for all t ≥ t1, F ′1(t) ≤ −kξ(t) (E(t) E(0) ) H ′ ( ε0 [E(t) E(0) ] m1 2m1−2 ) = −kξ(t)H3 (E(t) E(0) ) , (4.10) where H3(t) = tH ′ ( ε0t m1 2m1−2 ) . Using the strict convexity of H on (0, r], we find that H3(t), H ′3(t) > 0, and H3(t) ≤ tH ′(r) on (0, 1]. Thus, with R(t) = α1F1(t) E(0) , taking in account (4.9) and (4.10), we have R(t) ∼ E(t) (4.11) and, for some k1 > 0, R′(t) ≤ −k1ξ(t)H3(R(t)), ∀t ≥ t1. Then, the integration over (t1, t) yields, for some k2 > 0,∫ t t1 −R′(s) H3(R(s)) ds ≥ k1 ∫ t t1 ξ(s)ds =⇒ ∫ R(t1) R(t) 1 H3(s) ds ≥ k1 ∫ t t1 ξ(s)ds =⇒ R(t) ≤ H−1 0 (k1 ∫ t t1 ξ(s)ds) =⇒ by (4.11) E(t) ≤ k2H −1 0 (k1 ∫ t t1 ξ(s)ds), (4.12) where H0(t) = ∫ 1 t 1 H3(s)ds. Here, we have used, based on the properties of H3, the fact that H0 is a strictly decreasing function on (0, 1] and limt→0H0(t) = +∞. Also, it is easy to notice that we can start the integration inside at zero where if k1 < k1 is chosen so that k1 ∫ 2t1 0 ξ(s)ds = k1 ∫ 2t1 t1 ξ(s)ds, then, as H−1 0 is decreasing, E(t) ≤ k2H −1 0 (k1 ∫ t t1 ξ(s)ds) ≤ k2H −1 0 (k1 ∫ t 0 ξ(s)ds), ∀t ≥ 2t1 (4.13) and so (4.1)4 is established. Case 2: H1 is linear on [0, r] or m1 > 4/3. If H1 is linear then ξ(t) ∫ t 0 g(s) ∫ Ω |∇u(t)−∇u(t− s)|2 dx ds ≤ −cE′(t) or if m1 > 4/3 then I(t), used above, is itself uniformly bounded by some constant C because of (4.4)1. This enables us to repeat exactly the same steps starting at (4.8), but with H ′( ε0 E(t) E(0) ) instead of H ′( ε0 [E(t) E(0) ] m1 2m1−2 ) , and with H3(t) = tH ′(ε0t), and similarly obtain (4.13), and so (4.1)1,2,3. � EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 15 Applications. Here, we give applications of our result to some concrete examples. If assumption (A1) is satisfied with H1(t) = tp, 1 ≤ p < 2 and H2(t) = t m1 2m1−2 , then H(t) = min{H1(t), H2(t)} = tq on the interval (0, 1] where q = max{p, m1 2m1−2}. Hence, (4.1) and simple calculations lead to E(t) ≤  ke−k1 ∫ t 0 ξ(s)ds, if m1 ≥ 2 and p = 1 k2 ( 1 + ∫ t 0 ξ(s)ds )− 2m1−2 2−m1 , if 1 < m1 < 2 and p = 1 k3 ( 1 + ∫ t 0 ξ(s)ds ) −1 q−1 , if m1 > 4 3 and 1 < p < 2 k4 ( 1 + ∫ t 0 ξ(s)ds )− 2m1−2 m1(q−1) , if 1 < m1 ≤ 4 3 and 1 < p < 2. (4.14) • If g(t) = a (1+t)v , for v > 1, then g′(t) = −ξ(t)H1(g(t)) where H1(t) = tp, with p = v+1 v , and ξ(t) ≡ constant. By (4.14)3,4, E(t) ≤ { k3(1 + t) −1 q−1 , if m1 > 4 3 k4(1 + t) − 2m1−2 m1(q−1) , if 1 < m1 ≤ 4 3 . • If g(t) = a (t+e)[ln(t+e)]v , for v > 1, then g′(t) = −ξ(t)H1 ( g(t) ) where H1(t) = tp, with p = v+1 v , and ξ(t) = [ln(t+e)+v] a 1 v (t+e)1− 1 v . By (4.14)3,4, E(t) ≤ k ( 1 + ∫ t 0 ln(s+ e) + v a 1 v (s+ e)1− 1 v ds )−q0 ≤ for large t k ( (t+ e) 1 v ln(t+ e) )−q0 where q0 = { 1 q−1 , if m1 > 4 3 2m1−2 m1(q−1) , if 1 < m1 ≤ 4 3 . • If g(t) = a exp(−tv), for 0 < v ≤ 1, then g′(t) = −ξ(t)H1(g(t)) where H1(t) = t and ξ(t) = vtv−1. By (4.14)1,2, E(t) ≤ { ke−k1t v , if m1 ≥ 2 k2(1 + tv)− 2m1−2 2−m1 , if 1 < m1 < 2. (4.15) But, for 0 < v < 1, g′ can also be written as g′(t) = −H1(g(t)) where H1(t) = vt [ln(a/t)] 1 v −1 satisfies (A1) on the interval (0, r1] for any 0 < r1 < a. If 1 < m1 < 2, then m1 2m1−2 > 1 and H2(t) = t m1 2m1−2 is strictly convex and one can easily discover that