Viscoelastic plate equation with boundary feedback

In this paper we consider a viscoelastic plate equation with a nonlinear weakly dissipative feedback localized on a part of the boundary. Without imposing restrictive assumptions on the boundary frictional damping, we establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalizes previous results existing in the literature.

Here Ω is a bounded domain of IR 2 with a smooth boundary ∂Ω = Γ 0 ∪Γ 1 , where Γ 0 and Γ 1 are closed and disjoint with meas(Γ 0 ) > 0, and n = (ν 1 , ν 2 ) is the unit outward normal to ∂Ω, η = (−ν 2 , ν 1 ) is the unit tangent positively oriented on ∂Ω, the integral term in (1.1) 1 is the memory responsible for the viscoelastic damping where g is a positive function called the relaxation function, θ is a time dependent coefficient of the frictional damping, and h is a specific function. We are denoting by β 1 , β 2 the following differential operators: where B 1 u = 2ν 1 ν 2 u xy − ν 2 1 u yy − ν 2 2 u xx , B 2 u = ν 2 1 − ν 2 2 u xy + ν 1 ν 2 (u yy − u xx ) and µ ∈ 0, 1 2 represents the Poisson coefficient. This system describes the transversal displacement u = u(x, y, t) of a thin vibrating plate supjected to internal viscoelastic damping and time-dependent boundary frictional damping. u tt (t) + Au(t) − (g * A β u)(t) = 0, (1.2) where A is a strictly positive, self-adjoint operator with D(A) a subset of a Hilbert space and * denotes the convolution product in the variable t. The authors showed that solutions for (1.2) , when 0 < β < 1, decay polynomially even if the kernel g decays exponantially. While, in the case β = 1, the solution energy decays at the same decay rate as the relaxation function. Then, a natural question was raised: how does the energy behave as the kernel function does not necessarily decay polynomially or exponentially? Han and Wang gave an answer to the above quastion when treating (1.2), for β = 1, in [15]. They considerd relaxation functions satisfing where ξ : IR + → IR + is a nonincreasing differentiable function with for some constant k and showed that the rate of the decay of the energy is exactly the rate of decay of g which is not necessarily of polynomial or exponential decay type. These conditions (1.3) and (1.4) on g where first used by Messaoudi [33] and [34] in studying a viscoleastic wave equation. After that, Messaoudi and Mustafa [35] and Mustafa and Messaoudi [41] [19] to treat plate equation with memorytype boundary conditions and by Ferreira and Messaoudi [11] to treat a nonlinear viscoelastic plate equation with a − → p (x, t)-Laplacian operator. Another step forward is the work of Alabau-Boussouira and Cannarsa [1] who considered wave equation with memory whose relaxation function is satisfying where χ is a non-negative function, with χ(0) = χ (0) = 0, and χ is strictly increasing and strictly convex on (0, k 0 ], for some k 0 > 0. They also required that and proved an energy decay result . In addition to these assumptions, if then, in this case, an explicit rate of decay is given. Here, a new theorem was announced which was applied to some new examples giving optimal decay rates. These assumptions imposed on χ do not appear intrinsic to the result claimed, but rather to the method based on weighted energy inequalities with the use of convexity. Later on, Mustafa and Messaoudi [43] similarly used (1.5) and provided another variant of that approach which was able to remove some of the constraints imposed in [1] and obtain an explicit and general decay rate formula. The interaction between viscoelastic and frictional dampings was considered by several authors. Cavalcanti and Oquendo [10] looked into wave equation of the form and established exponential stability for g decaying exponentially and h linear and polynomial stability for g decaying polynomially and h having a polynomial growth near zero. For g of general-type decay and h having no restrictive growth assumption near the origin, Cavalcanti et al. [8] established decay rate estimates. Using (1.3), with time dependent coefficient and a (x) = b (x) = 1, Liu [29] proved a general decay result. Similarly, Guesmia and Messaoudi [12] studied Timoshenko systems with frictional versus viscoelastic damping and Messaoudi and Mustafa [36] studied viscoelastic wave equation with boundary feedback and obtained energy decay estimates. Once again, Kang [18] imposed the condition (1.3) on the relaxation function for an internal viscoelastic damping in a von Karman plate model which is also subject to a boundary frictional damping and they proved a general stability result.
Our aim in this work is to investigate (1.1) with both weak boundary frictional damping and internal viscoelastic damping. We obtain a general relation between the decay rate for the energy (when t goes to infinity) and the functions g, θ, and h without imposing any growth assumption near the origin on h and strongly weakening the usual assumptions on g. The result of this paper generalizes previous related results where it allows a larger class of functions g and h, from which the energy decay rates are not necessarily of exponential or polynomial types and takes into account the effect of a time dependent coefficient θ(t). The proof is based on the multiplier method and makes use of some properties of convex functions including the use of the general Young's inequality and Jensen's inequality. These convexity arguments were introduced by Lasiecka and Tataru [27] and used by Liu and Zuazua [30] and Alabau-Boussouira [3]. The paper is organized as follows. In section 2, we present some notation and material needed for our work. Some technical lemmas and the proof of our main result will be given in section 3.

