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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 905867, 7 pages
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.
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.
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  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  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  with positive initial energy. For a related study, we may see the work of Wu and Tsai . In , 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.  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.  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.  considered and obtained a global existence for and uniform exponential decay for . This work was extended by Messaoudi and Tatar  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  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  improved  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.  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  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 [14–24] 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.
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
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
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
Straightforward computations yield
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).
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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