Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 95, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ENERGY DECAY FOR VARIABLE COEFFICIENT VISCOELASTIC WAVE EQUATION WITH ACOUSTIC BOUNDARY CONDITIONS IN DOMAINS WITH NONLOCALLY REACTING BOUNDARY JIANGHAO HAO, MENGXIAN LV Abstract. In this article, we study a variable coefficients viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally re- acting boundary. By constructing suitable Lyapunov functionals and using the energy compensation method, we prove that under suitable conditions on the initial data and the relaxation function, the energy of the system has an explicit and general decay rate. 1. Introduction Let Ω ⊂ Rn (n ≥ 2) be an open bounded domain with smooth boundary Γ = Γ0 ∪ Γ1. Here, Γ0 and Γ1 are closed and disjoint with meas(Γ0) > 0. In this paper we consider the viscoelastic wave equation of variable coefficients with the acoustic boundary conditions u′′ − Lu+ ∫ t 0 g(t− τ)Lu(τ)dτ + ρ(u′) = 0 in Ω× (0,∞), u = 0 on Γ0 × (0,∞), ∂u ∂νL − ∫ t 0 g(t− τ) ∂u ∂νL (τ)dτ = z′ on Γ1 × (0,∞), fz′′ − p2∆Γz + qz′ + hz = −u′ on Γ1 × (0,∞), u(x, 0) = u0(x), u′(x, 0) = u1(x) in Ω, z(x, 0) = z0(x), z′(x, 0) = z1(x) on Γ1, (1.1) where Lu = div(A(x)∇u) = n∑ i,j=1 ∂ ∂xi ( aij(x) ∂u ∂xj ) , ∂u ∂νL = n∑ i,j=1 aij(x) ∂u ∂xj νi. 2010 Mathematics Subject Classification. 35L70, 35B35. Key words and phrases. Variable coefficients; viscoelastic wave equation; acoustic boundary conditions; nonlocally reacting boundary. c©2020 Texas State University. Submitted December 19, 2019. Published September 17, 2020. 1 2 J. HAO, M. LV EJDE-2020/95 The symbol ′ denotes the derivative with respect to time t, νL = Aν, where ν = (ν1, . . . , νn) represents the outward unit normal vector to Γ, and ∆Γ is the Laplace- Beltrami operator. In addition, p is a positive constant, ρ : R → R, g : R+ → R+ and f , q, h : Γ1 → R are functions. When g = 0 and p = 0, the boundary conditions (1.1)3 and (1.1)4 are the classical acoustic boundary conditions introduced by Morse and Ingard [23] and developed by Beale and Rosencrans [2, 3] via the assumption that each point on the boundary reacts to the excess pressure of the acoustic wave like a resistive harmonic oscillator or spring and each point of the boundary does not affect each other. The models usually are related to the problems of noise control and suppression in practical applications and have been studied by many authors, see [6, 7, 8] and the references therein. Ĺımaco et al. [16] investigated a nonlinear wave equation of Carrier type and established the existence of regular weak solution. Gao, Liang and Xiao [11] obtained the uniform stability of a nonlinear acoustic wave system with an internal localized damping term ω(x)ut. For the case f = 0, which means the material of surface is much lighter than the fluid medium, Hao and He [14, 15] studied two variable-coefficient wave equations with the acoustic boundary conditions, and they obtained the exponential decay result and general decay result respectively. On the other hand, when g = 0 and p > 0, the boundary conditions (1.1)3 and (1.1)4 are called acoustic boundary conditions to non-locally reacting boundary (see [9]), which models the surface Γ1 reacts to the excess pressure