Abstract and Applied Analysis

Volume 2013 (2013), Article ID 903625, 9 pages

http://dx.doi.org/10.1155/2013/903625

Research Article

Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

Yaojun Ye

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received 30 April 2013; Revised 1 September 2013; Accepted 2 September 2013

Academic Editor: T. Raja Sekhar

Copyright © 2013 Yaojun Ye. 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.

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.

1. Introduction

In this paper, we are concerned with the global solvability and decay stabilization for the following nonlinear wave equations system:

with the initial-boundary value conditions

where is a bounded open domain in with a smooth boundary , , and for and for .

When , Medeiros and Miranda [1] proved the existence and uniqueness of global weak solutions. Cavalcanti et al. in [2–4] considered the asymptotic behavior for wave equation and an analogous hyperbolic-parabolic system with boundary damping and boundary source term. In paper [5, 6], the authors dealt with the existence, uniform decay rates, and blowup for solutions of systems of nonlinear wave equations with damping and source terms.

Rammaha and Wilstein [7] and Yang [8] are concerned with the initial boundary value problem for a class of quasilinear evolution equations with nonlinear damping and source terms. Under appropriate conditions, by a Galerkin approximation scheme combined with the potential well method, they proved the existence and asymptotic behavior of global weak solutions when , where and are, respectively, the growth orders of the nonlinear strain terms and the source term.

Ono [9] considers the following initial-boundary value problem for nonlinear wave equations with nonlinear dissipative terms:

where , , and are constants. The author mainly investigates on the blowup phenomenon to problem (6). On the other hand, in the case of , he shows that the problem (6) admits a unique global solution, and its energy has some decay properties under some assumptions on and initial energy . In particular, when and in (6), the energy has some polynomial and exponential decay rates, respectively.

For the following strongly damped nonlinear wave equation

Dell’Oro and Pata [10] obtain the long-time behavior of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space. In addition, the existence of global and local solutions, decay estimates, and blowup for solutions of nonlinear wave equation with source and damping terms and exponential nonlinearities are studied in [11–14].

In this paper, we prove the global existence for the problem (1)–(5) by applying the potential well theory introduced by Sattinger [15] and Payne and Sattinger [16]. Meanwhile, we obtain the asymptotic stabilization of global solutions by using a difference inequality [17].

For simplicity of notations, hereafter we denote by the norm of ; denotes norm, and we write equivalent norm instead of norm . Moreover, denotes various positive constants depending on the known constants and may be different at each appearance.

2. Local Existence

In this section, we investigate the local existence and uniqueness of the solutions of the problem (1)–(5). For this purpose, we list up two useful lemmas which will be used later and give the definition of weak solutions.

Lemma 1. Let , then ; and the inequality holds with a constant depending on , , and , provided that , and , .

Lemma 2 (Young inequality). Let and for , ; then one has the inequality

where is an arbitrary constant, and is a positive constant depending on .

Definition 3. A pair of functions is said to be a weak solution of (1)–(5) on if , , , , and satisfies

for all test functions and for almost all .

The local existence and uniqueness of solutions for problem (1)–(5) can be proved through the use of Galerkin method. The result reads as follows.

Theorem 4 (local solution). Supposed that , , and if and for , then there exists such that the problem (1)–(5) has a unique local solution satisfying

where

Proof. Let be a basis for . Supposed that is the subspace of generated by . We are going to look for the approximate solution

which satisfies the following Cauchy problem:

Note that, we can solve the problem (14)–(19) by a Picard’s iteration method in ordinary differential equations. Hence, there exists a solution in for some , and we can extend this solution to the whole interval for any given by making use of the a priori estimates below.

Multiplying (14) by and (15) by and summing over from to , we obtain

By summing (20) and (21) and integrating the resulting identity over , we have

We estimate the right-hand terms of (22) as follows: we get from Hölder inequality and Lemmas 1 and 2 that

It follows from (22) and (23) that

which implies that

We get from (25) and Gronwall type inequality that

Thus, we deduce from (26) that there exists a time such that

where is a positive constant independent of .

