OPTIMAL ENERGY DECAY RATES

. In this article, we consider the viscoelastic wave equation u tt − ∆ u +


Introduction
In this article we consider the problem Here Ω is a bounded domain of R n (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 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 u tt − ∆u + α|u t | m(x)−1 u t = 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 where ξ is a non-increasing differentiable function, and established a more general decay result. Also, a condition of the form 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 where H is increasing and convex without any additional constraints, and established 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 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).
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 m 1 ≥ 2 and m 1 < 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].
Embedding Property: Let Ω be a bounded domain in R n with a smooth boundary ∂Ω. Assume that p, q ∈ C(Ω) such that and q(x) < p * (x) in Ω with then there is a continuous and compact embedding W 1,p(·) (Ω) → L q(·) (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 : is a C 1 function which is linear or strictly increasing and strictly convex 2 function on (0, 2m 1 −2 is strictly increasing and strictly convex when 1 < m 1 < 2. We will be using H(t) = min{H 1 (t), H 2 (t)} and r ≤ r 1 is small enough so that either H(t) = H 1 (t) or H(t) = H 2 (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 [d 1 , d 2 ], y : Ω → [d 1 , d 2 ] and w are integrable functions on Ω, w(x) ≥ 0, and Ω w(x) dx = d 3 > 0, then Jensen's inequality states and H * satisfies the generalized Young inequality At the end of this section, we state, without proof, the following existence and regularity result. 11,30]). Let (u 0 , u 1 ) ∈ H 1 0 (Ω) × L 2 (Ω) be given. If (A1) holds, then problem (1.1) has a unique global (weak) solution

Technical Lemmas
We introduce the energy functional 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 Proof. By multiplying equation (1.1) by u t and integrating over Ω, using integration by parts, hypothesis (A) and some manipulations, we obtain (3.1).
Lemma 3.3. The functional K 2 defined by satisfies, for any 0 < δ < 1, the estimate (3.7) Proof. By using (1.1) and integrating by parts, we have Using Young's and Poincaré's inequalities and similar calculations as in (3.5), we obtain On Ω * * , we use a similar argument as in Lemma 3.2 to obtain Combining the above estimates, (3.7) is established.
Next, we use the functional where f (t) = ∞ t g(s)ds.
Lemma 3.5. The functional L defined by for suitable choice of N, N 1 , N 2 > 0 and for all t ≥ t 1 , satisfies and which means that, for some constants a 1 , a 2 > 0, Proof. Let g 1 = t1 0 g(s)ds > 0 for some fixed t 1 > 0. By combining (3.1), (3.3), (3.7), recalling that g = (αg − h), and taking δ = 1/(8cN 2 ), we obtain that for all t ≥ t 1 , Now we choose N 1 large enough so that and N 2 large enough so that As δ now is fixed and C δ (x) is bounded, we have Next, as , it is easy to show, using the Lebesgue dominated convergence theorem, that Hence, there is 0 < α 0 < 1 such that if α < α 0 , then Let us choose N large enough and choose α satisfying So, we arrive at On the other hand, we find that Therefore, we can choose N even larger (if needed) so that (3.11) is satisfied.
Conclusions. In this paper, the important issue of stabilization of the wave equation 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, which is more useful from the physical point of view and needed in several applications. Our results combine the generality and optimality and improve earlier related results in the literature. We provided some numerical examples of exponential, 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.