as an elastic membrane. Later, Frota et al [10] studied the following semilinear wave equation u′′ −∆u+ αu′ + ρ(u′) = F. They proved the existence, uniqueness of solution by Galerkin’s method and ob- tained an exponential decay result. Moreover they also improved their previous results since estimates they made can be adapted to the problem treated in [9]. Frota and Vicente [26] took into account the dissipative term q(z′) in stead of qz′ and put a nonlinear internal localized damping term in the wave equation to achieve uniform stability successfully. Recently, Ha [13] considered the following wave equation of variable coefficients u′′ − Lu+ ρ(u′) = 0, where Lu = div(A(x)∇u) = n∑ i,j=1 ∂ ∂xi ( aij(x) ∂u ∂xj ) . Under suitable conditions on ρ, he improved his previous result [12] in which he focused on the case A = I, and obtained the general decay result. Liu [18] studied a variable coefficient wave equation with an acoustic undamped boundary condition and deduced the polynomial energy decay estimates by the Riemannian geometry method introduced by Yao [28]. In addition, the integral-differential term in (1.1) gives the memory effect to the problem, due to the mechanical response influenced by the history of the materials themselves. The study involving the wave equation with viscoelastic term and the acoustic boundary conditions can be found in [5, 19, 20]. For instance, Park and Park [25] studied the viscoelastic wave system u′′ −∆u+ ∫ t 0 g(t− τ)∆u(τ)dτ = 0 in Ω× (0,∞), EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 3 u = 0 on Γ0 × (0,∞), ∂u ∂ν − ∫ t 0 g(t− τ) ∂u ∂ν (τ)dτ = z′ on Γ1 × (0,∞), u′ + qz′ + hz = 0 on Γ1 × (0,∞), and deduced the energy decay rates under the assumption ∫∞ 0 g(s)ds < 1 2 . Later, without this assumption condition on the relaxation function g, Liu [17] generalized the work to an arbitrary decay rate which does not necessarily decay exponentially or polynomially. In presence of variable-coefficient matrices A(x), which reflects the inhomogeneous nature of the material in applications, Boukhatem and Benab- derrahmane [4] considered the damped semilinear viscoelastic wave system u′′ − Lu+ ∫ t 0 g(t− τ)Lu(τ)dτ = |u|p−2u in Ω× (0,∞), u = 0 on Γ0 × (0,∞), ∂u ∂νL − ∫ t 0 g(t− τ) ∂u ∂νL (τ)dτ = h(x)z′ on Γ1 × (0,∞), u′ + qz′ + hz = 0 on Γ1 × (0,∞). Instead of using the Riemannian geometry method, they obtained the local exis- tence of solution by combining the Faedo-Galerkin approximations and the con- traction mapping theorem. Furthermore, they proved the solution exists globally in time and established a uniform decay result. From the previous works with memory effect and the acoustic boundary conditions, we can see that most authors considered the porous case (f = 0). Motivated by the previous works, our goal of this paper is to prove the general decay estimates for problem (1.1). We consider the case f > 0, i.e., non-porous case and Γ1 is non-locally reacting. To the best of our knowledge, it is hardly seen in current literature on the study of variable-coefficient viscoelastic wave equation with acoustic boundary conditions to nonlocally reacting boundary. Therefore, the model is novel and the study on the asymptotic behavior of solutions for (1.1) is interesting