Journal of Applied Mathematics

Volume 2013, Article ID 353757, 8 pages

http://dx.doi.org/10.1155/2013/353757

## Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received 17 December 2012; Revised 15 May 2013; Accepted 30 May 2013

Academic Editor: Roberto Natalini

Copyright © 2013 Yu-Zhu Wang. 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

We consider the Cauchy problem for the damped nonlinear hyperbolic equation in *n*-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

#### 1. Introduction

We investigate the Cauchy problem for the following damped nonlinear hyperbolic equation: with the initial value Here is the unknown function of and , and are constants. The nonlinear term and is a positive integer.

Equation (1) is a model in variational form for the neo-Hookean elastomer rod and describes the motion of a neo-Hookean elastomer rod with internal damping; for more detailed physical background, we refer to [1]. In [1], the authors have studied a general class of abstract evolution equations where , , , and satisfy certain assumptions. For quite general conditions on the nonlinear term, global existence, uniqueness, regularity, and continuous dependence on the initial value of a generalized solution to (3) in a bounded domain of were obtained. Equation (1) fits the abstract framework of [1]. The local well-posedness for the Cauchy problem for (1), (2) in three-dimensional space was obtained by Chen and Da [2]. More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that , . Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that , . Their method is to first establish local-in-time well-posedness of a periodic version of (1), (2) and then construc a solution to (1), (2) as a limit of periodic solutions with divergent periods. This paper also arrived at some sufficient conditions for blow-up of the solution in finite time, and an example was given. Song and Yang [3] studied the existence and nonexistence of global solutions to the Cauchy problem for (1) in one-dimensional space. The boundary value problems for (1) are investigated (see [4, 5]). Equation (1) is a fifth-order wave equation. For more higher order wave equations, we refer to [6–8] and references therein.

The main purpose of this paper is to establish global existence and asymptotic behavior of solutions to (1), (2) by using the contraction mapping principle. Firstly, we consider the decay property of the following linear equation: We obtain the following decay estimate of solutions to (4), (2) Based on the above estimates, we define a solution space with time weighted norms, and then global existence and asymptotic behavior of solutions to (1), (2) are obtained by using the contraction mapping principle. More precisely, we prove global existence and the following decay estimate of solution to (1), (2): for , , and . Here , , and is assumed to be suitably small. When , our result allows for the initial data , . But in [2], the authors proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that , , so our result improves the regularity of the initial condition for the time derivative. This improvement is due to the strong damping term since the strong damping term has stronger dissipative effect than the damping . The stronger dissipative effect has been exhibited in the study of the strongly damped wave equation and related problems; see, for instance, [9].

The global existence and asymptotic behavior of solutions to hyperbolic-type equations have been investigated by many authors. We refer to [10–15] for hyperbolic equations, [16–21] for damped wave equation, and [22, 23] for various aspects of dissipation of the plate equation.

We give some notations which are used in this paper. Let denote the Fourier transform of defined by and we denote its inverse transform by .

For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of is defined by with the norm ; the homogeneous Sobolev space of is defined by with the norm ; especially , . Moreover, we know that for .

Finally, in this paper, we denote every positive constant by the same symbol or without confusion. is the Gauss symbol.

The paper is organized as follows. In Section 2 we derive the solution formula of our semilinear problem. We study the decay property of the solution operators appearing in the solution formula in Section 3. Then, in Section 4, we discuss the linear problem and show the decay estimates. Finally, we prove global existence and asymptotic behavior of solutions for the Cauchy problem (1), (2) in Section 5.

#### 2. Solution Formula

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate (4). Taking the Fourier transform, we have The corresponding initial value is given as The characteristic equation of (8) is Let be the corresponding eigenvalues of (10), and we obtain The solution to the problem (8)-(9) is given in the form where We define and by and , respectively, where denotes the inverse Fourier transform. Then, applying to (12), we obtain By the Duhamel principle, we obtain the solution formula to (1), (2)

#### 3. Decay Property

The aim of this section is to establish decay estimates of the solution operators and appearing in the solution formula (15).

Lemma 1. *The solution of the problem (8), (9) satisfies
**
for and , where
*

*Proof. *Multiplying (8) by and taking the real part yield
Multiplying (8) by and taking the real part, we obtain
Multiplying both sides of (19) and (20) by and and summing up the resulting equation yield
where
A simple computation implies that
where
Note that
It follows from (23) that
Using (21) and (26), we get
Thus
which together with (23) proves the desired estimates (17). Then we have completed the proof of the lemma.

Lemma 2. * Let and be the fundamental solution of (4) in the Fourier space, which are given in (13) and (14), respectively. Then one has the estimates
**
for and , where
*

* Proof. * If , from (12), we obtain
Substituting the equalities into (17) with , we get (29).

In what follows, we consider , and it follows from (12) that
Substituting the equalities into (17) with , we get the desired estimate (30). The lemma is proved.

Lemma 3. * Let and be nonnegative integer. Then one has
**
Here in (34) and in (38).*

* Proof. *By the Plancherel theorem and (29), Hausdorff-Young inequality, we obtain
Here and is a small positive constant in Lemma 1. Thus (34) follows.

Similarly, using (29) and (30), respectively, we can prove (35)–(37).

In what follows, we prove (38). By the Plancherel theorem, (29), and Hausdorff-Young inequality, we have
where is a small positive constant in Lemma 1. Thus (38) follows. Similarly, we can prove (39). Thus we have completed the proof of the lemma.

#### 4. Decay Estimate of Solutions to (4), (2)

Theorem 4. * Assume that , . Then the classical solution to (4), (2), which is given by the formula (15), satisfies the decay estimate
*

*Proof. * Firstly, we prove (42). Using (34) and (35), for , we obtain
For , it follows from (36) and (37) that
Equation (44) follows from (42) and Gagliardo-Nirenberg inequality. The lemma is proved.

#### 5. Global Existence and Asymptotic Behavior

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1), (2). We need the following lemma, which comes from [24] (see also [25]).

Lemma 5. * Let and be positive integers, , , , satisfy , and let . Assume that is class of and satisfies
**
If and , then for one has
*

Lemma 6. * Let and be positive integers, let , , , satisfy , and let . Let be a function that satisfies the assumptions of Lemma 5. Moreover, assume that
**
If , and , , then for , one has
**Based on the estimates (42)–(44) of solutions to the linear problem (4), (2), one defines the following solution space:
**
where
**
For , one defines
**
where depends on the initial value, which is chosen in the proof of main result. **For , using Gagliardo-Nirenberg inequality, one obtains
*

Theorem 7. * Assume that , , and integer . satisfies the assumptions of Lemmas 5 and 6. Put
**
If is suitably small, the Cauchy problem (1)-(2) has a unique global classical solution satisfying
**
Moreover, the solution satisfies the decay estimate
**
for and .*

* Proof. * Define the mapping
Using (34)-(35), (38), Lemma 5, and (54), for , we obtain
Thus
It follows from (58) that
Using (36)-(37), (39), Lemma 5, and (54), for , we have
Thus
Combining (60), (63) and taking and suitably small yield
For , , by using (58), we have
Exploiting (65), (38) Lemma 6, and (54), for , we obtain
which implies
Similarly for , from (61), (39), and (54), we have
which implies
Noting , by using (67), (69) and taking suitably small, yields

From (64) and (70), we know that is strictly contracting mapping. Consequently, we conclude that there exists a fixed point of the mapping , which is a classical solution to (1), (2). We have completed the proof of Theorem 7.

#### Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11101144) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.

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