We have from (24) and (26) that

It follows from (27) and (28) that

Using the same process as the proof of Theorem 2.1 in paper [18], we derive that is a local solution of the problem (1)–(5). By (20) and (21), we conclude that (11) is valid.

3. Global Existence

In order to state our main results, we first introduce the following functionals:

for .

We put that

Then, we are able to define the stable set as follows for problem (1)–(5):

We denote the total energy related to (1) and (2) by (12), and

is the total energy of the initial data.

Lemma 5. Let be a solution to problem (1)–(5); then, is a nonincreasing function for and

We have from (11) that is the primitive of an integrable function. Therefore, is absolutely continuous, and equality (35) is satisfied.

Lemma 6. Supposed that , and if ; if , then .

Proof. Since

so we get

In case , let , which implies that

As , an elementary calculation shows that . Therefore, we have that

It follows from Hölder inequality and Lemma 1 that

We get from (39) and (40) that

In case and or , then

Therefore, we have

We conclude from (41) and (43) that

Thus, we complete the proof of Lemma 6.

Lemma 7. Supposed that for and for , if , and , then for .

Proof. Assume that there exists a number such that on and . Then, in virtue of the continuity of , we see , where denotes the boundary of domain . From the definition of and the continuity of and in , we have either

or

It follows from (12) and (30) that

So, case (45) is impossible.

Assume that (46) holds; then, we get that

We obtain from that . Since

Consequently, we get from (47) that

which contradicts the definition of . Hence, case (46) is impossible as well. Thus we conclude that on .

Theorem 8 (global solution). Supposed that as and as , and is a local solution of problem (1)–(5) on . If , and , then is a global solution of problem (1)–(5).

Proof. It suffices to show that is bounded uniformly with respect to . Under the hypotheses in Theorem 8, we get from Lemma 7 that on . So the following formula holds on :

We have from (51) that

Hence, we get

The above inequality and the continuation principle lead to the global existence of the solution for problem (1)–(5).

4. Asymptotic Behavior of Global Solutions

The following lemma plays an important role in studying the decay estimate of global solutions for the problem (1)–(5).

Lemma 9 (see [9]). Suppose that is a nonincreasing nonnegative function on and satisfies

Then, has the decay property

where are constants and .

Lemma 10. Under the assumptions of Theorem 8, if initial value and are sufficiently small such that

then

where is a positive constant and is the optimal Sobolev’s constant from to .

Proof. We have from Lemma 1 and (52) that

Therefore, we get from (58) and (31) that

Let

then, we have from (59) that

Theorem 11. Under the assumptions of Theorem 8, if and (56) hold, then the global solution in of the problem (1)–(5) has the following decay property:

where is some constant depending only on and .

Proof. Multiplying (1) by and (2) by and integrating over , and summing up together, we get

Thus, there exists , such that

On the other hand, we multiply (1) by and (2) by and integrate over . Adding them together, we obtain

From (63), Sobolev inequality, and Hölder inequality, we have

We get from (52), (64), and Lemmas 1 and 2 that

From Hölder inequality and Lemma 2,we get

Since and the property of the function , , , we obtain

We conclude from (69), (70), , and Lemma 1 that

It follows from (63), (68), (69), and (71) that

and we obtain from (63), Sobolev inequality, Hölder inequality, and Lemma 2 that

Similarly, we have the following formula:

We get from (57), (73), and (74) that

Choosing small enough , we have from (65), (66), (67), (72), and (75) that

It follows from (30) and (31) that

On the other hand, from (12) and using (57) and (77), we deduce that

By integrating (78) over , we obtain

For small enough , we have from (76) and (79) that

Thus, there exists , such that

Multiplying (1) by and (2) by and integrating over , and summing up, we get

Therefore, we obtain from (63), (81), and (82) that

Choosing small enough , we have from (83) that

Since and , we get

Consequently,

Thus, applying Lemma 9 to (86), we get

where is some constant depending only on and .

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 61273016), The Natural Science Foundation of Zhejiang Province (no. Y6100016), The Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2012), and the Interdisciplinary Pre-research Project of Zhejiang University of Science and Technology (2010–2012).

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