and significant. Also, problem (1.1) in this paper is a improvement of [13], because we consider the viscoelastic damping effect and the assumptions on ρ allows a wider class of functions. Different from the method in [13], our strategy was to use the techniques of [21, 22, 27] with some necessary modifications due to the nature of problem (1.1). The main idea is to construct appropriate Lyapunov functionals and deduce the energy inequality which leads us to a general decay result. The paper is organized as follows. In Section 2, we present some assumptions and materials needed in our work and give the main results of this paper. Then, some estimates are given and the general decay of energy for (1.1) is derived in Section 3. 2. Preliminaries In this section, we present some assumptions and materials needed for our work. Throughout the paper Ci (i = 1, 2, . . . ) denote various positive constants which depend on the known constants. We consider the standard Sobolev spaces Lq(Ω) and Lq(Γ1) endowed with the usual inner products and norms. For simplicity, we 4 J. HAO, M. LV EJDE-2020/95 denote ‖ · ‖L2(Ω), ‖ · ‖Lq(Ω), ‖ · ‖L2(Γ1) and ‖ · ‖Lq(Γ1) by ‖ · ‖, ‖ · ‖q, ‖ · ‖Γ1 and ‖ · ‖q,Γ1 , respectively. Set H(L,Ω) = {u ∈ H1(Ω);Lu ∈ L2(Ω)} equipped with the norm ‖u‖H(L,Ω) = ( ‖u‖2H1(Ω) + ‖Lu‖2 )1/2 . Denoting γ0 : H1(Ω) → H1/2(Γ) and γ1 : H(L,Ω) → H−1/2(Γ) the trace map of order 0 and the Neumann trace map on H(L,Ω), respectively, we have γ0(u) = u|Γ and γ1(u) = ( ∂u ∂νL ) Γ . DefineW = {u ∈ V ∩H3(Ω); (γ1(u))|Γ1 ∈ H1 0 (Γ1)}, where V = {u ∈ H1(Ω); γ0(u) = 0 on Γ0} endowed with the norm ‖u‖V = ( N∑ i=1 ∫ Ω | ∂u ∂xi |2dx )1/2 . By Poincaré’s inequality and the continuity of the trace map, there exist positive constants k0 and k1 such that ‖u‖ ≤ k0‖∇u‖ and ‖γ0(u)‖Γ1 ≤ k1‖∇u‖, u ∈ V. (2.1) We consider the Sobolev space Hm(Γ1), m = 1, 2 with respect to the norm ‖z‖Hm(Γ1) = ( m∑ i=0 ‖∇iz‖2Γ1 )1/2 , m = 1, 2, where ∇i is the covariant derivative operator of order i. Let H1 0 (Γ1) be the closure of C∞0 (Γ1) in H1(Γ1). The Poincaré’s inequality holds in H1 0 (Γ1), thus there exists a constant k2 such that ‖z‖Γ1 ≤ k2‖∇τz‖Γ1 , z ∈ H1 0 (Γ1), (2.2) where ∇τ is the tangential gradient on Γ1. Therefore on H1 0 (Γ1) we have the inner product and norm (z, v)Γ1 = ∫ Γ1 〈∇τz(x),∇τv(x)〉dΓ1, ‖z‖Γ1 = ‖∇τz‖Γ1 , which is equivalent to the usual norm endowed by H1(Γ1). Next, we consider H1 0 (Γ1) ∩H2(Γ1) endowed with the norm ‖z‖H1 0 (Γ1)∩H2(Γ1) = ‖∆Γz‖Γ1 , here ∆Γz = div∇τz, which is equivalent to the usual norm endowed by H2(Γ1). We will use the following assumptions: (A1) The matrix A(x) = (aij(x)), with entires aij(x) ∈ C1(Ω̄), is symmetric and there exists a positive constant a0 such that for all x ∈ Ω̄ and ζ = (ζ1, ζ2, . . . , ζn) ∈ Rn, we have n∑ i,j=1 aij(x)ζjζi ≥ a0|ζ|2. EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 5 (A2) The relaxation function g : R+ → R+ is a bounded C1 function satisfying g(0) > 0, 1− ∫ ∞ 0 g(s)ds = l > 0, g′(t) ≤ −ξ(t)g(t), t ≥ 0, in which ξ : [0,∞) → [0,∞) is a positive nonincreasing C1 function satis- fying ∫ ∞ 0 ξ(s)ds =∞. (A3) ρ : R→ R is a nondecreasing C1 function and there exist positive constants ε, c1, c2 > 0 and an increasing function H1 : R+ → R+ of class C1(R+) ∩ C2(R+) satisfying