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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 905867, 7 pages
http://dx.doi.org/10.1155/2013/905867
Research Article

On Asymptotic Behavior and Blow-Up of Solutions for a Nonlinear Viscoelastic Petrovsky Equation with Positive Initial Energy

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 29 December 2012; Revised 9 April 2013; Accepted 7 May 2013

Academic Editor: L. E. Persson

Copyright © 2013 Gang Li et al. 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

This paper deals with the initial boundary value problem for the nonlinear viscoelastic Petrovsky equation . Under certain conditions on and the assumption that , we establish some asymptotic behavior and blow-up results for solutions with positive initial energy.

1. Introduction

In this work, we investigate the following nonlinear viscoelastic Petrovsky problem: where is a bounded domain in () with a smooth boundary , ; is the unit outer normal on ; and is a nonnegative memory term.

In the absence of viscoelastic term (i.e., ), Guesmia [1] considered the following equation: where is a bounded function and is continuous and increasing function satisfying . Under suitable growth conditions on , the author established global existence, uniqueness, and decay results by using the semigroup method. Messaoudi [2] investigated a nonlinearly damped semilinear Petrovsky equation where and and proved that the solution is global when while the solution blows up in finite time with negative initial energy when . Later, this blow-up result was improved by Chen and Zhou [3] with positive initial energy. For a related study, we may see the work of Wu and Tsai [4]. In [5], Amroun and Benaissa studied (3) by generalizing the damping term into the form of and obtained the global existence of the solutions by means of the stable set method combined with the Faedo-Galerkin procedure. Very recently, in the presence of the strong damping, Li et al. [6] considered the following Petrovsky equation: Without any interaction between and , the authors obtained the global existence and uniform decay of solutions when the initial data are in some stable set. And a blow-up result was established when and the initial energy is less than the potential well depth.

In the presence of the viscoelastic term (i.e., ), Muñoz Rivera et al. [7] studied the following equation: They proved that the memory effect produces strong dissipation capable of making uniform rate of decay for the energy. Later, in the presence of strong damping term, M. M. Cavalcanti et al. [8] considered and obtained a global existence for and uniform exponential decay for . This work was extended by Messaoudi and Tatar [9] to a situation where a nonlinear source term is competing with the damping induced by and the integral term. Then in the case of , the same authors [10] showed that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set by introducing a new functional and using the potential well method. Recently, Wu [11] improved [10] by considering the nonlinear equation: and a general decay result was obtained. In the presence of strong damping term and dispersive term , Xu et al. [12] considered the initial boundary value problem for the following viscoelastic wave equation: By introducing a family of potential wells, the authors not only obtained the invariant sets, but also proved the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, they established a blow-up result for certain solutions with arbitrary positive initial energy (high energy case). Very recently, Tahamtani and Peyravi [13] considered problem (1) and obtained the exponential decay of the energy under some assumptions on without any interaction between source term and damping term. Under an appropriate restriction on , they also proved that the norm of any solution grows as an exponential function if and the initial energy is negative. For other related works, we refer the readers to [1424] and the references therein.

Motivated by the above works, in this paper, we intend to consider problem (1) and establish some asymptotic behavior and blow-up results for solutions with positive initial energy. For our purpose, we use the functional and give a modified manner to estimate the term so that the appearance of the form like (for constants , , and ) which has been used in many earlier works (e.g., in [3, 19, 25]) can be avoided.

The paper is organized as follows. In Section 2 we present some assumptions and known results and state the main results. Section 3 is devoted to proof the the blow-up result—Theorem 4.

2. Preliminaries and Main Results

In this section, we first present some assumptions and known results which will be used throughout this work.

For the relaxation function , we give the following assumptions:(G1) is a function such that (G2), for all .

Lemma 1 (Sobolev-Poincaré inequality). Let be a number with or ; then for there exists a positive such that

We assume that satisfy We state a local existence theorem that can be established by adopting the arguments of [2, 5, 22].

Theorem 2. Assume that (11), (G1), and (G2) hold. Let be given. Then there exists a unique weak solution such that for small enough.

Next, we define the following functionals: where

Remark 3. A multiplication of (1) by and integration over easily yield since .
Our main results read as follows.

Theorem 4. Suppose that (11), (G1), and (G2) hold and . Let be the unique solution to problem (1) and denote . Then, for any fixed , , , and satisfies either there exists some such that the solution of problem (1) blows up in in the sense of , or one has .

3. Proof of the Main Results

We denote then we can prove the following lemma.

Lemma 5. For , we have

Proof. Obviously, Straightforward computations yield then Let which leads to An elementary calculation shows Using (G1) and Lemma 1 we arrive at which implies that .
To get (19), straightforward computations lead to which implies that . Also, for any , we note that Therefore we have for all . Hence, we complete the proof.

Lemma 6. Suppose that (11), (G1), and (G2) hold; ; and ; then we have , for all and

Proof. Using (15), we have , for all . To show that on , we proceed by contradiction. Assume there exists such that . Since , it follows that there exists some such that . Now, we define Then, we have and
Suppose that , by the regularity of , we have On the other hand, applying Lemma 1 to (31), we obtain From the above inequality, we can easily get , for all , and which implies and this contradicts to (32).
Suppose that . Applying Lemma 5, we see that is the infimum of over all functions in and , which contradicts to . Thus, we conclude that for all .
To get (29), using Lemma 5, (31) and the conclusion that , for all , we obtain which completes the proof.

Proof of Theorem 4. Suppose on the contrary that there exists a positive constant such that , for all . An integration of (15) over yields Set ; then we have We estimate (39) as follows. By using Cauchy-Schwarz inequality and Young's inequality, we get for any Using Young's inequality, we get for any By Hölder and Young inequalities, we arrive at where . Combining (40)–(42), we obtain where is the Poincaré constant. Since where will be chosen later. Then, we have Choosing and small enough and using (16), we can get a certain constant , such that Therefore, it follows from (G1) and Lemma 6 that since the choice of . Finally, we take small enough so that and use Lemma 6 to get Integrate twice the above inequality over and use (37); we have which implies that grows more quickly than the linear growth for .
On the other hand, by (37) and Hölder's inequality, we have Thus, where , , and are positive constants. Obviously, (51) contradicts (49).

Acknowledgments

This work was sponsored by Qing Lan Project of Jiangsu Province and was partly supported by the Tianyuan Fund of Mathematics (Grant no. 11026211), the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 09KJB110005), and the JSPS Innovation Program (Grant no. CXLX12_0490).

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