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Global Nonexistence of Positive Initial-Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms
This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.
In this paper we study the initial-boundary-value problem where is a bounded domain in with a smooth boundary , , and () are given functions to be specified later. We assume that is a function which satisfies for .
To motivate our work, let us recall some results regarding . The single-wave equation of the form in with initial and boundary conditions has been extensively studied, and many results concerning global existence, blow-up, energy decay have been obtained. In the absence of the source term, that is, (), it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial data (see ). In the absence of the damping term, the source term causes finite time blow-up of solutions with a large initial data (negative initial energy) (see [2, 3]). The interaction between the damping term and the source term makes the problem more interesting. This situation was first considered by Levine [4, 5] in the linear damping case and a polynomial source term of the form . He showed that solutions with negative initial energy blow up in finite time. The main tool used in [4, 5] is the “concavity method.” Georgiev and Todorova in  extended Levine's result to the nonlinear damping case . In their work, the authors considered problem (1.3) with and introduced a method different from the one known as the concavity method and showed that solutions with negative energy continue to exist globally in time if and blow up in finite time if and the initial energy is sufficiently negative. This latter result has been pushed by Messaoudi  to the situation where the initial energy and has been improved by the same author in  to accommodate certain solutions with positive initial energy.
In the case of being a given nonlinear function, the following equation: with initial and boundary conditions has been extensively studied. Equation of type of (1.4) is a class of nonlinear evolution governing the motion of a viscoelastic solid composed of the material of the rate type, see [9–12]. It can also be seen as field equation governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model, see . In two- and three-dimensional cases, they describe antiplane shear motions of viscoelastic solids. We refer to [14–16] for physical origins and derivation of mathematical models of motions of viscoelastic media and only recall here that, in applications, the unknown naturally represents the displacement of the body relative to a fixed reference configuration. When , there have been many impressive works on the global existence and other properties of solutions of (1.4), see [9, 10, 17, 18]. Especially, in  the authors have proved the global existence and uniqueness of the generalized and classical solution for the initial boundary value problem (1.4) when we replace and by and , respectively. But about the blow-up of the solution for problem, in this paper there has not been any discussion. Chen et al.  considered problem (1.4) and first established an ordinary differential inequality, next given the sufficient conditions of blow-up of the solution of (1.4) by the inequality. In , Hao et al. considered the single-wave equation of the form with initial and Dirichlet boundary condition, where satisfies condition (1.2) and The damping term has the form The source term is with for and for , are nonnegative constants, and . By using the energy compensation method [7, 8, 22], they proved that under some conditions on the initial value and the growth orders of the nonlinear strain term, the damping term, and the source term, the solution to problem (1.5) exists globally and blows up in finite time with negative initial energy, respectively.
Some special cases of system (1.1) arise in quantum field theory which describe the motion of charged mesons in an electromagnetic field, see [23, 24]. Recently, some of the ideas in [6, 22] have been extended to study certain systems of wave equations. Agre and Rammaha  studied the system of (1.1) with and proved several results concerning local and global existence of a weak solution and showed that any weak solution with negative initial energy blows up in finite time, using the same techniques as in . This latter blow-up result has been improved by Said-Houari  by considering a larger class of initial data for which the initial energy can take positive values. Recently, Wu et al.  considered problem (1.1) with the nonlinear functions and satisfying appropriate conditions. They proved under some restrictions on the parameters and the initial data several results on global existence of a weak solution. They also showed that any weak solution with initial energy blows up in finite time.
In this paper, we also consider problem (1.1) and improve the global nonexistence result obtained in , for a large class of initial data in which our initial energy can take positive values. The main tool of the proof is a technique introduced by Payne and Sattinger  and some estimates used firstly by Vitillaro , in order to study a class of a single-wave equation.
2. Preliminaries and Main Result
First, let us introduce some notation used throughout this paper. We denote by the norm for and by the Dirichlet norm in which is equivalent to the norm. Moreover, we set as the usual inner product.
Concerning the functions and , we take where are constants and satisfies One can easily verify that where
We have the following result.
Lemma 2.1 (see [30, Lemma 2.1]). There exist two positive constants and such that
In order to state and prove our result, we introduce the following function space:
Define the energy functional associated with our system A simple computation gives
Our main result reads as follows.
3. Proof of Theorem 2.2
In this section, we deal with the blow-up of solutions of the system (1.1). Before we prove our main result, we need the following lemmas.
Lemma 3.1. Let be a solution of the ordinary differential inequality where . If , then the solution ceases to exist for .
Lemma 3.2. Assume that (2.3) holds. Then there exists such that for any , one has
Proof. By using Minkowski's inequality, we get Also, Hölder's and Young's inequalities give us If , then we have If , from , we have . Since , we have which implies that (3.5) still holds for . Combining (3.3), (3.4) with (3.5) and the embedding , we have (3.2).
In order to prove our result and for the sake of simplicity, we take and introduce the following: where is the optimal constant in (3.2). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro .
Proof. We first note that, by (2.10), (3.2), and the definition of , we have
where . It is not hard to verify that is increasing for , decreasing for , as , and
where is given in (3.7). Since , there exists such that .
Set . Then by (3.11) we get , which implies that . Now, to establish (3.9), we suppose by contradiction that for some . By the continuity of , we can choose such that Again, the use of (3.11) leads to This is impossible since for all . Hence (3.9) is established.
To prove (3.10), we make use of (2.10) to get Consequently, (3.9) yields Therefore, (3.17) and (3.7) yield the desired result.
Proof of Theorem 2.2. We suppose that the solution exists for all time and we reach to a contradiction. Set
By using (2.10) and (3.18), we have
From (3.9), we have
Hence, by the above inequality and (2.6), we have
We then define
where small enough is to be chosen later and
Our goal is to show that satisfies the differential inequality (3.1) which leads to a blow-up in finite time. By taking a derivative of (3.23), we get
From the definition of , it follows that
which together with (3.25) gives
Then, using (3.10), we obtain
where . It is clear that , since . We now exploit Young's inequality to estimate the last two terms on the right side of (3.28)
where are parameters depending on the time and specified later. Inserting the last two estimates into (3.28), we have
By choosing and such that
where and are constants to be fixed later. Thus, by using (2.6) and (3.31), inequality (3.31) then takes the form
where and is a positive constant.
Since , taking into account (2.6) and (3.21), then we have for some positive constants and . By using (3.24) and the algebraic inequality we have where . Similarly, Also, since by using (3.24) and (3.34), we conclude that where is a generic positive constant. Taking into account (3.33)–(3.38), estimate (3.32) takes the form where , . At this point, and for large values of and , we can find positive constants and such that (3.39) becomes Once and are fixed, we pick small enough so that and Since , there exists such that (3.40) becomes Then, we have
Next, we have by Hölder's and Young's inequalities for . We take , to get . Here and in the sequel, denotes a positive constant which may change from line to line. By using (3.24) and (3.34), we have Therefore, (3.44) becomes Note that Combining (3.42) with (3.47), we have A simple application of Lemma 3.1 gives the desired result.
The authors are indebted to the referee for giving some important suggestions which improved the presentations of this paper. This work is supported in part by a China NSF Grant no. 10871097, Qing Lan Project of Jiangsu Province, the Foundation for Young Talents in College of Anhui Province Grant no. 2011SQRL115 and Program sponsored for scientific innovation research of college graduate in Jangsu province.
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Copyright © 2011 Liang Fei and Gao Hongjun. 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.