H1(0) = 0, and H1 is linear or H ′1(0) = 0 and H ′′1 (t) > 0 on (0, ε] such that c1|s| ≤ |ρ(s)| ≤ c2|s| if |s| ≥ ε, s2 + ρ2(s) ≤ H−1 1 (sρ(s)) if |s| ≤ ε. (A4) The positive functions f , q, h are essentially bounded and there exist pos- itive constants fi, qi, hi (i = 0, 1) such that f0 ≤ f ≤ f1, q0 ≤ q ≤ q1, h0 ≤ h ≤ h1, x ∈ Γ1. To simplify calculation in our analysis, we introduce the following notation (g � u)(t) = ∫ t 0 g(t− τ)a(u(t)− u(τ), u(t)− u(τ))dτ, where a(u(t), v(t)) = n∑ i,j=1 ∫ Ω aij(x) ∂u(t) ∂xj ∂v(t) ∂xi dx = ∫ Ω A∇u(t)∇v(t)dx. Lemma 2.1. For g ∈ C1(0, T ) and u ∈ C1(0, T ;V ), we have∫ t 0 g(t− τ)a(u(τ), u′(t))dτ = 1 2 (g′ � u)(t)− 1 2 g(t)a(u(t), u(t)) − 1 2 d dt ( (g � u)(t)− ∫ t 0 g(τ)dτa(u(t), u(t)) ) . (2.3) Similar to [10], a well posedness theorem can be derived by using Faedo-Galerkin method and we omit the proof. Theorem 2.2. Suppose that assumptions (A1)–(A4) hold and the initial data sat- isfies (u0, u1, z0) ∈W × V × (H1 0 (Γ1) ∩H2(Γ1)) (2.4) and the compatibility condition ∂u0 ∂ν = z1 in L2(Γ1). (2.5) Then, there exists a unique solution (u, z) to (1.1) satisfying u ∈ L∞loc(0,∞;V ), u′ ∈ L∞loc(0,∞;V ), u′′ ∈ L∞loc(0,∞;L2(Ω)), 6 J. HAO, M. LV EJDE-2020/95 z ∈ L∞loc(0,∞;H1 0 (Γ1) ∩H2(Γ1)), z′ ∈ L∞loc(0,∞;H1 0 (Γ1)), z′′ ∈ L∞loc(0,∞;L2(Γ1)). We denote the modified energy functional E(t) associated with problem (1.1) by E(t) = 1 2 ‖u′‖2 + 1 2 ( 1− ∫ t 0 g(τ)dτ ) a(u(t), u(t)) + 1 2 (g � u)(t) + 1 2 ‖f1/2z′‖2Γ1 + p2 2 ‖∇τz‖2Γ1 + 1 2 ‖h1/2z‖2Γ1 . (2.6) Multiplying the first equation in (1.1) by ut and the fourth equation by zt, integrating over Ω and Γ1 respectively, using integration by parts and (2.3), we obtain the following lemma. Lemma 2.3. Suppose that assumptions (A1)–(A4), (2.4) and (2.5) hold. Then E(t) is nonincreasing and satisfies E′(t) = −‖q1/2z′‖2Γ1 − 1 2 g(t)a(u(t), u(t)) + 1 2 (g′ � u)(t)− ∫ Ω u′ρ(u′)dx. (2.7) Now we can state the main result of this paper. Theorem 2.4. Suppose that assumptions (A1)–(A4), (2.4) and (2.5) hold. Then there exist positive constants ε0, t0, µ1, µ2 and nonnegative constant µ3 such that the solution of system (1.1) satisfies E(t) ≤ µ1H −1 ( µ2 ∫ t 0 ξ(s)ds+ µ3 ) , t ≥ t0, (2.8) where H(r) = ∫ 1 r 1 H0(s) ds and H0(r) = rH ′1(ε0r). Here, H is strictly decreasing and convex on (0, 1], with limr→0H(r) = +∞. 3. Decay estimate In this section we give the proof of our main result. To do this, we define the functional L(t) := E(t) + εψ(t) + ηφ(t), (3.1) where ε and η are positive constants to be chosen later and ψ(t) := ∫ Ω uu′dx+ ∫ Γ1 fzz′dΓ + ∫ Γ1 uz dΓ, (3.2) φ(t) := − ∫ Ω u′ ∫ t 0 g(t− τ)(u(t)− u(τ)) dτ dx. (3.3) It is easy to obtain the following result, i.e. the functional L is equivalent to the energy functional E. Lemma 3.1. Suppose that assumptions (A1)–(A4), (2.4) and (2.5) hold. Then for ε, η > 0 small enough, there exist two positive constants λ1 and λ2 such that λ1E(t) ≤ L(t) ≤ λ2E(t). EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 7 Lemma 3.2. Let assumptions (A1)–(A4), (2.4) and (2.5) hold. Then there exists some constant C1 such that the functional ψ(t) satisfies ψ′(t) ≤ ‖u′‖2 + C1‖z′‖2Γ1 − ‖h1/2z‖2Γ1 − l 4 a(u(t), u(t)) −p 2 2 ‖∇τz‖2Γ1 + 1− l 2l (g � u)(t) + 1 4α1 ∫ Ω ρ2(u′)dx. (3.4) Proof. By differentiating ψ and