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Abstract and Applied Analysis
Volume 2013, Article ID 484596, 7 pages
http://dx.doi.org/10.1155/2013/484596
Research Article

Exponential Decay for Nonlinear von Kármán Equations with Memory

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 5 September 2013; Accepted 8 November 2013

Academic Editor: Valery Y. Glizer

Copyright © 2013 Jum-Ran Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the nonlinear von Kármán equations with memory term. We show the exponential decay result of solutions. Our result is established without imposing the usual relation between and its derivative. This result improves on earlier ones concerning the exponential decay.

1. Introduction

In this paper we consider the exponential decay rate of solutions for the nonlinear von Kármán equations with memory term: and the boundary conditions where is an open bounded set of , with a sufficiently smooth boundary . Here, and are closed and disjoint. The constants . Let us denote by the external unit normal to and by the corresponding unit tangent vector. The von Kármán bracket is given by

Here, we are denoting by , the following differential operators: where and are given by and the constant represents Poisson’s ratio.

This system describes the transversal displacement and the Airy stress function of a vibrating plate. The dissipation in (1) is due to the term , where is positive real function and the convolution product is given by . A material whose contained term is is called viscoelastic and is said to be “endowed with long-range memory” since the stress at any instant depends on the complete history of strain that the material has undergone.

Problems related to are interesting not only from the point of view of PDE general theory, but also due to its applications in Mechanics. For instance, when the material density, , is equal to 1, (8) describes the extensional vibrations of thin rods; see Love [1] for the physical details. When the material density is not constant, we are dealing with a thin rod which possesses a rigid surface and whose interior is somehow permissive to slight deformations such that the material density varies according to the velocity.

On the other hand, the problem of stability of the solutions to the following wave equation with memory was studied by many authors [26]: Cavalcanti et al. [2] showed an exponential and polynomial decay for the viscoelastic wave equation (9) with under the usual conditions for some . Han and Wang [3] proved the uniform decay for the nonlinear viscoelastic equation under condition where . Park and Kang [7] studied the uniform decay for a nonlinear viscoelastic problem with damping. They obtained the exponential decay estimate under condition (11). Later, this assumption was relaxed by several authors. Messaoudi and Tatar [6] investigated exponential and polynomial decay for a quasilinear viscoelastic equation under condition on such as where , by choosing a suitable perturbed energy. Liu [5] showed exponential and polynomial decay for the system of two coupled quasilinear viscoelastic equation, under condition (12). Messaoudi and Tatar [8] proved the exponential decay rate for a quasilinear viscoelastic equation under the conditions They improved some earlier results concerning the exponential decay. Han and Wang [4] studied the general decay rate for the nonlinear viscoelastic equations under the more general conditions on such as When , the problem of stability of the solutions to the viscoelastic system with memory has been studied by many authors. In [9, 10], the authors proved exponential and polynomial decay for the viscoelastic wave equation under conditions (10). Berrimi and Messaoudi [11] studied exponential and polynomial decay rates under condition (12). Messaoudi [12] investigated the general decay rate for the viscoelastic equations under general conditions (14). Guesmia and Messaoudi [13] obtained general stability for the Timoshenko system under weaker condition on such as where is a nonincreasing and positive function. As for problem of stability of the solutions to a viscoelastic system under condition (15), we also refer the reader to [1416] and references therein. These general decay estimates extended and improved on some earlier results—exponential or polynomial decay rates.

The problem of stability of the solutions to a von Kármán system with dissipative effects has been studied by several authors. For example, in [17, 18] the authors studied the von Kármán equation in the presence of thermal effects. In [1923] the authors considered the von Kármán system with frictional dissipations effective in the boundary. It is shown in these works that these dissipations produce uniform rate of decay of the solution when goes to infinity. Rivera and Menzala [24] and Rivera et al. [25] studied the stability of the solutions to a von Kármán system for viscoelastic plates with memory and boundary memory conditions. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation function. Later, Santos and Soufyane [26] generalized the decay result of [24]. Raposo and Santos [27] considered the general decay of the solutions to a von Kármán plate model (1)–(4) for . They showed that the energy decays with a similar rate of decay of the relaxation function, which is not necessarily decaying in a polynomial or exponential fashion. Kang [28] investigated the general decay of the solution to a von Kármán system with memory and boundary damping. Recently, Kang [29] proved that solutions for a von Kármán plate with memory decay exponentially to zero as time goes to infinity in case for all provided that for some .