H(t) = min{H1(t), H2(t)} = H2(t) near the origin. By Theorem 4.1, we obtain E(t) ≤  ke−k1t v , if m1 ≥ 2 k2(1 + t)− 2m1−2 2−m1 , if 4 3 < m1 < 2 k3(1 + t) − (2m1−2)2 m1(2−m1) , if 1 < m1 ≤ 4 3 which gives better rates than (4.15) in the case 4/3 < m1 < 2. Conclusions. In this paper, the important issue of stabilization of the wave equa- tion was addressed. The main contribution of this work is studying the competition between two different types of dissipative mechanisms and establishing, by carefully tailored techniques, explicit formulae for the energy decay rates with very general assumptions on the relaxation function and with variable exponent of the feedback, 16 M. I. MUSTAFA EJDE-2023/53 which is more useful from the physical point of view and needed in several ap- plications. Our results combine the generality and optimality and improve earlier related results in the literature. We provided some numerical examples of expo- nential, polynomial or logarithmic energy decay estimates and all the rates in these examples are faster than the rates obtained in [14]. Our paper opens the door for further research and new suggestions that may be addressed in the future, for instance the control of the system by such types of damping but located on the boundary. Acknowledgments. This work was supported bythe MASEP Research Group in the Research Institute of Sciences and Engineering at University of Sharjah. References [1] F. Alabau-Boussouira, P. Cannarsa; A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris, Ser. I 347(2009), 867–872. [2] S. Antontsev; Wave equation with p(x, t)-Laplacian and damping term: blow-up of solutions, C. R.Mec. 339(12) (2011), 751–755. [3] S. Antontsev; Wave equation with p(x, t)-Laplacian and damping term: existence and blow- up, Differ. Equ. Appl. 3(4) (2011), 503–525. [4] S. Antontsev, S. Shmarev; Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math. 234(9) (2010), 2633–2645. [5] S. Antontsev, V. Zhikov; Higher integrability for parabolic equations of p(x, t)-Laplacian type, Adv. Differ. Equ. 10(9) (2005), 1053–1080. [6] V. I. Arnold; Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989. [7] F. Belhannache, M. M. Algharabli, S. A. Messaoudi; Asymptotic stability for a viscoelas- tic equation with nonlinear damping and very general type of relaxation functions, J. Dyn. Control Syst. (2019), 1-23. [8] A. Benaissa, S. Mimouni; Energy decay of solutions of a wave equation of p-Laplacian type with a weakly nonlinear dissipation, JIPM.J. Inequal. Pure Appl. Math. 7(1) (2006), Article 15, pp 8. [9] M. M. Cavalcanti, H. P. Oquendo; Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42(4) (2003), 1310-1324. [10] M. Fabrizio, S. Polidoro; Asymptotic decay for some differential systems with fading memory, Appl. Anal. 81(6) (2002), 1245-1264. [11] X. Fan, D. Zhao; On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263(2) (2001), 424–446. [12] Y. Gao, W. Gao; Existence of weak solutions for viscoelastic hyperbolic equations with vari- able exponents, Boundary Value Problems 2013 no. 1 (2013), 1-8. [13] B. Guo, W. Gao; Blow-up of solutions to quasilinear hyperbolic equations with p(x, t)- Laplacian