using (1.1), we obtain ψ′(t) = ‖u′‖2 + ‖f1/2z′‖2Γ1 − p2‖∇τz‖2Γ1 − ‖h1/2z‖2Γ1 − a(u(t), u(t)) + ∫ Γ1 uz′dΓ− ∫ Ω uρ(u′)dx− ∫ Γ1 zu′dΓ− ∫ Γ1 qzz′dΓ + d dt ∫ Γ1 uzdΓ + ∫ t 0 g(t− τ) ∫ Ω A∇u(t)∇u(τ) dx dτ. (3.5) Now we estimate the last term on the right-hand side of (3.5). By (A2), Young’s inequality and Hölder’s inequality, we obtain∫ t 0 g(t− τ) ∫ Ω A∇u(t)∇u(τ) dx dτ ≤ 1 2 a(u(t), u(t)) + 1 2 ∫ Ω A (∫ t 0 g(t− τ)(|∇u(τ)−∇u(t)|+∇u(t))dτ )2 dx ≤ 1 2 ( 1 + (1 + λ)(1− l)2 ) a(u(t), u(t)) + 1 2 ( 1 + 1 λ ) (1− l)(g � u)(t). (3.6) Using (2.1), (2.2), (2.3), (A1), (A4) and Cauchy’s inequality, we arrive at∣∣ ∫ Γ1 uz′dΓ ∣∣ ≤ α1k 2 1 a0 a(u(t), u(t)) + 1 4α1 ‖z′‖2Γ1 , (3.7)∫ Ω uρ(u′)dx ≤ α1k 2 0 a0 a(u(t), u(t)) + 1 4α1 ∫ Ω ρ2(u′)dx, (3.8) − ∫ Γ1 zu′dΓ ≤ − d dt ∫ Γ1 uzdΓ + α1k 2 1 a0 a(u(t), u(t)) + 1 4α1 ‖z′‖2Γ1 , (3.9)∫ Γ1 qzz′dΓ ≤ α2k 2 2q 2 1‖∇τz‖2Γ1 + 1 4α2 ‖z′‖2Γ1 . (3.10) Substituting (3.6)–(3.10) into (3.5) and taking λ = l 1− l , α1 = a0l 4(2k2 1 + k2 0) , α2 = p2 2k2 2q 2 1 , we obtain (3.4) with C1 = f1 + 1 2α1 + 1 4α2 . This completes the proof. � Lemma 3.3. Suppose that assumptions (A1)–(A4), (2.4) and (2.5) hold, then there exist two positive constants C2, C3 such that the functional φ(t) satisfies φ′(t) ≤ ( µ− ∫ t 0 g(τ)dτ ) ‖u′‖2 + µ(1 + 2(1− l)2)a(u(t), u(t)) + ‖z′‖2Γ1 +C2(1− l)(g � u)(t) + µ ∫ Ω ρ2(u′)dx− C3(g′ � u)(t). (3.11) 8 J. HAO, M. LV EJDE-2020/95 Proof. Differentiating φ and using (1.1), we obtain φ′(t) = ∫ Ω A∇u ∫ t 0 g(t− τ)(∇u(t)−∇u(τ)) dτ dx − ∫ Ω (∫ t 0 g(t− τ)A∇u(τ)dτ )(∫ t 0 g(t− τ)(∇u(t)−∇u(τ))dτ ) dx − ∫ Γ1 z′ ∫ t 0 g(t− τ)(u(t)− u(τ))dτdΓ + ∫ Ω ρ(u′) ∫ t 0 g(t− τ)(u(t)− u(τ)) dτ dx − ∫ Ω u′ ∫ t 0 g′(t− τ)(u(t)− u(τ)) dτ dx− ∫ t 0 g(τ)dτ‖u′‖2 := I1 + I2 + I3 + I4 + I5 − ∫ t 0 g(τ)dτ‖u′‖2. (3.12) Now, we estimate the terms on the right-hand side of (3.12). By (2.1), (2.2), (A2) and Cauchy’s inequality, we obtain for any µ > 0 |I1| ≤ µa(u(t), u(t)) + 1 4µ (1− l)(g � u)(t), |I2| ≤ 2µ(1− l)2a(u(t), u(t)) + ( 2µ+ 1 4µ ) (1− l)(g � u)(t), |I3| ≤ ‖z′‖2Γ1 + k2 1 4a0 (1− l)(g � u)(t), |I4| ≤ µ ∫ Ω ρ2(u′)dx+ k2 0 4µa0 (1− l)(g � u)(t), |I5| ≤ µ‖u′‖2 − k2 0g(0) 4µa0 (g′ � u)(t). Taking into account these estimates, (3.12) yields (3.11) with C2 = 2µ+ 1 2µ + k2 0 4µa0 + k2 1 4a0 , C3 = k2 0g(0) 4µa0 . This completes the proof. � Next we prove our main result. Proof of Theorem 2.4. For a fixed positive number t0, we define g0 := ∫ t0 0 g(τ)dτ . Since g is nonincreasing and g(0) > 0, we have ∫ t 0 g(τ)dτ ≥ g0, t ≥ t0. Then combining (A4), (2.7), (3.1), (3.4) and (3.11), we deduce that L′(t) ≤ − ( η(g0 − µ)− ε ) ‖u′‖2 − ( lε 4 − ηµ(1 + 2(1− l)2) ) a(u(t), u(t)) −(q0 − C1ε− η)‖z′‖2Γ1 − p2ε 2 ‖∇τz‖2Γ1 + (1 2 − C3η ) (g′ � u)(t) + ( ε 2l + C2η ) (1− l)(g � u)(t)− ε‖h1/2z‖2Γ1 − 1 2 g(t)a(u(t), u(t)) − ∫ Ω u′ρ(u′)dx+ ( ε 4α1 + ηµ )∫ Ω ( u′2 + ρ2(u′) ) dx. (3.13) EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 9 At this point, we choose µ > 0 such that g0 − µ > g0 2 , 4µ l ( 1 + 2(1− l)2) < g0 4 . Then, (3.13) yields L′(t) ≤ − (g0η 2 − ε ) ‖u′‖2 − l 4 ( ε− g0η 4 ) a(u(t), u(t))− p2ε 2 ‖∇τz‖2Γ1 −(q0 − C1ε− η)‖z′‖2Γ1 + ( ε 2l + C2η ) (1− l)(g � u)(t) + (1 2 − C3η ) (g′ � u)(t)− ε‖h1/2z‖2Γ1 − 1 2 g(t)a(u(t), u(t)) − ∫ Ω u′ρ(u′)dx+ ( ε 4α1 + ηµ )∫ Ω ( u′2 + ρ2(u′) ) dx. (3.14) Taking ε and η