- Research Article
- Open Access
- Published:

# Global Existence and Asymptotic Behavior of Solutions for Some Nonlinear Hyperbolic Equation

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 895121 (2010)

## Abstract

The initial boundary value problem for a class of hyperbolic equation with nonlinear dissipative term in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in and show the asymptotic behavior of the global solutions through the use of an important lemma of Komornik.

## 1. Introduction

We are concerned with the global solvability and asymptotic stability for the following hyperbolic equation in a bounded domain

with initial conditions

and boundary condition

where is a bounded domain in with a smooth boundary , and are real numbers, and is a divergence operator (degenerate Laplace operator) with , which is called a -Laplace operator.

Equations of type (1.1) are used to describe longitudinal motion in viscoelasticity mechanics and can also be seen as field equations governing the longitudinal motion of a viscoelastic configuration obeying the nonlinear Voight model [1–4].

For , it is well known that the damping term assures global existence and decay of the solution energy for arbitrary initial data [4–6]. For , the source term causes finite time blow-up of solutions with negative initial energy if [7].

The interaction between the damping and the source terms was first considered by Levine [8, 9] in the case . He showed that solutions with negative initial energy blow up in finite time. Georgiev and Todorova [10] extended Levine's result to the nonlinear damping case . In their work, the authors considered (1.1)–(1.3) with and introduced a method different from the one known as the concavity method. They determined suitable relations between and , for which there is global existence or alternatively finite time blow-up. Precisely, they 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. Vitillaro [11] extended these results to situations where the damping is nonlinear and the solution has positive initial energy. For the Cauchy problem of (1.1), Todorova [12] has also established similar results.

Zhijian in [13–15] studied the problem (1.1)–(1.3) and obtained global existence results under the growth assumptions on the nonlinear terms and initial data. These global existence results have been improved by Liu and Zhao [16] by using a new method. As for the nonexistence of global solutions, Yang [17] obtained the blow-up properties for the problem (1.1)–(1.3) with the following restriction on the initial energy , where and , and are some positive constants.

Because the -Laplace operator is nonlinear operator, the reasoning of proof and computation is greatly different from the Laplace operator . By mean of the Galerkin method and compactness criteria and a difference inequality introduced by Nakao [18], the author [19, 20] has proved the existence and decay estimate of global solutions for the problem (1.1)–(1.3) with inhomogeneous term and .

In this paper we are going to investigate the global existence for the problem (1.1)–(1.3) by applying the potential well theory introduced by Sattinger [21], and we show the asymptotic behavior of global solutions through the use of the lemma of Komornik [22].

We adopt the usual notation and convention. Let denote the Sobolev space with the norm , and let denote the closure in of . For simplicity of notation, hereafter we denote by the Lebesgue space norm, and denotes norm and write equivalent norm instead of norm . Moreover, denotes various positive constants depending on the known constants and it may be different at each appearance.

## 2. Main Results

In order to state and study our main results, we first define the following functionals:

for . Then we define the stable set by

We denote the total energy associated with (1.1)–(1.3) by

for , , and is the total energy of the initial data.

For latter applications, we list up some lemmas.

Lemma 2.1.

Let , then and the inequality holds with a constant depending on , and , provided that (i) if ; (ii) , .

Lemma 2.2 (see [22]).

Let be a nonincreasing function and assume that there are two constants and such that

then , for all , if , and , for all , if , where and are positive constants independent of .

Lemma 2.3.

Let be a solutions to problem (1.1)–(1.3). Then is a nonincreasing function for and

Proof.

By multiplying (1.1) by and integrating over , we get

Therefore, is a nonincreasing function on .

We need the following local existence result, which is known as a standard one (see [13–15]).

Theorem 2.4.

Suppose that , and , . If , , then there exists such that the problem (1.1)–(1.3) has a unique local solution in the class

Lemma 2.5.

Assume that the hypotheses in Theorem 2.4 hold, then

for .

Proof.

By the definition of and , we have the following identity:

Since , so we have . Therefore, we obtain from (2.9) that

Lemma 2.6.

Suppose that and . If and such that

then , for each .

Proof.

Since , so . Then there exists such that for all . Thus, we get from (2.3) and (2.8) that

and it follows from Lemma 2.3 that

Next, we easily arrive at from Lemma 2.1, (2.11), and (2.13) that

Hence

which implies that , for all . By noting that

we repeat the steps (2.12)–(2.14) to extend to . By continuing the procedure, the assertion of Lemma 2.6 is proved.

Theorem 2.7.

Assume that , and , . is a local solution of problem (1.1)–(1.3) on . If and satisfy (2.11), then the solution is a global solution of the problem (1.1)–(1.3).

Proof.

It suffices to show that is bounded independently of .

Under the hypotheses in Theorem 2.7, we get from Lemma 2.6 that on . So the formula (2.8) in Lemma 2.5 holds on . Therefore, we have from (2.8) and Lemma 2.3 that

Hence, we get

The above inequality and the continuation principle lead to the global existence of the solution, that is, . Thus, the solution is a global solution of the problem (1.1)–(1.3).

The following theorem shows the asymptotic behavior of global solutions of problem (1.1)–(1.3).

Theorem 2.8.

If the hypotheses in Theorem 2.7 are valid, and , and , , then the global solutions of problem (1.1)–(1.3) have the following asymptotic behavior:

Proof.

