Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 42, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY OF ENERGY FOR VISCOELASTIC WAVE EQUATIONS WITH BALAKRISHNAN-TAYLOR DAMPING AND MEMORIES FEI WANG, JIANGHAO HAO Abstract. In this article, we consider a viscoelastic wave equation with Balakrishnan- Taylor damping, and finite and infinite memory terms in a bounded domain. Under suitable assumptions on relaxation functions and with certain initial data, by adopting the perturbed energy method, we establish a decay of en- ergy which depends on the behavior of the relaxation functions. 1. Introduction In this article, we study the following viscoelastic problem with Balakrishnan- Taylor damping, a nonlinear source term and finite and infinite memories: |ut|ρutt −M(t)∆u−∆utt −∆ut + ∫ t 0 g1(t− s) div(a1(x)∇u(s))ds + ∫ ∞ 0 g2(s) div(a2(x)∇u(t− s))ds+ γ(t)h(ut) = |u|p−1u, in Ω× (0,∞), u(x, t) = 0, on ∂Ω× (0,∞), u(x,−t) = u0(x, t), in Ω× (0,∞), ut(x, 0) = u1(x), in Ω, (1.1) where M(t) = a+ b‖∇u‖22 + σ ∫ Ω ∇u · ∇ut dx, a, b, σ are positive constants, Ω is a bounded domain of Rn (n ≥ 1) with smooth boundary ∂Ω, g1 and g2 are positive non-increasing functions defined on R+, a1(x) and a2(x) are essentially bounded non-negative functions, h is a non-decreasing function, p and ρ satisfy 1 < p ≤ n n− 2 , for n ≥ 3, 1 ≤ p <∞, for n = 1, 2, 0 < ρ ≤ 2 n− 2 , for n ≥ 3, 0 ≤ ρ <∞, for n = 1, 2. (1.2) From the physical point of view, equation (1.1) is related to the panel flutter equation and spillover problem with memories and damping. The case of σ = 0, 2010 Mathematics Subject Classification. 35B35, 93D20. Key words and phrases. Viscoelasticity; Balakrishnan-Taylor damping; decay; memory. c©2020 Texas State University. Submitted December 23, 2019. Published May 7, 2020. 1 2 F. WANG, J. HAO EJDE-2020/42 in the absence of the Balakrishnan-Taylor damping, equation (1.1) can be used to describe the motion of viscoelastic materials. It is well known that viscoelastic materials have a wide application in science and engineering because they have the capacity of storage and dissipation of mechanical energy, which is modeled by the convolution terms (as in (1.1)). Many authors have considered the behavior of the partial differential equations (PDEs) with convolution term, see for example [5, 14, 7, 16, 8, 4, 18] and references therein. Guesmia and Messaoudi [7] discussed the problem utt −∆u+ ∫ t 0 g1(t− s) div(a1(x)∇u(s))ds + ∫ ∞ 0 g2(s) div(a2(x)∇u(t− s))ds = 0, in Ω× (0,∞), u(x, t) = 0, on ∂Ω× (0,∞), u(x,−t) = u0(x, t), in Ω× (0,∞), ut(x, 0) = u1(x), in Ω. (1.3) Under suitable conditions on a1, a2 and for a wide class of relaxation functions g1 and g2, they established a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Guesmia and Messaoudi [8] were concerned with the long-time behavior of the solution of the Timoshenko system ρ1ϕtt − k1(ϕx + ψ)x + b(x)h(ϕt) + ∫ ∞ 0 g(s)(a(x)ϕx(t− s))xds = 0, in (0, L)× (0,∞), ρ2ψtt − k2ψxx + k1(ϕx + ψ) = 0, in (0, L)× (0,∞), ϕ(0, t) = ψx(0, t) = ϕ(L, t) = ψx(L, t) = 0, in (0,∞), ϕ(x,−t) = ϕ0(t), ϕt(x, 0) = ϕ1(x), in (0, L)× (0,∞), ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), in (0, L). (1.4) They showed that the dissipation generated by these two complementary controls guarantees the stability of the system in case of the equal-speed propagation as well as in the opposite case. Mustafa [18] studied the following equation with the Dirichlet boundary condition utt−∆u+ ∫ t 0 