small enough such that Lemma 3.1 remains valid, we pick g0η 4 < ε < g0η 2 , q0 − C1ε− η > 0, 1 2 − C3η > 0. Hence, we have g0η 2 − ε > 0 and l 4 ( ε− g0η 4 ) > 0. Whence, it follows from (A2), (2.6), (2.7) that L′(t) ≤ −C4E(t) + C5(g � u)(t) + C6 ∫ Ω ( u′2 + ρ2(u′) ) dx (3.15) where C4 is a positive constant and C5 := ( ε 2l + C2η ) (1− l), C6 := ε 4α1 + ηµ. Multiplying (3.15) by ξ(t) and applying (A2), (2.7), we have ξ(t)L′(t) ≤ −C4ξ(t)E(t) + C5ξ(t)(g � u)(t) + C6ξ(t) ∫ Ω ( u′2 + ρ2(u′) ) dx ≤ −C4ξ(t)E(t)− C5(g′ � u)(t) + C6ξ(t) ∫ Ω ( u′2 + ρ2(u′) ) dx ≤ −C4ξ(t)E(t)− 2C5E ′(t) + C6ξ(t) ∫ Ω ( u′2 + ρ2(u′) ) dx. (3.16) Exploiting the fact that ξ is a nonincreasing continuous function and defining F (t) := ξ(t)L(t) + 2C5E(t), we see from Lemma 3.1 and (3.16) that F (t) ∼ E(t), and F ′(t) ≤ −C4ξ(t)E(t) + C6ξ(t) ∫ Ω ( u′2 + ρ2(u′) ) dx. (3.17) To obtain our desired result, we shall estimate the last term on the right-hand side of (3.17). For this purpose, we adapt the arguments in [24]. Case 1. H1 is linear on [0, ε]. Then, by (A2), (A3) and (2.7), we deduce that there exists some positive constant C7 such that F ′(t) ≤ −C4ξ(t)E(t) + C7 ∫ Ω u′ρ(u′)dx ≤ −C4ξ(t)E(t)− C7E ′(t), 10 J. HAO, M. LV EJDE-2020/95 which together with (3.17) give, as J(t) := F (t) + C7E(t) and J ′(t) ≤ −C4ξ(t)E(t). Hence, using that J(t) ∼ E(t), we easily obtain for t ≥ t0, E(t) ≤ C8e −C4 ∫ t 0 ξ(s)ds := C8H −1 ( C4 ∫ t 0 ξ(s)ds ) . (3.18) Case 2. H ′1(0) = 0 and H ′′1 > 0 on (0, ε]. In this case, we choose 0 < ε1 < ε such that sρ(s) ≤ min{ε,H1(s)}, s ≤ ε1, Then, it is easy to show that c1|s| ≤ |ρ(s) ≤ c2|s| if |s| ≥ ε1, s2 + ρ2(s) ≤ H−1 1 (sρ(s)) if |s| ≤ ε1. Next we consider a partition of Ω, Ω1 = {x ∈ Ω : |u′| ≤ ε1} and Ω2 = {x ∈ Ω : |u′| > ε1}. To estimate the last term on the right side of (3.17), we set S(t) := 1 |Ω1| ∫ Ω1 u′ρ(u′)dx. By Jensen’s inequality, we obtain H−1 1 (S(t)) ≥ C9 ∫ Ω1 H−1 1 (u′ρ(u′))dx. From this and (2.7), we have ξ(t) ∫ Ω (u′2 + ρ2(u′))dx = ξ(t) ∫ Ω1 (u′2 + ρ2(u′))dx+ ξ(t) ∫ Ω2 (u′2 + ρ2(u′))dx ≤ ξ(t) ∫ Ω1 H−1 1 ( u′ρ(u′))dx− C10E ′(t) ≤ 1 C9 ξ(t)H−1 1 (S(t))− C10E ′(t). Therefore, (3.17) yields F ′(t) ≤ −C4ξ(t)E(t) + C11ξ(t)H −1 1 (S(t))− C6C10E ′(t), (3.19) which gives R′0(t) ≤ −C4ξ(t)E(t) + C11ξ(t)H −1 1 (S(t)), (3.20) where R0(t) := F (t) + C6C10E(t), and R0(t) ∼ E(t) because of Lemma 3.1. Now, for ε0 < ε and c0 > 0, we define R1(t) := H ′1 ( ε0 E(t) E(0) ) R0(t) + c0E(t). Then, it is easy to show that for a1, a2 > 0, a1R1(t) ≤ E(t) ≤ a2R1(t). Recalling that E′(t) ≤ 0, H ′1(r) > 0, H ′′1 (r) > 0 on (0, ε], and using (3.20), we obtain R′1(t) = ε0 E′(t) E(0) H ′′1 ( ε0 E(t) E(0) ) R0(t) +H ′1 ( ε0 E(t) E(0) ) R′0(t) + c0E ′(t) EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 11 ≤ −C4ξ(t)E(t)H ′1 ( ε0 E(t) E(0) ) + C11ξ(t)H ′ 1 ( ε0 E(t) E(0) ) H−1 1 (S(t)) + c0E ′(t). On the other hand, thanks to the argument given in [1], we have H∗1 (s) = s(H ′1)−1(s)−H1((H ′1)−1(s)), if s ∈ (0, H ′1(ε)], where H∗1 is the Legendre transform of the convex function H1 defined by H∗1 (s) := sup t∈R+ (st−H1(t)). Then, the fact that H ′1(0) = 0 and H, (H ′1)−1 are increasing functions yields H∗1 (s) ≤ s(H ′1)−1(s), if s ∈ (0, H ′1(ε)]. (3.21) Using Young’s inequality, we obtain AB ≤ H∗1 (A) +H1(B) if A ∈ (0, H ′1(ε)], B ∈ (0, ε]. (3.22) Taking A = H ′1(ε0 E(t) E(0) ) and B = H−1 1 (S(t)), from (2.7), (3.20), (3.21) and (3.22) it follows that R′1(t) ≤ −C4ξ(t)E(t)H ′1 ( ε0 E(t) E(0) ) + C11ξ(t)H ∗ 1 ( H ′1 ( ε0 E(t) E(0) )) + C11ξ(t)S(t) + c0E ′(t) ≤ −C4ξ(t)E(t)H ′1 ( ε0 E(t) E(0) ) + C11ε0ξ(t) E(t) E(0) H ′1 ( ε0 E(t) E(0) ) − C12E ′(t) + c0E ′(t), where C12 := C11ξ(0) |Ω1| . Choosing ε0 small enough such that C13 := C4E(0)− C11ε0 > 0 and taking c0 > C12, we arrive at R′1(t) ≤ −C13ξ(t) E(t) E(0) H ′1 ( ε0 E(t) E(0) ) = −C13ξ(t)H0 (E(t) E(0) ) , (3.23) where H0(r) = rH ′1(ε0r). By the strict convexity of H1 on (0, ε], we can see that H ′0(t) and H0(t) > 0 on (0, 1]. Thus, setting R(t) := a1R1(t) E(0) , which satisfies R(t) ∼ E(t), and using (3.23), we have R′(t) ≤ −a1C13 E(0) ξ(t)H0 (E(t) E(0) ) = −µ2ξ(t)H0(R(t)). A simple integration over (t0, t) yields R(t) ≤ H−1(µ2 ∫ t t0 ξ(s)ds+ µ3), t ≥ t0. (3.24) Combining (3.18) and (3.24), we obtain the desired result. The proof is complet. � Acknowledgements. The authors are very grateful to Dr. Zhaosheng Feng and the referees for their careful reading and valuable comments and suggestions, which improved this article. This work is partially supported by the NNSF of China (11871315, 61374089), and by the Natural Science Foundation of Shanxi Province of China (201801D121003, 201901D111021). 12 J. HAO, M. LV EJDE-2020/95 References [1] V. I. Arnold; Mathematical methods of classical mechanics. Springer-Verlag, New York (1989). [2] J. T. Beale; Spectral properties of an acoustic boundary condition, Indiana University Math- ematics Journal, 25(9) (1976), 895-917. [3] J. T. Beale, S. I. Rosencrans; Acoustic boundary conditions, Bulletin of the American Math- ematical Society, 80(6) (1974), 1276-1278. [4] Y. Boukhatem, B. Benabderrahmane; Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions, Nonlinear Analysis: Theory, Methods & Applications, 97 (2014), 191-209. [5] Y. Boukhatem, B. Benabderrahmane; Polynomial decay and blow up of solutions for variable coefficients viscoelastic wave equation with acoustic boundary conditions, Acta Mathematica Sinica, English Series, 32(2) (2016), 153-174. [6] A. T. Cousin, C. L. Frota, N. A. Larkin; On a system of Klein-Gordon type equations with acoustic boundary conditions, Journal of Mathematical Analysis and Applications, 293(1) (2004), 293-309. [7] S. Frigeri; Attractors for semilinear damped wave equations with an acoustic boundary con- dition, Journal of Evolution Equations, 10(1) (2010), 29-58. [8] C. L. Frota, J. A. Goldstein; Some nonlinear wave equations with acoustic boundary condi- tions, Journal of Differential Equations, 164(1) (2000), 92-109. [9] C. L. Frota, L. A. Medeiros, A. Vicente; Wave equation in domains with non-locally reacting boundary, Differential and Integral Equations, 24(11-12) (2011), 1001-1020. [10] C. L. Frota, L. A. Medeiros, A. Vicente; A mixed problem for semilinear wave equations with acoustic boundary conditions in domains with non-locally reacting boundary, Electronic Journal of Differential Equations, 243 (2014), 1-14. [11] Y. Gao, J. Liang, T. J. Xiao; A new method to obtain uniform decay rates for multidimen- sional wave equations with nonlinear acoustic boundary conditions, SIAM Journal on Control