2.
Preliminaries. We use the standard Lebesgue and Sobolev spaces with their usual scalar products and norms. Throughout this paper, c is used to denote a generic positive constant. We first consider the following hypotheses (A1) θ : and there exists a positive function H ∈ C 1 (IR + ) and H is linear or strictly increasing and strictly convex C 2 function on (0, r], r < 1,with In the sequel we assume that system (1.1) has a unique solution This result can be proved, for initial data in suitable function spaces, using standard arguments such as the Galerkin method (see [45]).
Let us define the bilinear form a(., .) as follows and, as measΓ 0 > 0, we know that a(u, u) is an equivalent norm on W ; that is, for some positive constants α and β, Young's inequality gives, for any ε > 0, Also, we mention the following useful identity, see [23], Now, we introduce the energy functional Our main stability result is the following Theorem 2.1. Assume that (A1)-(A3) hold. Then there exist positive constants k 1 , k 2 , k 3 and ε 0 such that the solution of (1.1) satisfies where provided that D is a positive C 1 function, with D(0) = 0, for which H 0 is strictly increasing and strictly convex C 2 function on (0, r] and where In particular, this last estimate is valid for the special case H(t) = ct p , for 1 ≤ p < 2. Hypothesis (A3) implies that sh(s) > 0, for all s = 0. 3. The condition (A3), with r = 1 and θ ≡ 1, was introduced and employed by Lasiecka and Tataru [27] in their study of the asymptotic behavior of solutions of nonlinear wave equations with nonlinear frictional boundary damping where they obtained decay estimates that depend on the solution of an explicit nonlinear ordinary differential equation. It was also shown there that the monotonicity and continuity of h guarantee the existence of the function H with the properties stated in (A3). In our present work, we study the plate equation with both boundary frictional damping, modulated by a time dependent coefficient θ(t), and viscoelastic damping. We investigate the influence of these simultaneous damping mechanisms on the decay rate of the energy and establish an explicit and general energy decay formula, depending on g, h, and θ. 4. The usual exponential and polynomial decay rate estimates, already proved for g satisfying (2.1) and H(t) = ct p , 1 ≤ p < 3/2, are special cases of our result. We will provide a "simpler" proof for these special cases. We should refer here to the references [7], [26] and [28] where the optimal polynomial decay was pushed up to p < 2.

5.
Our result allows decay rates which are not necessarily of exponential or polynomial decay. For instance, if for 0 < q < 1 and a chosen so that g satisfies (2.1), then g (t) = −H(g(t)) where, for t ∈ (0, r], r < a, which satisfies hypothesis (A2). Also, by taking D(t) = t α , (2.8) is satisfied for any α > 1. Therefore, if h satisfies (A3) with this function H, then we can use Theorem 2.1 and do some calculations (see [43]) to deduce that the energy decays at the rate One can show that this example of g does not satisfy (1.6), and so no explicit rate of decay for this case is given in [1]. 6. The well-known Jensen's inequality will be of essential use in establishing our main result. If F is a convex function on [a, b], f : Ω → [a, b] and z are integrable functions on Ω, z(x) ≥ 0, and Ω z(x)dx = k > 0, then Jensen's inequality states that 7. By (A2), we easily deduce that lim t→+∞ g(t) = 0. Similarly, assuming the existence of the limit, we find that lim t→+∞ (−g (t)) = 0. Hence, there is t 1 > 0 large enough such that g(t 1 ) > 0 and max{g(t), −g (t)} < min{r, H(r), H 0 (r)}, ∀ t ≥ t 1 . (2.10) As g is nonincreasing, g(0) > 0 and g(t 1 ) > 0, then g(t) > 0 for any t ∈ [0, t 1 ] and Therefore, since H is a positive continuous function, then for some positive constants a and b. Consequently, for all t ∈ [0, t 1 ], which gives, for some positive constant d, 8. If different functions H 1 and H 2 have the properties mentioned in (A2) and (A3) such that g (t) ≤ −H 1 (g(t)) and s 2 + h 2 (s) ≤ H −1 2 (sh(s)), then there is r < min{r 1 , r 2 } small enough so that, say, H 1 (t) ≤ H 2 (t) on the interval (0, r]. Thus, the function H(t) = H 1 (t) satisfies both (A2) and (A3), ∀ t ≥ t 1 .
3. Proof of the main result. In this section we prove Theorem 2.1. For this purpose, we establish several lemmas. Lemma 3.1. Under the assumptions (A1)-(A3), the energy functional satisfies, along the solution of (1.1), the estimate Proof. By multiplying equation (1.1) 1 by u t and integrating over Ω, using integration by parts, the boundary conditions, (2.5), hypotheses (A1)-(A3) and some manipulations, we obtain (3.1). Now we are going to construct a Lyapunov functional L equivalent to E, with which we can show the desired result.

Similarly, using (2.3), Hölder and Poincaré inequalities, and the Trace Theorem yield
By combining all the above estimates, the assertion of Lemma 3.3 is proved.