div[a(x)g(t−τ)∇u(τ)]dτ +η(t)b(x)h(ut) = 0, in Ω× (0,∞), (1.5) and established an explicit and general decay rate result, using some properties of convex functionals. The model in hand, with the Balakrishnan-Taylor damping (σ 6= 0) and in the absence of ∆utt, the strong damping ∆ut and ρ = g1 = g2 = h = 0, was proposed by Balakrishnan and Taylor [1], and Bass and Zes [2]. The model is used to solve the overflow problem, that is, to set up an appropriate feedback control function, which consists of a limited number of modes, to achieve a high performance of the closed-loop systems. So far, there are many stability results for the problem having the Balakrishnan-Taylor damping and memory term see for example [3, 9, 11, 17, 23, 24]. Mu and Ma [17] considered the wave equations with EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 3 Balakrishnan-Taylor memory terms and source terms: utt − (a+ b‖∇u‖2 + σ ∫ Ω ∇u · ∇ut dx) + ∫ t 0 g1(t− s)∆u(s)ds = f1(u, v), t > 0, x ∈ Ω, vtt − (a+ b‖∇v‖2 + σ ∫ Ω ∇v · ∇vt dx) + ∫ t 0 g2(t− s)∆v(s)ds = f2(u, v), t > 0, x ∈ Ω, u(x, t) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ Ω, u(x, t) = v(x, t) = 0, (x, t) ∈ Γ× [0,∞), (1.6) and proved that for a certain class of relaxation functions and certain initial data, the decay rate of the solution energy is similar to that of relaxation functionals which is not necessarily of exponential or polynomial type. In addition, they considered problem (1.6) with the added terms ∆2u+ ∆2ut and ∆2v + ∆2vt, namely, utt − (a+ b‖∇u‖2 + σ ∫ Ω ∇u∇ut dx) + ∆2u+ ∆2ut + ∫ t 0 g1(t− s)∆u(s)ds = f1(u, v), t > 0, x ∈ Ω, vtt − (a+ b‖∇v‖2 + σ ∫ Ω ∇v∇vt dx) + ∆2v + ∆2vt + ∫ t 0 g2(t− s)∆v(s)ds = f2(u, v), t > 0, x ∈ Ω, u(x, t) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, v(x, 0) = v0(x), vt(x, 0) = v1(x), x ∈ Ω, u(x, t) = v(x, t) = 0, (x, t) ∈ Γ× [0,∞) . (1.7) They established the blow-up of the solution for (1.7) when relaxation functionals and initial data satisfy some conditions even in presence of strong damping. There are some methods developed to analyze the stability of the PDEs such as Lyapunov’s energy method (see [10, 19]), Riesz basis approach (see [21, 13]), frequency multiplier technique (see [15, 20]), Carleman estimates (see [6]) and so on. Motivated by [7, 8, 18], we consider (1.1). Our major contributions in this article are the following: (1) We put together several useful models of viscoelasticity: dispersion term |ut|ρ, the Balakrishna-Taylor damping ∫ Ω ∇u∇ut dx, strong damping term ∆ut, infinite and finite time history memories ∫∞ 0 g2(s) div(a2(x)∇u(t− s))ds, ∫ t 0 g1(t− s) div(a1(x)∇u(s))ds, and a nonlinear source term |u|p−1u. (2) Give the model in (1.1) which is a unified treatment. (3) Our technical treatment offers, as far as we know, the sharpest assump- tions/conditions for the study of the viscoelastic wave equation. The rest of this article is organized as follows. In section 2, we present preliminary material needed for our work. In section 3, we prove the global existence and the uniform decay of energy. 