Abstract and Applied Analysis

Volume 2012 (2012), Article ID 419717, 25 pages

doi:10.1155/2012/419717

Research Article

Asymptotic Energy Estimates for Nonlinear Petrovsky Plate Model Subject to Viscoelastic Damping

Xiuli Lin and Fushan Li

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received 1 July 2012; Accepted 16 September 2012

Academic Editor: Liming Ge

Copyright © 2012 Xiuli Lin and Fushan Li. 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 Petrovsky plate model under the presence of long-time memory. Under suitable conditions, we show that the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity.

1. Introduction

Let be a bounded domain of with a sufficiently smooth boundary . We will assume that and have positive measures. The vector is the unit exterior normal and represents the tangential direction to . Here, the variable represents displacement of a plate occupying the domain . The governing equation is given by

where the notations . The positive constant is the proportional to the thickness of the plate, is relaxation function, div stands for scalar divergence of a vector field, the stress resultant is given by

and the nonlinear function is defined as for all . is a linear operator defined on with value on , the space of symmetric matrices, and defined as

for any , where is the identity matrix and denotes the trace of . Moreover, is the density of the shell, denotes the Young modulus, and is the Poisson’s ratio.

With (1.1), we associate the boundary conditions on the portion of the boundary ,

and the boundary conditions on the remaining portion of the boundary ,

where

and the boundary operators and are defined by

With (1.1), we also associate the initial conditions

where

the symbol denotes the subspace of such that there exists a constant such that the functions vanish as .

This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials, to retain a memory of their past history. From the mathematical point of view, these damping effects are modeled by integro-differential operators. Therefore, the dynamics of viscoelastic materials are of great importance and interest as they have wide applications in natural sciences. From the physical point of view, the problem (1.1) describes the position of the material particle at time , which is clamped in the portion of its boundary.

Models of Petrovsky type are of interest in applications in various areas in mathematical physics, as well as in geophysics and ocean acoustics [1, 2]. The Petrovsky type models without memory were discussed in [3, 4]. Messaoudi [3] considered the initial-boundary value problem

established an existence result for (1.10), and showed that the solution continues to exist globally if , however if and the initial energy is negative, the solution blows up in finite time. Chen and Zhou [4] proved that the solution of (1.10) blows up with positive initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing initial energy under the condition by different method.

The Petrovsky type equations with memory arouse the attention of mathematicians to study them. Alabau-Boussouira et al. [5] discussed the initial-boundary value problem of linear Petrovsky equation related to a plate model with memory,

and showed that the solution decays exponentially or polynomially as if the initial data is sufficient small. Yang [6] considered the problem in -dimensional space,

and proved that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as in the states of large initial data and small initial energy. In particular, in the case of space dimension , the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. Muñoz Rivera et al. [7] considered the initial-boundary problem for viscoelastic plate equation,

and proved that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel satisfies

then the energy decays as . On the recently related papers concerning the Petrovsky type models, the readers can see references [8–12].

In [13–15], Li et al. proved the existence uniqueness, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system, respectively. To our best knowledge, we do not find the research report on the problem (1.1) which is considered in this paper.

Motivated by the above work, we obtain the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity. The main contribution of this paper are as follows. (a) The problem considered in this paper is nonlinear equation with integral dissipation, to our knowledge this model has not been considered; (b) the hypothesis on and initial data are weaker; (c) we naturally define the energy by simple computation and only define simple auxiliary functionals to prove our result by precise priori estimates.

The outline of this paper is the following. In Section 2, we present some material needed to be proved. Section 3 contains the statement and the proof of our results.

2. Notations and Preliminaries

In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue space and Sobolev space and adopt the following notations

We denote

for any pair of tensors and . We introduce the following space

Define the bilinear form as follows

For simplicity, we denote by . For the relaxation function , we assume that

is a function satisfying

Both and exist, and

Hypotheses () assure that the viscoelastic energy (defined below) is nonincreasing and the assumption () means that the material behaves like an viscoelastic solid at (cf. [16]).

Our results are based on the following existence theorem.

Theorem 2.1. One assumes that satisfies conditions ()-(). For any initial data

subject to the compatibility conditions satisfied on the boundary ,

Then for any , there exists a unique global solution to (1.1)–(1.10) satisfying

Proof. We can use the method used in [13, 17] to show the existence and uniqueness as well as the regularity of the global solution.

At First, the energy of the system must be properly defined. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression of energy functional, using assumption (), we rewrite system (1.1) and (1.5) in the form

In order to define the energy functional, we give the following lemma.

Lemma 2.2. Let and be the functions in . Then, one has

Proof.. The definition of gives

Using Green’s formula, we see

Using

we get

Inserting (2.14) and (2.15) into (2.17) and noting

we have

we obtain the conclusion.

In order to define the energy function of the problem (1.1)–(1.8), we give the following computations. Multiplying equation (2.10) by , integrating the result over and adding Green’s formula, we get from Lemma 2.2 and (1.4) that

Clearly

By applying Lemma 2.2, the first term on the right-hand side of (2.21) equals

In the same way, the second term on the right-hand side of (2.21) equals

Therefore, from (2.21)–(2.23), we have

Note

and (by )

Consequently, we conclude form (2.24)–(2.26) that

Summing (2.20) and (2.27), using the boundary conditions (2.11) together with the symmetric of yields

Note the relation

In fact

with . Denote . Hence

which implies that

From (2.30) and (2.32), we conclude that (2.29) holds. Combining (2.28) and (2.29), we conclude that

The relation (2.33) inspires us to define energy functional as the following

From (2.33) and (2.34), we obtain the following lemma.