and positive initial energy, C. R. Mec. 342(9) (2014), 513–519. [14] J. H. Hassan, S. A. Messaoudi; General decay results for a viscoelastic wave equation with a variable exponent nonlinearity, Asymptot. Anal. 125 (2021), 365–388. [15] A. Haraux, E. Zuazua; Decay estimates for some semilinear damped hyperbolic problems, Arch. Rational Mech. Anal. (1988), 191-206. [16] W. J. Hrusa; Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM Journal of Math. Anal. 16(1) (1985), 110-134. [17] V. Komornik; Decay estimates for the wave equation with internal damping, International Series of Numerical Mathematics 118 (1994), 253-266. [18] D. Lars, P. Harjulehto, P. Hasto, M. Ruzicka; Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Vol. 2017, 2017. [19] S. A. Messaoudi; General decay of solutions of a viscoelastic equation, J. Math. Anal. Appl. 341 (2008), 1457-1467. [20] S. A. Messaoudi; General stability in viscoelasticity, in: Viscoelastic Viscoplastic Materials, InTech (2016), DOI: 10.5772/64217. [21] S. A. Messaoudi, A. A. Talahmeh; A blow-up result for a quasilinear wave equation with variable-exponent nonlinearities, Math. Methods Appl. Sci. 40 (2017), 6976-6986. EJDE-2023/53 ENERGY DECAY FOR VISCOELASTIC WAVE EQUATIONS 17 [22] S. A. Messaoudi, A. A. Talahmeh; On wave equation: review and recent results, Arab. J. Math. 7(2) (2018), 113–145. [23] S. A. Messaoudi, A. A. Talahmeh, J. H. Al-Smail; Nonlinear damped wave equation: existence and blow-up, Comput.Math. Appl. 74 (2017), 3024-3041. [24] J. E. Munoz Rivera; Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math. 52(4) (1994), 628-648. [25] M. I. Mustafa; Energy estimates to the Cauchy problem of a weakly damped Klein-Gordon equation with variable-exponent nonlinearity, Math. Methods Appl. Sci. 44 (11) (2021), 8999- 9011. [26] M. I. Mustafa; Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci. 41 (1) (2018), 192–204. [27] M. I. Mustafa; Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems, Z. Angew. Math. Phys. 72 (2), 67 (2021), 1-15. [28] M. I. Mustafa, S. A. Messaoudi, M. Zahri; Theoretical and computational results of a wave equation with variable exponent and time dependent nonlinear damping, Arab. J. Math. 10 (2) (2021), 443–458. [29] M. Nakao; Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann. 305 (1996), 403-417. [30] S.-H. Park, J.-R. Kang; Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Methods Appl. Sci. 42 (2019), 2083–2097. [31] M. L. Santos; Asymptotic behavior of solutions to wave equations with a memory conditions at the boundary, Electron. J. Differ. Equ. 2001(73) (2001), 1-11. [32] L. Sun, Y. Ren, W. Gao; Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources, Comput. Math. Appl. 71(1) (2016), 267–277. [33] E. Zuazua; Exponential decay for the semilinear wave equation with locally distributed damp- ing, Comm. Partial Differential Equations 15 (1990), 205-235. Muhammad I. Mustafa Department of Mathematics, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates Email address: mmustafa@sharjah.ac.ae 1. Introduction 2. Preliminaries Remarks 3. Technical Lemmas 4. Main result Applications Conclusions Acknowledgments References