and Optimization, 56(2) (2018), 1303-1320. [12] T. G. Ha; General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Applied Mathematics Letters, 60 (2016), 43-49. [13] T. G. Ha; Energy decay for the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary, Applied Mathematics Letters, 76 (2018), 201-207. [14] J. H. Hao, W. H. He; Energy decay of variable-coefficient wave equation with acoustic bound- ary conditions and delay, Applicable Analysis, 98(3) (2019), 499-515. [15] J. H. Hao, W. H. He; Energy decay of variable-coefficient wave equation with nonlinear acoustic boundary conditions and source term, Mathematical Methods in the Applied Sci- ences, 42(6) (2019), 2109-2123. [16] J. Ĺımaco, H. R. Clark, C. L. Frota, L. A. Medeiros; On an evolution equation with acoustic boundary conditions, Mathematical Methods in the Applied Sciences, 34(16) (2011), 2047- 2059. [17] W. J. Liu; Arbitrary rate of decay for a viscoelastic equation with acoustic boundary condi- tions, Applied Mathematics Letters, 38 (2014), 155-161. [18] Y. X. Liu; Polynomial decay of a variable coefficient wave equation with an acoustic un- damped boundary condition, Journal of Mathematical Analysis and Applications, 479(2) (2019), 1641-1652. [19] W. J. Liu, K. W. Chen; Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions, Zeitschrift für angewandte Mathematik und Physik, 66(4) (2015), 1595-1614. [20] W. J. Liu, Y. Sun; General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions, Zeitschrift für angewandte Mathematik und Physik, 65(1) (2014), 125- 134. [21] S. A. Messaoudi; General decay of the solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Analysis: Theory, Methods & Applications, 69(8) (2008), 2589- 2598. EJDE-2020/95 ENERGY DECAY FOR WAVE EQUATION 13 [22] S. A. Messaoudi, M. I. Mustafa; On convexity for energy decay rates of a viscoelastic equation with boundary feedback, Nonlinear Analysis: Theory, Methods Applications, 72(9-10) (2010), 3602-3611. [23] P. M. Morse, K. U. Ingard; Theoretical Acoustics. McGraw-Hill, New York, 1968. [24] J. Y. Park, S. H. Park; General decay for quasilinear viscoelastic equations with nonlinear weak damping, Journal of Mathematical Physics, 50(8) (2009), 083505. [25] J. Y. Park, S. H. Park; Decay rate estimates for wave equations of memory type with acoustic boundary conditions, Nonlinear Analysis: Theory, Methods Applications, 74(3) (2011), 993- 998. [26] A. Vicente, C. L. Frota; Uniform stabilization of wave equation with localized damping and acoustic boundary condition, Journal of Mathematical Analysis and Applications, 436(2) (2016), 639-660. [27] F. Wang, J. H. Hao; Decay of energy for viscoelastic wave equations with Balakrishnan-Taylor damping and memories, Electronic Journal of Differential Equations, 2020(42) (2020), 1-19. [28] P. F. Yao; On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM Journal on Control and Optimization, 37(5) (1999), 1568-1599. Jianghao Hao (corresponding author) School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China Email address: hjhao@sxu.edu.cn Mengxian Lv School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China Email address: 1550432308@qq.com 1. Introduction 2. Preliminaries 3. Decay estimate Acknowledgements References