In this paper, we establish an exponential decay of the solutions to the nonlinear von Kármán plate model (1)–(4) without assumption (15), which is the usual relation between and its derivative. Instead of (15), we require the function to have sufficiently small -norms on for some . This result improves on earlier ones concerning the exponential decay of the solutions to the von Kármán equations.

The organization of this paper is as follows. In Section 2, we give some notations and introduce the relative results of Airy stress function and von Kármán bracket. In Section 3, we prove that the energy decreases exponentially. The construction of the Lyapunov function is inspired in multiplier techniques that was used in [8].

2. Preliminaries

In this section, we present some material needed in the proof of our result and state the main result. Throughout this paper we denote and define

For a Banach space , denotes the norm of . For simplicity, we denote by . We define for all

A simple calculation, based on the integration by parts formula, yields where the bilinear symmetric form is given by where . Since , we know that is equivalent to the norm; that is, where and are generic positive constants. This and Sobolev embedding theorem imply that for some positive constants and

We establish the following hypotheses on the relaxation function (see [8]). The relaxation function is nonincreasing function satisfying

To simplify calculation in our analysis, we introduce the following notation:

From the symmetry of , we have that, for any ,

Now, we introduce the relative results of the Airy stress function and von Kármán bracket .

Lemma 1 (see [30]). Let be functions in and in , where is an open bounded and connected set of with regular boundary. Then,

Lemma 2 (see [20, 31]). If , then and satisfies

The energy of problem (1)–(4) is given by

The existence of solutions can be proved by the Faedo-Galerkin method; see [2, 9].

Theorem 3. Assume that the kernel is a positive continuous function satisfying (23). Let . Then, the system (1)–(4) has a unique weak solution such that

3. Exponential Decay of the Energy

In this section we will prove the exponential decay rates. To demonstrate the stability of the system (1)–(4), the lemmas below are essential. The following result shows the dissipative property of the system (1)–(4). Multiplying (1) by , we get the identity

Define the modified energy by and applying (26) to (31), we have

This implies that is nonincreasing, and one easily sees that

Therefore, it is enough to obtain the desired decay for the modified energy , which will be done below. The key point for showing our desired result is finding a Lyapunov functional which is equivalent to . First, we introduce three functionals and establish several lemmas. So, let with

We define the modified energy by for some positive constants is to be specified later.

Lemma 4. Assume that satisfies (23) and (24). For large enough, there exist and such that

Proof. From Young inequality, we deduce
Considering the embedding and taking (22) into account, it holds that where comes from the inequality for all . On the other hand, by Young inequality, Hölder inequality and (22) can be estimated as
Thus, from (40) and (41) we obtain where is a positive constant depending on , and . Choosing large, we complete the proof of Lemma 4.

Lemma 5. For each and sufficiently large , there exists positive constant such that

Proof. By differentiating and using Young inequality, we get where , and . Using (1)–(4), we have
We use the following inequality: then we obtain where . Similarly we deduce
Now, we estimate the terms in the right hand side of (48). The Young and Hölder inequalities and (22) give that where . From Lemmas 1 and 2 and (22), we obtain
Summarizing these estimates with (48), we deduce that
Since is continuous and positive, for any we have
Thus, making use of (52) and combining (33), (37), (44), (47), and (51), we obtain
We first take and so small that respectively. And then, we choose and so small that respectively. We then pick large enough so that
Finally, taking large enough and by (53), we conclude that for some .

Our main result reads as follows.

Theorem 6. Suppose that satisfies (23) and (24). Then, for each , there exist two positive constants and such that

Proof. From (38) and (43), we have
Integrating this over , we obtain with . Consequently, (34), (38), and (60) yield the result in Theorem 6.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2012R1A1A3011630).

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