4 F. WANG, J. HAO EJDE-2020/42 2. Preliminaries In this section, we present some materials needed for our main results. Through- out this article, we use the following assumptions and notation. We shall write ‖ · ‖ and ‖ · ‖p to denote the usual L2(Ω) norm and Lp(Ω) norm respectively, (·, ·) de- notes the usual inner product in L2(Ω). We denote by c and ci (i ∈ N+) various positive constants, which may be different at different occurrences. Weuse the following hypotheses. (A1) γ(t): R+ → R+ is a non-increasing continuous function. (A2) gi(s): R+ → R+ (i = 1, 2) are differentiable non-increasing functions such that gi(0) > 0, a− ‖a1‖∞ ∫ ∞ 0 g1(s)ds− ‖a2‖∞ ∫ ∞ 0 g2(s)ds := l > 0. (A3) There exists a positive differentiable non-increasing function ξ : R+ → R+ satisfying g′1(t) ≤ −ξ(t)g1(t), t ≥ 0. (A4) There exists a positive constant κ and an increasing strictly convex function G : R+ → R+ of class C1(R+) ∩ C2(R+) satisfying G(0) = G′(0) = 0, lim t→+∞ G′(t) = +∞, such that g′2(t) ≤ −κg2(t), t ≥ 0, (2.1) or ∫ ∞ 0 g2(t) G−1(−g′2(t)) dt+ sup t∈R+ g2(t) G−1(−g′2(t)) < +∞. (2.2) (A5) h(s): R → R is a non-decreasing function with sh(s) ≥ 0 for all s ∈ R and there exists a convex and increasing function H : R+ → R+ of class C1(R+)∩C2(R+) satisfyingH(0) = 0, andH is linear on [0, ε1] orH ′(0) = 0 and H ′′ > 0 on (0, ε1] such that m1|s| ≤ |h(s)| ≤M1|s|, if |s| ≥ ε1, (2.3) h2(s) ≤ H−1(sh(s)), if |s| < ε1, (2.4) where ε1, m1 and M1 are positive constants. (A6) There exists a positive constant m0, such that ‖∇u0(., s)‖2 ≤ m0, s ∈ R+. (2.5) (A7) ai(x) : Ω→ R+ (i = 1, 2) are in C1(Ω) such that, for positive constants ε2 and ε3 and for Γ1, Γ2 ⊂ ∂Ω with meas (Γi > 0), inf x∈Ω (a1(x) + a2(x)) ≥ ε2, ai = 0 or inf Γi ai(x) ≥ 2ε3, i = 1, 2. As in [7], let d = min{ε2, ε3} and let αi ∈ C1(Ω) (i = 1, 2), be such that 0 ≤ αi(x) ≤ ai(x), αi(x) = 0, if ai(x) ≤ d 4 , αi(x) = ai(x), if ai(x) ≥ d 2 . (2.6) EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 5 Assumption (2.4) was introduced for the first time in [12]. Lemma 2.1 (Sobolev-Poincare inequality). Let q be a number with 2 ≤ q <∞ for n = 1, 2, and 2 ≤ q ≤ 2n n−2 for n ≥ 3. Then there exists a constant c∗ = c∗(Ω, q) such that ‖u‖q ≤ c∗‖∇u‖2, u ∈ H1 0 (Ω). From this lemma and (1.2) we know that there exists some positive constant η such that for any u ∈ H1 0 (Ω) one has ‖u‖p+1 p+1 ≤ η(l‖∇u‖22) p+1 2 . (2.7) Using the Faedo-Galerkin method, we can obtain the following local solution. We omit the proof. Theorem 2.2. Suppose that (A1)–(A7) hold, and let (u0, u1) ∈ H1 0 (Ω)×L2(Ω) be given. Then there exists a unique weak solution u of (1.1) such that u ∈ C([0, T ), H1 0 (Ω)) ∩ Lp+1(Ω), ut ∈ C([0, T );H1 0 (Ω)) ∩ Lρ+2(Ω), for some T > 0. Now, for (1.1), we consider the functionals I(t) := ∫ Ω [ a− a1(x) ∫ t 0 g1(s)ds− a2(x) ∫ ∞ 0 g2(s)ds ] |∇u|2dx+ b‖∇u‖42 + (g1 ◦ ∇u)(t) + (g2 }∇u)(t)− ‖u‖p+1 p+1, (2.8) and J(t) := 1 2 ∫ Ω [ a− a1(x) ∫ t 0 g1(s)ds− a2(x) ∫ ∞ 0 g2(s)ds ] |∇u|2dx+ b 4 ‖∇u‖42 + 1 2 (g1 ◦ ∇u)(t) + 1 2 (g2 }∇u)(t)− 1 p+ 1 ‖u‖p+1 p+1. (2.9) We define the energy functional of problem (1.1) as E(t) := 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) =− σ (1 2 d dt ‖∇u‖22 )2 − ‖∇ut‖22 + 1 2 (g′1 ◦ ∇u)(t) + 1 2 (g′2 }∇u)(t) − 1 2 g1(t) ∫ Ω a1(x)|∇u|2dx− ∫ Ω γ(t)uth(ut)dx ≤ 0, (2.11) where (g′1 ◦ ∇u)(t) = ∫ Ω a1(x) ∫ t 0 g′1(t− s)|∇u(t)−∇u(s)|2 ds dx, (g′2 }∇u)(t) = ∫ Ω a2(x) ∫ ∞ 0 g′2(s)|∇u(t)−∇u(t− s)|2 ds dx. 6 F. WANG, J. HAO EJDE-2020/42 Proof. Multiplying the first equation in (1.1) by ut, integrating over Ω and using integration by parts and hypotheses (A1)–(A7), we obtain (2.10). � 3. Global solution and energy decay results Lemma 3.1. Suppose that (A1)–(A7) hold. Let (u0, u1) ∈ H1 0 (Ω)×L2(Ω), I(0) > 0 and η (2(p+ 1) p− 1 E(0) ) p−1 2 < 1. Then I(t) > 0 for all t ≥ 0. Proof. Since I(0) > 0, by continuity, there exists T∗ ≤ T such that I(t) ≥ 0 for all t ∈ [0, T∗). Using