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Journal of Function Spaces
Volume 2014 (2014), Article ID 284809, 21 pages
General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping
College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Received 8 May 2013; Revised 28 September 2013; Accepted 16 November 2013; Published 4 February 2014
Academic Editor: William Ziemer
Copyright © 2014 Wenjun Liu 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.
The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.
In this work, we investigate the following system of viscoelastic equations of Kirchhoff type: where is a bounded domain with smooth boundaryis a positive locally Lipschitz function, andare given functions to be specified later.
To motivate our work, let us recall some previous results regarding viscoelastic equations of Kirchhoff type. The following problem: is a model to describe the motion of deformable solids as hereditary effect is incorporated. It was first studied by Torrejón and Yong  who proved the existence of a weakly asymptotic stable solution for large analytical datum. Later, Muñoz Rivera  showed the existence of global solutions for small datum and the total energy decays to zero exponentially under some restrictions. Then, Wu and Tsai  treated problem (2) for and proved the global existence, decay, and blow-up with suitable conditions on initial data. They obtained the blow-up properties of local solution with small positive initial energy by the direct method of . To obtain the decay result, they assumed that the nonnegative kernelfor someThis energy decay result was recently improved by Wu in  under a weaker condition on (i.e.,for ). For a single wave equation of Kirchhoff type without the viscoelastic term, we refer the reader to Matsuyama and Ikehata  and Ono [7–10].
Many results concerning local existence, global existence, decay, and blow-up of solutions for a system of wave equations of Kirchhoff type without viscoelastic terms (i.e., ) have also been extensively studied. For example, Park and Bae  considered the system of wave equations with nonlinear dampings forand, and showed the global existence and asymptotic behavior of solutions under some restrictions on the initial energy. Later, Benaissa and Messaoudi  discussed blow-up properties for negative initial energy. Recently, Wu and Tsai  studied the system (1) for. Under some suitable assumptions on they proved local existence of solutions by applying the Banach fixed point theorem and the blow-up of solutions by using the method of Li and Tsai in , where three different cases on the sign of the initial energyare considered.
In the case ofand in the presence of viscoelastic term (i.e.,), Cavalcanti et al.  studied the equation that was subject to a locally distributed dissipation with the same initial and boundary conditions as that of (2), and proved an exponential decay rate. This work extended the result of Zuazua , in which he considered (3) with and the localized linear damping. By using the piecewise multipliers method, Cavalcanti and Oquendo  investigated the equation with the same initial and boundary conditions as that of (2). Under the similar conditions on the relaxation functionas above, and for all , they improved the results of  by establishing exponential stability for exponential decay functionand linear function and polynomial stability for polynomial decay functionand nonlinear function, respectively.
Concerning blow-up results, Messaoudi  considered the equation He proved that any weak solution with negative initial energy blows up in finite time ifand while exists globally for any initial data in the appropriate space if This result was improved by the same author in  for positive initial energy under suitable conditions on , and. Recently, Liu  studied the equation with the same initial and boundary condition as that of (2). By virtue of convexity technique and supposing that where, he proved that the solution with nonpositive initial energy as well as positive initial energy blows up in finite time.
We should mention that the following system: was considered by Han and Wang in , where is a bounded domain with smooth boundary in . Under suitable assumptions on the functions , the initial data and the parameters in the above problem established local existence, global existence, and blow-up property (the initial energy). This latter blow-up result has been improved by Messaoudi and Said-Houari  into certain solutions with positive initial energy. Recently, Liang and Gao in  investigated the following problem: with the same initial and boundary conditions as that of (9). Under suitable assumptions on the functionsand certain initial data in the stable set, they proved that the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, they proved that there are solutions with positive initial energy that blow up in finite time. It is also worth mentioning the work  in which we studied system (1). Under suitable assumptions on the functionsand certain initial conditions, we showed that the solutions are global in time and the energy decays exponentially. For other papers related to existence, uniform decay, and blow-up of solutions of nonlinear wave equations, we refer the reader to [14, 24–29] for existence and uniform decay, to [17, 30–34] for blow-up, and to [35–40] for the coupled system. To the best of our knowledge, the general decay and blow-up of solutions for systems of viscoelastic equations of Kirchhoff type with strong damping have not been well studied.
Motivated by the above mentioned research, we consider in the present work the coupled system (1) with nonzeroand nonconstant. We note that in such a coupled system case we should overcome the additional difficulties brought by the treatment of the nonlinear coupled terms. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy, the other is for certain solutions with arbitrarily positive initial energy. Then, we give a decay result of global solutions under a weaker assumption on the relaxation functions.