Multiplying by on both sides of (1.1) and integrating over , we obtain that

where .

Since

so, substituting the formula (2.21) into the right-hand side of (2.20), we get that

We obtain from (2.14) and (2.12) that

It follows from (2.22), (2.23), and (2.24) that

We have from Hölder inequality, Lemma 2.1, and (2.17) that

and similarly, we have

Substituting the estimates (2.26) and (2.27) into (2.25), we conclude that

It follows from that .

We get from Young inequality and Lemma 2.3 that

From Young inequality, Lemmas 2.1 and 2.3, and (2.17), We receive that

Choosing small enough and , such that

then, substituting (2.29) and (2.30) into (2.28), we get

Therefore, we have from Lemma 2.2 that

where is a positive constant depending on .

We conclude from (2.17) and (2.33) that and

The proof of Theorem 2.8 is thus finished.

## References

- 1.
Andrews G: On the existence of solutions to the equation .

*Journal of Differential Equations*1980, 35(2):200–231. 10.1016/0022-0396(80)90040-6 - 2.
Andrews G, Ball JM: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity.

*Journal of Differential Equations*1982, 44(2):306–341. 10.1016/0022-0396(82)90019-5 - 3.
Ang DD, Dinh PN: Strong solutions of quasilinear wave equation with non-linear damping.

*SIAM Journal on Mathematical Analysis*1985, 19: 337–347. - 4.
Kawashima S, Shibata Y: Global existence and exponential stability of small solutions to nonlinear viscoelasticity.

*Communications in Mathematical Physics*1992, 148(1):189–208. 10.1007/BF02102372 - 5.
Haraux A, Zuazua E: Decay estimates for some semilinear damped hyperbolic problems.

*Archive for Rational Mechanics and Analysis*1988, 100(2):191–206. 10.1007/BF00282203 - 6.
Kopackova M: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation.

*Commentationes Mathematicae Universitatis Carolinae*1989, 30(4):713–719. - 7.
Ball JM: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations.

*The Quarterly Journal of Mathematics*1977, 28(112):473–486. - 8.
Levine HA: Instability and nonexistence of global solutions to nonlinear wave equations of the form .

*Transactions of the American Mathematical Society*1974, 192: 1–21. - 9.
Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.

*SIAM Journal on Mathematical Analysis*1974, 5: 138–146. 10.1137/0505015 - 10.
Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source terms.

*Journal of Differential Equations*1994, 109(2):295–308. 10.1006/jdeq.1994.1051 - 11.
Vitillaro E: Global nonexistence theorems for a class of evolution equations with dissipation.

*Archive for Rational Mechanics and Analysis*1999, 149(2):155–182. 10.1007/s002050050171 - 12.
Todorova G: Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms.

*Journal of Mathematical Analysis and Applications*1999, 239(2):213–226. 10.1006/jmaa.1999.6528 - 13.
Zhijian Y: 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*2002, 25(10):795–814. 10.1002/mma.306 - 14.
Zhijian Y, Chen G: Global existence of solutions for quasi-linear wave equations with viscous damping.

*Journal of Mathematical Analysis and Applications*2003, 285(2):604–618. 10.1016/S0022-247X(03)00448-7 - 15.
Zhijian Y: Initial boundary value problem for a class of non-linear strongly damped wave equations.

*Mathematical Methods in the Applied Sciences*2003, 26(12):1047–1066. 10.1002/mma.412 - 16.
Liu YC, Zhao JS: Multidimensional viscoelasticity equations with nonlinear damping and source terms.

*Nonlinear Analysis: Theory, Methods & Applications*2004, 56(6):851–865. 10.1016/j.na.2003.07.021 - 17.
Yang Z: Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms.

*Mathematical Methods in the Applied Sciences*2002, 25(10):825–833. 10.1002/mma.312 - 18.
Nakao M: A difference inequality and its application to nonlinear evolution equations.

*Journal of the Mathematical Society of Japan*1978, 30(4):747–762. 10.2969/jmsj/03040747 - 19.
Ye Y: Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term.

*Journal of Zhengzhou University. Natural Science Edition*1997, 29(3):18–23. - 20.
Ye Y: On the decay of solutions for some nonlinear dissipative hyperbolic equations.

*Acta Mathematicae Applicatae Sinica. English Series*2004, 20(1):93–100. - 21.
Sattinger DH: On global solution of nonlinear hyperbolic equations.

*Archive for Rational Mechanics and Analysis*1968, 30: 148–172. - 22.
Komornik V:

*Exact Controllability and Stabilization, The Multiplier Method, Research in Applied Mathematics*. Masson, Paris, France; 1994:viii+156.

## Acknowledgments

This Research was supported by the Natural Science Foundation of Henan Province (no. 200711013), The Science and Research Project of Zhejiang Province Education Commission (no. Y200803804), The Research Foundation of Zhejiang University of Science and Technology (no. 200803) and the Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2012).

## Author information

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Ye, Y. Global Existence and Asymptotic Behavior of Solutions for Some Nonlinear Hyperbolic Equation.
*J Inequal Appl* **2010, **895121 (2010) doi:10.1155/2010/895121

#### Received

#### Accepted

#### Published

#### DOI

### Keywords

- Asymptotic Behavior
- Global Solution
- Global Existence
- Hyperbolic Equation
- Longitudinal Motion