In this paper we consider a plate equation with internal feedback and viscoelastic damping localized on a part of the boundary. Without imposing restrictive assumptions on the time-dependent 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.
In this paper we are concerned with the following problem
The uniform stabilization of Kirchhoff plates with linear or nonlinear internal feedback, with , was investigated by several authors. In Ammari and Tucsnak , Cavalcanti et al. , Guzman and Tucsnak , Komornik , Pazoto et al. , and Vasconcellos and Teixeira , it was proved that if h satisfies
On the other hand, when the unique damping mechanism is given by memory conditions, we refer to Lagnese  and Rivera et al.  who considered internal viscoelastic damping and proved that the energy decays exponentially if the relaxation function g decays exponentially and polynomially if g decays polynomially. The same results were obtained by Alabau-Boussouira et al.  for a more general abstract equation. For boundary viscoelastic damping, if is the resolvent kernel of for , Santos and Junior  showed that the energy decays exponentially (polynomially), provided the resolvent kernels also decay exponentially (polynomially). In Rivera et al. [33,34] investigated a class of abstract viscoelastic systems of the form
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 question when treating (1.6), for , in . They considered relaxation functions satisfying
The interaction between viscoelastic and frictional dampings was considered by several authors. Cavalcanti and Oquendo  looked into wave equation of the form
Our aim in this work is to investigate (1.1)–(1.5) with both weak frictional damping and boundary viscoelastic damping. We obtain a general relation between the decay rate for the energy (when t goes to infinity) and the functions , θ, and h using resolvent kernels of general-type decay and without imposing any growth assumption near the origin on h. 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 . 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  and used by Liu and Zuazua  and Alabau-Boussouira . 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.
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 hypothesis
Ω is a bounded domain of with a smooth boundary , where and are closed and disjoint, with , ν is the unit outward normal to is the unit tangent positively oriented on , and there exists a fixed point such that, for , on and on .
Hypothesis (A1) implies that there exist constants and R such that
, for , are functions such that
is a nondecreasing function and there exist constants such that
is a nonincreasing function.
Let us define the bilinear form as follows
Let u and v be functions in and . Then we have
Our main stability result is the following
1. Using the properties of H, one can show that the function is strictly decreasing and convex on , with . Therefore, Theorem 2.1 ensures
2. Hypothesis (A3) implies that , for all .
3. The condition (A3), with and , was introduced and employed by Lasiecka and Tataru  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 frictional damping, modulated by a time dependent coefficient , and boundary 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 the resolvent kernels and , h, and θ.
4. The usual exponential and polynomial decay rate estimates, already proved for , , are special cases of our result. We will provide a “simpler” proof for these special cases.
5. The condition , , assumes when and when . Our result allows resolvent kernels whose derivatives are not necessarily of exponential or polynomial decay. For instance, if
6. The well-known Jensen’s inequality will be of essential use in establishing our main result. If F is a convex function on , and j are integrable functions on Ω, , and , then Jensen’s inequality states that
7. Since and is nonnegative and nonincreasing, then we can easily deduce that . Similarly, assuming the existence of the limit, we find that . Hence, there is large enough such that and
8. If different functions , , and have the properties mentioned in (A2) and (A3) such that , , and , then there is small enough so that, say, on the interval . Thus, the function satisfies both (A2) and (A3), .
3.Proof of the main result
In this section we prove Theorem 2.1. For this purpose, we establish several lemmas.
Now we are going to construct a Lyapunov functional equivalent to E, with which we can show the desired result.
Direct computations, taking in (2.6), we get
Proof of Theorem 2.1.
For , we define
(I) and : This means, using Holder’s inequality, that
Case 1. : Estimate (3.14) yields
Case 2. : One can easily show that for any and . Using this fact (3.1), and the trace theory and choosing even larger if needed, we deduce that, for all ,
(II) The general case: We define by
Since , then, using the strict convexity of on , we find that , on . Thus, with
Moreover, if , then , and so, by (2.7), . Then, we have
The author thanks University of Sharjah for its continuous support.
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