that a− ‖a1‖∞ ∫∞ 0 g1(s)ds− ‖a2‖∞ ∫∞ 0 g2(s)ds = l > 0, for any t ∈ [0, T∗), we have J(t) = p− 1 2(p+ 1) ∫ Ω ( a− a1(x) ∫ t 0 g1(s)ds− a2(x) ∫ ∞ 0 g2(s)ds ) |∇u|2dx + p− 3 4(p+ 1) b‖∇u‖42 + p− 1 2(p+ 1) ( (g1 ◦ ∇u)(t) + (g2 }∇u)(t) ) + 1 p+ 1 I(t) ≥ p− 1 2(p+ 1) l‖∇u(t)‖22. From the above inequality, and (2.7)–(2.11), we have l‖∇u‖22 ≤ 2(p+ 1) p− 1 J(t) ≤ 2(p+ 1) p− 1 E(t) ≤ 2(p+ 1) p− 1 E(0), (3.1) and ‖u‖p+1 p+1 ≤ η(l‖∇u‖22) p+1 2 ≤ η (2(p+ 1) p− 1 E(0) ) p−1 2 l‖∇u‖22 < l‖∇u‖22 < ∫ Ω ( a− a1(x) ∫ t 0 g1(s)ds− a2(x) ∫ ∞ 0 g2(s)ds ) |∇u|2dx. (3.2) This shows that I(t) > 0 for all t ∈ [0, T∗). By repeating this procedure, we can extended T∗ to T . This completes the proof of Lemma 3.1. � Theorem 3.2. Suppose that (A1)–(A7) hold. If (u0, u1) ∈ H1 0 (Ω) × L2(Ω), then the solution of (1.1) is global and bounded. Proof. Using (2.11) and Lemma 3.1, we have E(0) ≥ E(t) = 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + 1 2 ‖∇ut‖22 + J(t) ≥ 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + 1 2 ‖∇ut‖22 + p− 1 2(p+ 1) l‖∇u‖22. Therefore, ‖ut‖ρ+2 ρ+2 + ‖∇ut‖22 + ‖∇u‖22 ≤ cE(0), where c is a positive constant that depends on l, ρ and p. This completes the proof. � By using the Hölder inequality and the properties of the functions α1 and α2, we easily obtained the following Lemma. We omit the proof. EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 7 Lemma 3.3. The following inequalities hold,∫ Ω α1(x) (∫ t 0 g1(t− s)(u(t)− u(s))ds )2 dx ≤ c(g1 ◦ ∇u)(t), (3.3)∫ Ω α1(x) (∫ t 0 g1(t− s)(∇u(t)−∇u(s))ds )2 dx ≤ c(g1 ◦ ∇u)(t), (3.4)∫ Ω |∇α1(x)| (∫ t 0 g1(t− s)(u(t)− u(s))ds )2 dx ≤ c(g1 ◦ ∇u)(t), (3.5)∫ Ω |∇α1(x)| (∫ t 0 g1(t− s)(∇u(t)−∇u(s))ds )2 dx ≤ c(g1 ◦ ∇u)(t), (3.6)∫ Ω α2(x) (∫ ∞ 0 g2(s)(u(t)− u(t− s))ds )2 dx ≤ c(g2 }∇u)(t), (3.7)∫ Ω α2(x) (∫ ∞ 0 g2(s)(∇u(t)−∇u(t− s))ds )2 dx ≤ c(g2 }∇u)(t), (3.8)∫ Ω |∇α2(x)| (∫ ∞ 0 g2(s)(u(t)− u(t− s))ds )2 dx ≤ c(g2 }∇u)(t), (3.9)∫ Ω |∇α2(x)| (∫ ∞ 0 g2(s)(∇u(t)−∇u(t− s))ds )2 dx ≤ c(g2 }∇u)(t). (3.10) We define the perturbed energy functional L(t) = ME(t) + εψ(t) + χ1(t) + χ2(t), (3.11) where M and ε are positive constants that will be specified later, and ψ(t) = 1 ρ+ 1 ∫ Ω |ut|ρutu dx+ σ 4 ‖∇u‖42 + ∫ Ω ∇u · ∇ut dx+ 1 2 ‖∇u‖22, χ1(t) = ∫ Ω α1(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx, χ2(t) = ∫ Ω α2(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut) ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx. Lemma 3.4. There exist two positive constants β1 and β2 such that the relation β1L(t) ≤ E(t) ≤ β2L(t), holds for ε small enough while M is large enough. Proof. By using Young’s inequality, Hölder inequality and Lemma 3.3, we obtain∣∣ 1 ρ+ 1 ∫ Ω |ut|ρutu dx ∣∣ ≤ 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + 1 (ρ+ 1)(ρ+ 2) ‖u‖ρ+2 ρ+2 ≤ 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + cρ+2 ∗ (ρ+ 1)(ρ+ 2) ‖∇u‖ρ+2 2 ≤ 1 ρ+ 2 ‖ut‖ρ+2 ρ+2 + cρ+2 ∗ (ρ+ 1)(ρ+ 2) (2(p+ 1) (p− 1)l E(0) )ρ/2 ‖∇u‖22, 8 F. WANG, J. HAO EJDE-2020/42 ∣∣ ∫ Ω α1(x)∆u(t) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx ∣∣ ≤ ∣∣ ∫ Ω ∇α1(x)∇u(t) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx ∣∣ + ∣∣ ∫ Ω α1(x)∇u(t) ∫ t 0 g1(t− s)(∇u(t)−∇u(s)) ds dx ∣∣ ≤ δ‖∇u‖22 + c δ (g1 ◦ ∇u)(t),∣∣ ∫ Ω α1(x)∆ut(t) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx ∣∣ ≤ δ‖∇ut‖22 + c δ (g1 ◦ ∇u)(t), ∣∣ 1 ρ+ 1 ∫ Ω α1(x)|ut|ρut ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx ∣∣ ≤ 1 (ρ+ 1)(ρ+ 2) ∫ Ω α1(x) (∫ t 0 g1(t− s)(u(t)− u(s))ds )ρ+2 dx + 1 ρ+ 2 ∫ Ω α1(x)|ut|ρ+2dx ≤ c ρ+ 2 ‖ut‖ρ+2 ρ+2 + ccρ+2 ∗ (ρ+ 1)(ρ+ 2) (4(p+ 1) (p− 1)l E(0) )ρ/2 (g1 ◦ ∇u)(t), ∣∣ ∫ Ω α2(x)∆u(t) ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx ∣∣ ≤ δ‖∇u‖22 + c δ (g2 }∇u)(t), ∣∣ ∫ Ω α2(x)∆ut ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx ∣∣ ≤ δ‖∇ut‖22 + c δ (g2 }∇u)(t), ∣∣ 1 ρ+ 1 ∫ Ω α2(x)|ut|ρut ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx ∣∣ ≤ 1 ρ+ 2 ( 1 ρ+ 1 ∫ Ω α2(x) (∫ ∞ 0 g2(s)|u(t)− u(t− s)|ds )ρ+2 dx + ∫ Ω α2(x)|ut|ρ+2dx ) ≤ c ρ+ 2 ‖ut‖ρ+2 ρ+2 + cρ+2 ∗ c (ρ+ 1)(ρ+ 2) ∫ ∞ 0 g2(s)‖∇u(t)−∇u(t− s)‖ρ+2 2 ds. Now, using (3.1) and (A6) we obtain ‖∇u(t)−∇u(t− s)‖22 ≤ 2‖∇u(t)‖22 + 2‖∇u(t− s)‖22 ≤ 4 sup s>0 ‖∇u(s)‖22 + 2 sup τ<0 ‖∇u(τ)‖22 ≤ 4 sup s>0 ‖∇u(s)‖22 + 2 sup τ>0 ‖∇u0(τ)‖22 ≤ 8(p+ 1) (p− 1)l E(0) + 2m2 0 := N1,∣∣ 1 ρ+ 1 ∫ Ω α2(x)|ut|ρut ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx ∣∣ ≤ c ρ+ 2 ‖ut‖ρ+2 ρ+2 + cρ+2 ∗ N ρ/2 1 c (ρ+ 1)(ρ+ 2) (g2 }∇u)(t). EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 9 Therefore, |L(t)−ME(t)| ≤ cE(t). The prof is complete. � Lemma 3.5. Suppose that (1.2) and (A1)–(A7) hold. Then ψ(t) = 1 ρ+ 1 ∫ Ω |ut|ρutu dx+ σ 4 ‖∇u‖42 + ∫ Ω ∇u · ∇ut dx+ 1 2 ‖∇u‖22 along the solution of (1.1), and for any ε1 > 0, ψ′(t) ≤ 1 ρ+ 1 ‖ut‖ρ+2 ρ+2 + ‖∇ut‖22 − b‖∇u‖42 + c ε1 ∫ Ω h2(ut)dx + c 2ε1 (g1 ◦ ∇u)(t) + c 2ε1 (g2 }∇u)(t) + ‖u‖p+1 p+1 − ∫ Ω [a− a1(x) ∫ t 0 g1(s)ds− a2(x) ∫ ∞ 0 g2(s)ds− ε1]|∇u|2dx. (3.12) Proof. By taking the time derivative of ψ(t) and using problem (1.1), we obtain ψ′(t) = 1 ρ+ 1 ‖ut‖ρ+2 ρ+2 + ‖∇ut‖22 − a‖∇u‖22 − b‖∇u(t)‖42 − ∫ Ω u(t) ∫ t 0 g1(t− s) div(a1(x)∇u(s)) ds dx− γ(t) ∫ Ω u(t)h(ut)dx − ∫ Ω u(t) ∫ ∞ 0 g2(s) div(a2(x)∇u(t− s)ds)dx+ ‖u(t)‖p+1 p+1. (3.13) For the fifth term, by using Young’s inequality and Hölder inequality, we obtain − ∫ Ω u(t) ∫ t 0 g1(t− s) div(a1(x)∇u(s)) ds dx = ∫ Ω ∇u(t) ∫ t 0 g1(t− s)a1(x)∇u(s) ds dx = ∫ Ω a1(x) ∫ t 0 g1(t− s)(∇u(s)−∇u(t))ds∇u(t)dx + ∫ t 0 g1(s)ds ∫ Ω a1(x)|∇u(t)|2dx ≤ ε1 2 ‖∇u(t)‖22 + 1 2ε1 ∫ Ω a1(x)( ∫ t 0 g1(t− s)(∇u(s)−∇u(t))ds)2dx + ∫ t 0 g1(s)ds ∫ Ω a1(x)|∇u(t)|2dx ≤ ε1 2 ‖∇u(t)‖22 + 1 2ε1 ∫ t 0 g1(s)ds(g1 ◦ ∇u)(t) + ∫ t 0 g1(s)ds ∫ Ω a1(x)|∇u(t)|2dx ≤ ε1 2 ‖∇u(t)‖22 + c 2ε1 (g1 ◦ ∇u)(t) + ∫ t 0 g1(s)ds ∫ Ω a1(x)|∇u(t)|2dx. (3.14) Similarity, for the sixth term we obtain − ∫ Ω γ(t)u(t)h(ut)dx ≤ cε1‖∇u(t)‖22 + c ε1 ∫ Ω h2(ut)dx. (3.15) 10 F. WANG, J. HAO EJDE-2020/42 For the seventh term, we have − ∫ Ω u(t) ∫ ∞ 0 g2(s) div(a2(x)∇u(t− s)) ds dx ≤ ε1 2 ‖∇u(t)‖22 + c 2ε1 (g2 }∇u)(t) + ∫ ∞ 0 g2(s)ds ∫ Ω a2(x)|∇u(t)|2dx. (3.16) By using (3.14)-(3.16) in (3.13), estimate (3.12) follows. � Lemma 3.6. Suppose that (1.2) and (A1)–(A7) hold. Then χ1(t) = ∫ Ω α1(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx, (3.17) along the solution of (1.1), and for any ε2, ε3 > 0, χ′1(t) ≤− [ ∫ t 0 g1(s)ds− cε2 ] ∫ Ω α1(x)|∇ut|2dx+ c(ε2 + ε3) ∫ Ω |∇u|2dx − 1 ρ+ 1 ∫ t 0 g1(s)ds ∫ Ω α1(x)|ut|ρ+2dx+ c ε3 (g1 ◦ ∇u)(t) + ε3(g2 }∇u)(t) − c ε2 (g′1 ◦ ∇u)(t)− σ 4(p+ 1) p− 1 E(0)E′(t) + ε3 ∫ Ω h2(ut)dx. Proof. Taking the derivative of χ1 and using (1.1), we obtain χ′1(t) = − ∫ Ω α1(x)M(t)∆u ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1(x) ∫ t 0 g1(t− s) div(a1(x)∇u(s))ds ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1(x) ∫ ∞ 0 g2(s) div(a2(x)∇u(t− s))ds ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1(x)γ(t)h(ut) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx − ∫ Ω α1(x)|u|p−1u ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut)( ∫ t 0 g′1(t− s)(u(t)− u(s)) ds