Lemma 2.3. Under the above notations and assumptions () and (), one has that

3. Main Results

In this section, , and denote some positive constants. Here, we need to point out that denotes the small enough different positive constant and denotes the different positive constant depending on in a different place, respectively. The main goal of this paper is to show, respectively, that the solution decays exponentially or polynomially to zero as the time goes to infinity under suitable conditions. Our main results are formulated below.

Theorem 3.1. Let be the global solution of the problem (1.1)–(1.8) with the conditions

Then the energy functional decays exponentially to zero as the time goes to infinity.

To prove our main result, we will give some important preliminaries.

Lemma 3.2. Assume , functional and tensor are defined as above. Then

(1)(2)(3)(4).

Proof. In fact

Combining (3.2)–(3.4) and noting , we finish the desire conclusion (1).

By , the definition of , and Poincaré inquality, we know

Using Poincaré inequality, we get

which with conclusion (2) shows the desire result.

Denote . Hence

with , and

which imply

From (3.9) and noting , we obtain the desire conclusion (4).

The proof is completed.

Lemma 3.3. Suppose and is defined as above, then

Proof. In fact, for small enough,

Denote , then . Owing to , from (3.3) and (3.11), we obtain the conclusion.

Lemma 3.4. Define the functional

then

Proof. Multiplying (1.1) with and integrating the result over , we obtain

which with Lemma 2.2 and (1.5) implies

By Lemma 3.3, we know

Using (3.15)-(3.16), we get the conclusion.

Lemma 3.5. Suppose is the solution of (1.1)–(1.8) and (), (), and (3.1) hold, then

(1); (2); (3).

Proof. (1) By the formula of integral by part and (), we deduce

(2) Using () and (3.1), we infer the conclusion.

(3) By and the formula of integral by part, we get the conclusion (3).

The proof is completed.

Lemma 3.6. Define the functional

then for some small enough, there exist and as the following inequality holds

Proof. Multiplying (1.1) with and integrating the result over , we have

which with Lemma 3.5 gives

By Green formula, we have

Clearly

Inserting (3.22) and (3.23) into (3.21) yields

Applying Lemma 2.2, Lemma 3.5 and noting , we have

Inserting (3.25) into (3.24) and owing to (1.5) yield

Now, we estimate some terms on the right-hand side of (3.26). The Lemma 3.3 and (3.1) imply

Using Cauchy inequality, Hölder inequality, and Lemma 3.2 and noting (3.1), we obtain

Applying Hölder inequality, Lemma 3.2, and Lemma 2.3, we conclude

Combining (3.26)–(3.29), we conclude that

The proof is completed.

proof of Theorem 3.1. Let

By Lemma 2.3, Lemma 3.4, and Lemma 3.6 and noting (3.1), we have

with

Fixing , we obtain . Under the conditions, we obtain from (2.34) and (3.32) that there exists a positive such that

Fixing a large enough positive constant , we prove and are equivalent, that is,

with . Indeed, using Cauchy inequality, Lemma 3.2, and (2.34), we have

Using the definition of and Green’s formula, we infer

Applying Cauchy inequality, Lemma 3.2 and noting (3.1), we conclude that

Relations (3.37)-(3.39) imply that

Fixing a large enough positive constant , inequalities (3.36) and (3.40) show (3.35) hold. Relations (3.34) and (3.35) imply that there exists a positive constant such that

which with Gronwall’s inequality gives

Combining (3.35) and (3.42), we finish the proof.

Theorem 3.7. Assume that is the global solution of (1.1)–(1.8) with condition . then for some there exists a positive constant , such that

In order to prove Theorem 3.7, we quote the following technical lemma.

Lemma 3.8. Suppose that and is a continuous function and there exists , such that

Then, there exists such that

Proof. Applying Hölder inequality, we obtain

This completes the proof of Lemma 3.8.

Proof of Theorem 3.7. From the hypothesis , we have . Integrating the above inequality over , we get

Fixing in Lemma 3.8, then , hence we obtain

Using this estimate into (3.46), noting (2.34) and Lemma 3.2, we get

Denote . Applying (3.32) and (3.49), we arrive at

Since , applying Young inequality and from Lemma 3.2, (3.50), we deduce

Since , then

By Lemma 3.2, we have

Hence, we get

Taking small enough, from (3.51) and (3.54), we get

Thus, we have

Let be a positive constant and define

Since , for large enough, we get

From Lemma 3.2 and (3.56), taking large enough, we obtain

From (3.58) and (3.59), we have

Applying Gronwall inequality, we get

Hence, we obtain

This completes the proof of Theorem 3.7.

4. Comment

In this paper, the key point is to construct suitable energy functional , find the appropriate auxiliary functionals and by multiplier method, and prove that and are equivalent and satisfies by precise apriori estimates. Thanks to Lemma 3.8 and auxiliary functional , we show the polynomial decay result under suitable conditions on relaxation .

Acknowledgment

The authors would like to thank the referee for his/her careful reading and kind suggestions. The paper was supported by the National Natural Science Foundation of China (11201258), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2011AQ006, and ZR2012AM010), the STPF of University in Shandong Province of China (J09LA04), and the Science Foundation of Qufu Normal University of China (XJ201114).

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