The Rate at Which the Energy of Solutions for a Class of -Laplacian Wave Equation Decays

Abstract

We will investigate the decay estimate of the energy of the global solutions to the p-Laplacian wave equation with dissipation of the form under suitable assumptions on the positive function . For this end we use the multiplier method combined with nonlinear integral inequalities given by Martinez; the proof is based on the construction of a special weight function that depends on the behavior of .

1. Introduction

In this paper we are concerned with the energy decay rate of the -Laplacian type wave equation of the form where is a bounded domain in with smooth boundary , , are real numbers, and is a positive function satisfying some conditions to be specified later.

Problem can be considered as a system describing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Voigt model. The bibliography of works in this direction is so long. We mention, for instance, the works of Andrews [1], Andrews and Ball [2], Ang and Dinh [3], and Kawashima and Shibata [4].

Existence of global solution and the decay property of the energy for the wave equation related to the problem have been investigated by many authors through various approaches (see [5–8]).

In [5], Benaissa and Messaoudi considered the following problem: where , . They showed that, for suitably chosen initial data, the problem has a global weak solution, which decays exponentially even if . Further they proved the global existence by using the potential well theory introduced by Sattinger [9].

Similar results have been established by Ye [7, 10]. In these works the author used the Faedo-Galerkin approximation together with compactness criteria and difference inequality introduced by Nakao [11].

More related studies of the nonlinear -Laplacian wave equation type with damping term can be found in the papers [6, 8, 12, 13].

In [14], with considering instead of the damping term , we have obtained global existence result by using the argument in [15] combined with the concepts of so-called stable sets due to Sattinger [9]. We have also shown the asymptotic behavior of global solutions through the use of the integral inequality given by Komornik [16].

The purpose of this paper is to give an energy decay estimate of the solution of problem . Our proof is based on the multiplier method combined with nonlinear integral inequalities given by Martinez [17].

The plan of the paper is as follows. In the next section we present some assumptions, technical lemmas, and main result. Then in Section 3 we are devoted to the proof of decay estimate.

For simplicity of notation, we denote by the Lebesgue space norm. In particular denotes and the inner product of . We also write equivalent norm instead of norm As usual, we write, respectively, and instead of and . Furthermore, throughout this paper the functions considered are all real valued.

2. Preliminaries and Main Result

First assume that is a nonincreasing positive function of class on , satisfying Let us define the energy equality associated with the solution of the problem by the following formula: for and .

We state, without proof, a global existence result for the problem . For more details we refer the reader to [6].

Theorem 1. Let and assume that . Then, for any , the problem has a unique strong solution on in the class

We now present the following well-known lemmas which will be needed later.

Lemma 2 (energy identity). Let be a global solution to the problem on . Then one has for all

Lemma 3 (Sobolev-Poincaré inequality). Let and with or . Then there is a constant such that The case gives the known Poincaré’s inequality.

Before stating our main result, we introduce the following lemma which plays an important part in studying the decay estimate of energy associated with the solution of the problem .

Lemma 4 (see [17]). Let be a nonincreasing function and an increasing function such that Assume that there exist and such that Then one has

Now we are in position to state and prove our main result.

Theorem 5. Let and Suppose that (2) holds. Then the solution of the problem satisfies the following energy decay estimates.(1)If , then there exists a positive constant such that (2)If , then there exists a positive constant depending continuously on such that

3. Proof of Main Result

From now on, we denote by various positive constants depending on the known constants and they may be different at each appearance.

Multiplying by on both sides of the first equation of and integrating over , where is a function satisfying all the hypotheses of Lemma 4 and , we obtain that By an integration by parts we see that Hence from the definition of energy and a simple argument we can obtain Now we must estimate both sides of (14) to arrive at a similar inequality as (8).

Define It is clear that is a nondecreasing function of class on and hypothesis (2) ensures that Since is nonincreasing, is a bounded nonnegative function on (we denote by its maximum) and, using the definition of energy, Cauchy-Schwartz inequality, and Sobolev-Poincaré inequality, we have where the above estimate follows from the fact that Again, exploiting Cauchy-Schwartz inequality, Sobolev-Poincaré inequality, the definition of energy, and (18), we obtain where the fact that is nonincreasing is used.

Furthermore, by using Lemma 2, we have We then estimate the last term in (14) as follows: which implies that Using Hölder’s and Sobolev-Poincaré’s inequalities, we see that We also have This gives Further, by Young inequality, we have for We choose such that Thus

Combining estimates (17)–(26), (14) becomes where , , and are different positive constants independent of .

Letting , this yields the following estimate: and we conclude from Lemma 4 that It is clear that, for , we have and the energy associated with the solution of the problem satisfies the decay property in (10).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.