dx + ∫ t 0 g1(s)ds ∫ Ω α1(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut)ut dx. Therefore, χ′1(t) = ∫ Ω b‖∇u‖22∇α1∇u(t) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx − ∫ t 0 g1(s)ds ∫ Ω α1|∇ut|2dx− 1 ρ+ 1 ∫ t 0 g1(s)ds ∫ Ω α1|ut|ρ+2dx + ∫ Ω α1b‖∇u‖22∇u(t) ∫ t 0 g1(t− s)(∇u(t)−∇u(s)) ds dx EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 11 + ∫ Ω ∇α1 ( a− a1 ∫ t 0 g1(s)ds− a2 ∫ ∞ 0 g2(s) ) ∇u(t) × ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1 ( a− a1 ∫ t 0 g1(s)ds− a2 ∫ ∞ 0 g2(s) ) ∇u(t) × ∫ t 0 g1(t− s)(∇u(t)−∇u(s)) ds dx + ∫ Ω (σ ∫ Ω ∇u∇ut dx)∇α1∇u(t) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω (σ ∫ Ω ∇u∇ut dx)α1∇u(t) ∫ t 0 g1(t− s)(∇u(t)−∇u(s)) ds dx + ∫ Ω ∇α1a1(x) ∫ t 0 g1(t− s)(∇u(t)−∇u(s))ds × ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω α1a1(x)( ∫ t 0 g1(t− s)(∇u(t)−∇u(s))ds)2dx + ∫ Ω a2∇α1 (∫ ∞ 0 g2(s)(∇u(t)−∇u(t− s))ds ) × ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx + ∫ Ω a2α1 (∫ ∞ 0 g2(s)(∇u(t)−∇u(t− s))ds ) × ∫ t 0 g1(t− s)(∇u(t)−∇u(s)) ds dx + ∫ Ω α1γh(ut) ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx − ∫ Ω α1|u|p−1u ∫ t 0 g1(t− s)(u(t)− u(s)) ds dx − ∫ Ω ∇α1∇u ∫ t 0 g′1(t− s)(u(t)− u(s)) ds dx − ∫ Ω α1∇u ∫ t 0 g′1(t− s)(∇u(t)−∇u(s)) ds dx − ∫ Ω ∇α1∇ut ∫ t 0 g′1(t− s)(u(t)− u(s)) ds dx − ∫ Ω α1∇ut ∫ t 0 g′1(t− s)(∇u(t)−∇u(s)) ds dx − 1 ρ+ 1 ∫ Ω α1|ut|ρut ∫ t 0 g′1(t− s)(u(t)− u(s)) ds dx 12 F. WANG, J. HAO EJDE-2020/42 − ∫ t 0 g1(s)ds ∫ Ω ut∇α1∇u dx − ∫ t 0 g1(s)ds ∫ Ω α1∇ut∇u dx− ∫ t 0 g1(s)ds ∫ Ω ut∇α1∇ut dx. Using Young’s inequality, Poincare inequality and that |∇α1| ≤ ca1(x), we obtain (3.17). � With a similar proof as that of Lemma 3.6 we can obtain the following Lemma. Lemma 3.7. Suppose that (1.2) and (A1)-(A7) hold. Then χ2(t) = ∫ Ω α2(x)(∆u+ ∆ut − 1 ρ+ 1 |ut|ρut) ∫ ∞ 0 g2(s)(u(t)− u(t− s)) ds dx along the solution of (1.1), and for any ε2, ε3 > 0, χ′2(t) ≤− [ ∫ ∞ 0 g2(s)ds− cε2 ] ∫ Ω α2(x)|∇ut|2dx+ ε3(g1 ◦ ∇u)(t) − 1 ρ+ 1 ∫ ∞ 0 g2(s)ds ∫ Ω α2(x)|ut|ρ+2dx+ c(ε2 + ε3) ∫ Ω |∇u|2dx + c ε3 (g2 }∇u)(t)− c ε2 (g′2 }∇u)(t)− σ 4(p+ 1) p− 1 E(0)E′(t) + ε3 ∫ Ω h2(ut)dx. (3.18) Assumption (A2) guarantees that for any t0 > 0, we have g0 := min {∫ t0 0 g1(s)ds, ∫ ∞ 0 g2(s)ds } . A differentiation of L, together with Lemmas 3.5, 3.5 and 3.7, give L′(t) ≤− 1 ρ+ 1 ∫ Ω [g0(α1 + α2)− ε]|ut|ρ+2dx+ ( M 2 − c ε2 )(g′1 ◦ ∇u+ g′2 }∇u) − ∫ Ω [(g0 − cε2)(α1 + α2)− ε+M ]|∇ut|2dx+ ε‖u‖p+1 p+1 + ( cε ε1 + c ε3 + ε3)(g1 ◦ ∇u+ g2 }∇u)− σ 8(p+ 1) p− 1 E(0)E′(t) + (2ε3 + cε ε1 ) ∫ Ω h2(ut)dx− bε‖∇u‖42 − [(l − ε1)ε− cε3] ∫ Ω |∇u|2dx. We choose ε small enough and M large enough such that cε3 l − ε1 < ε < (g0 − cε2)(α1 + α2), M > 2c ε2 . Therefore, there exist positive constants κ1, κ2, κ3, and κ4 such that for all t ≥ t0 we have L′(t) ≤ −κ1E(t) + κ2(g1 ◦ ∇u+ g2 }∇u) + κ3 ∫ Ω h2(ut)dx− κ4E ′(t). We define F1(t) = L(t) + κ4E(t). Thus, we have F ′1(t) ≤ −κ1E(t) + κ2(g1 ◦ ∇u+ g2 }∇u) + κ3 ∫ Ω h2(ut)dx. (3.19) EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 13 Now, we are ready to prove our main results by adopting and modifying the arguments in [22]. Firstly we give the following Lemma. Lemma 3.8. If condition (2.2) holds, then for any ε0 > 0, we have G′(ε0E(t))(g2 }∇u) ≤ −cE′(t) + cε0E(t)G′(ε0E(t)). (3.20) Proof. Since E(t) is non-increasing and the assumption (A6), we have∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx ≤ 2‖a2‖∞ ∫ Ω |∇u(t)|2dx+ 2‖a2‖∞ ∫ Ω |∇u(t− s)|2dx ≤ { cE(0), if 0 ≤ s < t, cE(0) + c ∫ Ω |∇u0(s− t)|2dx, if s ≥ t, ≤ A, in