This paper is organized as follows. In the next section we present some assumptions, notations and known results and state the main results: Theorems 4, 5, 6, and 7. The two blow-up results, Theorems 5 and 6, are proved in Sections 3 and 4, respectively. Section 5 is devoted to the proof of the decay result—Theorem 7.
2. Preliminaries and Main Result
In this section we present some assumptions, notations, and known results and state the main results. First, we make the following assumptions.(A1)is a positive locally Lipschitz function forwith the Lipschitz constantsatisfying (A2)are strictly decreasingfunctions such that (A3)There exist two positive differentiable functions and such that andsatisfies (A4)We make the following extra assumption on: where.
Remark 1. It is clear that (which occurs physically in the study of vibrations of damped flexible space structures in a bounded domain in) satisfies (A4) for as long as Indeed, by straightforward calculations, we obtain
Next, we introduce some notations. Consider the Hilbert spaceendowed with the inner product and the functionsand(see also ): whereare constants andsatisfies
One can easily verify that where
Then, we give two lemmas which will be used throughout this work.
Lemma 2 (Sobolev-Poincaré inequality ). If , then holds with some constant.
We introduce where is the optimal constant in (24).
Our first result is concerned with the blow-up for certain local solutions with nonpositive initial energy as well as positive initial energy.
Our second result shows that certain local solutions with arbitrarily positive initial energy can also blow up.
Theorem 6. Suppose that (A1)-(A2), (A4), (20), and hold. Assume further that and and satisfy then the solution of problem (1) blows up at a finite timein the sense of (119) below. Moreover, the upper bounds forcan be estimated by whereandis given in (116) below.
Finally, we state the general decay result. For convenience, we choose especially, and Then the energy functional , defined by (27), becomes
To achieve general decay result we will use a Lyapunov type technique for some perturbation energy following the method introduced in . This result improves the one in Li et al.  in which only the exponential decay rates are considered.
3. Blow-Up of Solutions with Initial Data in the Unstable Set
In this section, we prove a finite time blow-up result for initial data in the unstable set. We need the following lemmas.
Proof. We first note that, by (27), (24) and the definition of we have
where we have usedand
It is easy to verify that is increasing in , decreasing in , and that , as , and
whereis given in (30). Since there exists such that . Set , then from (41) we can get , which implies that .
Now we establish (40) by contradiction. First we assume that (40) is not true over then there existssuch that By the continuity ofwe can choosesuch that Again, the use of (41) leads to This is impossible sincefor all .
Hence (40) is established.
Proof of Theorem 5. Assume by contradiction that the solutionis global. Then we consider defined by
where , , and are positive constants to be chosen later. Then for all . Furthermore
for almost every Testing the first equation of system (1) with and the second equation of system (1) with , integrating the results over , using integration by parts, and summing up, we have
Therefore, we have
where are the functions defined by
Using the Cauchy-Schwarz inequality, we obtain
Similarly, we have By Hlder’s inequality and Young’s inequality, we obtain The previous inequalities imply thatfor everyUsing (53), we get for almost every , where is the map defined by For the fourth term on the right hand side of (59), we have Similarly, Combining (59), (60) with (61), we get Since, we have for any , inserting (63) into (62) and utilizing (27), we have Using (29) and (A4), we have
If , that is, , we choose in (65) and small enough such that . Then by (32), we have
If , that is, , we choose and in (65). Then, we get By (32), we have (notice that due to ) that is, for . Therefore, from the above two inequalities and (A2), we can get Since by Lemma 8, there exists a constant such that which implies It follows from (67) and (73) that
Therefore, by (58), (66), and (74), we obtain for almost everyBy (49), we then choosesufficiently large such that consequently, Then by (76) and (77), we choose large enough so that which ensures that. As , letting , we can select such that . By using Lemma 9, we get . This implies that which is a contradiction. Thus,
4. Blow-Up of Solutions with Arbitrarily Positive Initial Energy
In this section, we prove the second blow-up result (Theorem 6) for solutions with arbitrarily positive initial energy. In order to attain our aim, we need the following three lemmas.
Lemma 10 (see ). Let and be a nonnegative function satisfying If then for , where is a constant and is the smallest root of the equation
Lemma 11 (see ). If is a nonincreasing function on and satisfies the differential inequality where , and , then there exists a finite time such that and the upper bound of is estimated by where.
For the next lemma, we define
Proof. By (86), we have Testing the first equation of system (1) with and testing the second equation of system (1) with and plugging the results into the expression of we obtain By (27), (29), and (90), we get