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Abstract and Applied Analysis

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

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

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

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).

#### References

- L. A. Medeiros and M. M. Miranda, “Weak solutions for a system of nonlinear Klein-Gordon equations,”
*Annali di Matematica Pura ed Applicata*, vol. 146, pp. 173–183, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. P. Filho, and J. A. Soriano, “Existence and uniform decay of solutions of a parabolic-hyperbolic equation with nonlinear boundary damping and boundary source term,”
*Communications in Analysis and Geometry*, vol. 10, no. 3, pp. 451–466, 2002. View at Zentralblatt MATH · View at MathSciNet - M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term,”
*Journal of Differential Equations*, vol. 203, no. 1, pp. 119–158, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. M. Cavalcanti and V. N. Domingos Cavalcanti, “Existence and asymptotic stability for evolution problems on manifolds with damping and source terms,”
*Journal of Mathematical Analysis and Applications*, vol. 291, no. 1, pp. 109–127, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Agre and M. A. Rammaha, “Systems of nonlinear wave equations with damping and source terms,”
*Differential and Integral Equations*, vol. 19, no. 11, pp. 1235–1270, 2006. View at Zentralblatt MATH · View at MathSciNet - C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha, and D. Toundykov, “On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms,”
*Discrete and Continuous Dynamical Systems*, vol. 2, no. 3, pp. 583–608, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Rammaha and Z. Wilstein, “Hadamard well-posedness for wave equations with $p$-Laplacian damping and supercritical sources,”
*Advances in Differential Equations*, vol. 17, no. 1-2, pp. 105–150, 2012. View at MathSciNet - Z. Yang, “Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms,”
*Mathematical Methods in the Applied Sciences*, vol. 25, no. 10, pp. 795–814, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Ono, “Blow up phenomenon for nonlinear dissipative wave equatins,”
*Journal of Mathematics, Tokushima University*, vol. 30, pp. 19–43, 1996. - F. Dell'Oro and V. Pata, “Long-term analysis of strongly damped nonlinear wave equations,”
*Nonlinearity*, vol. 24, no. 12, pp. 3413–3435, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. F. Ma and J. A. Soriano, “On weak solutions for an evolution equation with exponential nonlinearities,”
*Nonlinear Analysis*, vol. 37, no. 8, Ser. A: Theory Methods, pp. 1029–1038, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Rammaha, “The influence of damping and source terms on solutions of nonlinear wave equations,”
*Boletim da Sociedade Paranaense de Matemática*, vol. 25, no. 1-2, pp. 77–90, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. O. Alves and M. M. Cavalcanti, “On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source,”
*Calculus of Variations and Partial Differential Equations*, vol. 34, no. 3, pp. 377–411, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Bociu, M. Rammaha, and D. Toundykov, “On a wave equation with supercritical interior and boundary sources and damping terms,”
*Mathematische Nachrichten*, vol. 284, no. 16, pp. 2032–2064, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. H. Sattinger, “On global solution of nonlinear hyperbolic equations,”
*Archive for Rational Mechanics and Analysis*, vol. 30, pp. 148–172, 1968. View at Zentralblatt MATH · View at MathSciNet - L. E. Payne and D. H. Sattinger, “Saddle points and instability of nonlinear hyperbolic equations,”
*Israel Journal of Mathematics*, vol. 22, no. 3-4, pp. 273–303, 1975. View at Publisher · View at Google Scholar · View at MathSciNet - M. Nakao, “A difference inequality and its application to nonlinear evolution equations,”
*Journal of the Mathematical Society of Japan*, vol. 30, no. 4, pp. 747–762, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Piskin and N. Polat, “Global existence, decay and blow-up solutions for coupled nonlinear wave equations with damping and source terms,”
*Turkish Journal of Mathematics*, vol. 30, pp. 1–19, 2013.