which A is a positive constant. Let ε0, δ1, δ2 > 0. Then (g2 }∇u)(t) = ∫ Ω a2(x) ∫ ∞ 0 g2(s)(∇u(t)−∇u(t− s))2 ds dx = 1 δ1G′(ε0E(t)) ∫ ∞ 0 G−1 ( − δ2g′2(s) ∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx ) × δ1G ′(ε0E(t))g2(s) ∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx G−1 ( −δ2g′2(s) ∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx ) ds ≤ 1 δ1G′(ε0E(t)) ∫ ∞ 0 G−1 ( − δ2g′2(s) ∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx ) × Aδ1G ′(ε0E(t))g2(s) G−1(−Aδ2g′2(s)) ds. Let G∗ be the dual function of the convex function G defined by G∗(t) = sup s≥0 {ts−G(s)}, t ∈ R+. Obviously, G′ is increasing and defines a bijection from R+ to R+, and then, for any t ∈ R+, the function s 7→ ts−G(s) reaches its maximum on R+ at the unique point (G′)−1(t). Therefore G∗(t) = t(G′)−1(t)−G((G′)−1(t)), t ∈ R+. Using the general Young’s inequality: s1s2 ≤ G(s1) +G∗(s2), for s1 = G−1 ( − δ2g′2(s) ∫ Ω a2(x)(∇u(t)−∇u(t− s))2dx ) , s2 = Aδ1G ′(ε0E(t))g2(s) G−1(−Aδ2g′2(s)) , we obtain (g2 }∇u)(t) ≤ 1 δ1G′(ε0E(t)) ∫ ∞ 0 G∗( Aδ1G ′(ε0E(t))g2(s) G−1(−Aδ2g′2(s)) )ds− δ2 δ1G′(ε0E(t)) (g′2 }∇u)(t). 14 F. WANG, J. HAO EJDE-2020/42 Using that G∗(s) ≤ s(G′)−1(s) and the definition of E′(t), we obtain (g2 }∇u)(t) ≤ ∫ ∞ 0 Ag2(s) G−1(−Aδ2g′2(s)) (G′)−1( Aδ1G ′(ε0E(t)g2(s)) G−1(−Aδ2g′2(s)) )ds− 2δ2 δ1G′(ε0E(t)) E′(t). Condition (2.2) implies sup s∈R+ g2(s) G−1(−g′2(s)) = m2 < +∞. Then, using that (G′)−1 is non-decreasing, for δ2 = 1 A we obtain (g2 }∇u)(t) ≤ ∫ ∞ 0 Ag2(s) G−1(−g′2(s)) (G′)−1(m2Aδ1G ′(ε0E(t)))ds− 2 Aδ1G′(ε0E(t)) E′(t). Choosing δ1 = 1 m2A and using that∫ ∞ 0 Ag2(s) G−1(−g′2(s)) ds = m3 < +∞, we obtain (g2 }∇u)(t) ≤ −2m1 G′(ε0E(t)) E′(t) +m3ε0E(t). Thus, (3.20) holds. � We define the following partition of Ω Ω+ := {x ∈ Ω : |ut| ≥ ε1}, Ω− := {x ∈ Ω : |ut| < ε1}. Now we state our main result of this article. Theorem 3.9. Assume that (1.1) and (A1)–(A7) are satisfied. Then there exist positive constants ε0, τ0, c′ and c′′ such that the solution of (1.1) satisfies E(t) ≤ c′′(G2)−1 (∫ t 0 c′ζ(s)ds ) , t ≥ 0, (3.21) where G2(t) = ∫ 1 t 1 H1(s) ds, G1(s) = { s, if (2.1) holds, sG′(ε0s), if (2.2) holds, H1(s) = { G1(s), if H is linear on [0, ε1], G1(s)H ′(τ0G1(s)), otherwise, ζ(t) = min{ξ(t), γ(t)}. Proof. Case (2.1) holds. At this point we use (2.10) to obtain (g2 }∇u)(t) ≤ − 1 κ (g′2 }∇u)(t) ≤ − 2 κ E′(t). (3.22) Case (2.2) holds. We have (3.20). EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 15 For the two cases, from (3.20) and (3.22) we deduce that G1(E(t)) E(t) (g2 }∇u)(t) ≤ −cE′(t) + cε0G1(E(t)). (3.23) Therefore, multiplying (3.19) by G1(E(t)) E(t) , and using (3.23) we obtain G1(E(t)) E(t) F ′1(t) ≤− κ1G1(E(t)) + κ2 G1(E(t)) E(t) (g1 ◦ ∇u)(t)− κ2cE ′(t) + κ3 G1(E(t)) E(t) ∫ Ω h2(ut)dx+ κ2cε0G1(E(t)). Choosing ε0 small enough, we arrive at G1(E(t)) E(t) F ′1(t) + κ2cE ′(t) ≤− cG1(E(t)) + κ2 G1(E(t)) E(t) (g1 ◦ ∇u)(t) + κ3 G1(E(t)) E(t) ∫ Ω h2(ut)dx. (3.24) Let F2(t) = G1(E(t)) E(t) F1(t) + κ2cE(t). By recalling that t → G1(E(t)) E(t) is non-increasing, we deduce that F2(t) ∼ E(t) and by exploiting (3.24), we conclude that for t ≥ t0, F ′2(t) ≤ −cG1(E(t)) + κ2 G1(E(t)) E(t) (g1 ◦ ∇u)(t) + κ3 G1(E(t)) E(t) ∫ Ω h2(ut)dx. (3.25) We use (A3) to obtain ζ(t)(g1 ◦ ∇u)(t) ≤ ξ(t)(g1 ◦ ∇u)(t) = ∫ Ω a1(x) ∫ t 0 ξ(t)g1(t− s)(∇u(t)−∇u(s))2 ds dx ≤ ∫ Ω a1(x) ∫ t 0 ξ(t− s)g1(t− s)(∇u(t)−∇u(s))2 ds dx ≤ −c(g′1 ◦ ∇u)(t) ≤ −cE′(t). Multiplying (3.25) by ζ(t) and using that t→ G1(E(t)) E(t) is non-increasing, we obtain ζ(t)F ′2(t) ≤− cζ(t)G1(E(t)) + κ2 G1(E(t)) E(t) ζ(t)(g1 ◦ ∇u)(t) + κ3 G1(E(t)) E(t) ζ(t) ∫ Ω h2(ut)dx ≤− cζ(t)G1(E(t))− cE′(t) + cζ(t) ∫ Ω h2(ut)dx. (3.26) Using that ζ(t) is a non-increasing continuous function, ξ(t) and η(t) are non- increasing, and ζ(t) is differentiable, with ζ ′(t) ≤ 0, then the functional F3(t) := ζ(t)F2(t) + cE(t) 16 F. WANG, J. HAO EJDE-2020/42 satisfies F3(t) ∼ E(t), and F ′3(t) ≤ −cζ(t)G1(E(t)) + cζ(t) ∫ Ω h2(ut)dx. (3.27) Case 1. H is linear on [0, ε1]. In this case, there exists κ5 > 0 such that ζ(t) ∫ Ω h2(ut)dx ≤ κ5γ(t) ∫ Ω uth(ut)dx ≤ −κ5E ′(t), which together with (3.27) implies (F3(t) + cE(t))′ ≤ −cζ(t)G1(E(t)), (3.28) which gives J(t) := (F3(t) + cE(t))δ0 ∼ E(t) and J ′(t) ≤ −cδ0ζ(t)G1(J(t)) =: −c′ζ(t)H1(J(t)). (3.29) We choose δ0 small enough so that J(t) ≤ E(t) and G2(J(t0))− c′ ∫ t0 0 ζ(s)ds > 0. By integrating (3.29) over (t0, t) and noting that G2 is non-increasing, we deduce that J(t) ≤ G−1 2 (G2(J(t0)) + c′ ∫ t 0 ζ(s)ds− c′ ∫ t0 0 ζ(s)ds) ≤ G−1 2 (c′ ∫ t 0 ζ(s)ds). Consequently, the relation between of J(t) and E(t) yields E(t) ≤ c′′G−1 2 (c′ ∫ t 0 ζ(s)ds). Case 2. H ′(0) = 0 and H ′′ > 0 on (0, ε1]. In this case, we first estimate ∫ Ω h2(ut)dx on the right-hand of (3.27). Noting that H−1 is concave and increasing, and using Jensen’s inequality, (A5) and (2.10), we deduce that∫ Ω h2(ut)dx = ∫ Ω+ h2(ut)dx+ ∫ Ω− h2(ut)dx ≤M1 ∫ Ω+ uth(ut)dx+ ∫ Ω− h2(ut)dx ≤ −M1E ′(t) + ∫ Ω− H−1(uth(ut))dx ≤ −M1E ′(t) + cH−1(S(t)), (3.30) where S(t) = 1 vol(Ω−) ∫ Ω− uth(ut)dx. Hence (3.27) becomes F ′3(t) ≤ −cζ(t)G1(E(t))− cM1ζ(t)E′(t) + cζ(t)H−1(S(t)). (3.31) Now, we define F4(t) := F3(t) + cM1ζ(t)E(t). Then, we have F ′4(t) ≤ −cζ(t)G1(E(t)) + cζ(t)H−1(S(t)). (3.32) We define F5(t) := H ′(τ0G1(E(t)))F4(t) + κ6E(t), (3.33) EJDE-2020/42 VISCOELASTIC WAVE EQUATIONS 17 where τ0 > 0 and κ6 > 0 to be determined later. Then, using E′(t) ≤ 0, G′1(t) ≥ 0, H ′′(t) ≥ 0, and (3.31), we obtain F ′5(t) = τ0G ′ 1(E(t))E′(t)H ′′(τ0G1(E(t)))F4(t) +H ′(τ0G1(E(t)))F ′4(t) + κ6E ′(t) ≤ H ′(τ0G1(E(t)))F ′4(t) + κ6E ′(t) ≤ −cζ(t)G1(E(t))H ′(τ0G1(E(t))) + cζ(t)H−1(S(t))H ′(τ0G1(E(t))) + κ6E ′(t). (3.34) Let H∗ be the convex conjugate of H in the sense of Young, then H∗(s) = s(H ′)−1(s)−H ( (H ′)−1(s) ) , s ∈ R+, (3.35) and H∗ satisfies AB ≤ H∗(A) +H(B), A,B ≥ 0. (3.36) Furthermore, using (3.35) and H ′(0) = 0, (H ′)−1 is increasing, and H is also increasing yield H∗(s) ≤ s(H ′)−1(s), s ≥ 0. (3.37) Taking H ′(τ0G1(E(t))) = A and H−1(S′(t)) = B in (3.34), applying (3.36) and (3.37), and noting that 0 ≤ H ′(τ0G1(E(t))) ≤ H ′(τ0G1(E(0))) due to H ′ is in- creasing, we obtain F ′5(t) ≤− cζ(t)G1(E(t))H ′(τ0G1(E(t))) + cζ(t)H∗(H ′(τ0G1(E(t)))) + cζ(t)S(t) + κ6E ′(t) ≤− cζ(t)G1(E(t))H ′(τ0G1(E(t))) + cζ(t)τ0G1(E(t))H ′(τ0G1(E(t))) − cE′(t) + κ6E ′(t). Consequently, with a suitable choice of τ0 and κ6, we obtain F ′5(t) ≤ −cζ(t)G1(E(t))H ′(τ0G1(E(t)) =: −cζ(t)H1(E(t)). (3.38) Thus, by defining R(t) = δ3F5(t) ∼ E(t), and by computation, we have R′(t) ≤ −cδ3ζ(t)G1(R(t))H ′(τ0G1(R(t))) =: −c′ζ(t)H1(R(t)). (3.39) We choose δ3 small enough so that R(t) ≤ E(t) and G2(R(t0))− c′ ∫ t0 0 ζ(s)ds > 0. By integrating (3.39) over (t0, t) and noting that G2 is non-increasing, we deduce R(t) ≤ G−1 2 (G2(R(t0) + c′ ∫ t 0 ζ(s)ds− c′ ∫ t0 0 ζ(s)ds) ≤ G−1 2 (c′ ∫ t 0 ζ(s)ds). Consequently, the equality relation between of R(t) and E(t) yields E(t) ≤ c′′G−1 2 (c′ ∫ t 0 ζ(s)ds). 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