Journal of Applied Mathematics

Volume 2012 (2012), Article ID 805158, 15 pages

doi:10.1155/2012/805158

Research Article

Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

Zhigang Pan,¹ Hong Luo,² and Tian Ma¹

¹Yangtze Center of Mathematics, Sichuan University, Chengdu, Sichuan 610041, China

²College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China

Received 20 February 2012; Revised 8 May 2012; Accepted 9 May 2012

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2012 Zhigang Pan et al. 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 global existence of strong solution , corresponding to a class of fully nonlinear wave equations with strongly damped terms in a bounded and smooth domain in , where is a given monotone in nonlinearity satisfying some dissipativity and growth restrictions and is in a sense subordinated to . By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution .

1. Introduction

We are concerned with the following mixed problem for a class of fully nonlinear wave equations with strongly damped terms in a bounded and domain :

where

Equations of type (1.1) are a class essential nonlinear wave equations describing the speed of strain waves in a viscoelastic configuration (e.g., a bar if the space dimension and a plate if ) made up of the material of the rate type [1, 2]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model [3]. Concerning damped cases, there is much to the global existence of solutions for the problem:

they discussed the global existence of weak solutions and regularity in and [4–8]. On the other hand, Ikehata and Inoue [9] considered the global existence of weak solutions for two-dimensional problem in an exterior domain with a compact smooth boundary for a semilinear strongly damped wave equation with a power-type nonlinearity and :

Cholewa and Dlotko [10] discussed the global solvability and asymptotic behavior of solutions to semilinear Cauchy problem for strongly damped wave equation in the whole of . They assume the nonlinear term grows like and if . Similar problems attracted attention of the researchers for many years [11–13]. Especially, Yang [14] studied the global existence of weak solutions to the more general equation including (1.4), but he did not discuss the regularity of weak solution for the quasilinear wave equation. We are interested in discussing the global existence and regularity of weak solutions for strongly damped wave equation with the dissipative terms containing and the nonlinear terms containing . Here is a given monotone in nonlinearity satisfying some dissipativity and growth restrictions and is in a sense subordinated to .

In [15], we have investigated the existence of global solutions to a class of nonlinear damped wave operator equations. In this paper, our first aim is to study the global existence of strong solutions to the more general equation including (1.4), which is the motivation that we establish our abstract strongly damped wave equation model with. The second aim is to deal with the global existence of strong solutions to a class of fully nonlinear wave equations with strongly damped terms under some weakly growing conditions.

This paper is organized as follows:

(i)in Section 2, we recall some preliminary tools and definitions;(ii)in Section 3, we put forward our abstract strongly damped wave equation model and proof the global existence of strong solution of it;(iii)in Section 4, we provide the proof of the main results about the mixed problem (1.1).

2. Preliminaries

We introduce two spatial sequences:

where , , , and are Hilbert spaces, is a linear space, and are Banach spaces.

All embeddings of (2.1) are dense. Let

Furthermore, has eigenvectors satisfying

and constitutes common orthogonal basis of and .

We consider the following abstract wave equation model:

where is a map, , and is a bounded linear operator, satisfying

Definition 2.1. We say that is a global weak solution of the (2.4) provided for

for each and .

Definition 2.2. Let , . We say that in is uniformly weakly convergent if is bounded, and

Lemma 2.3 (see [16]). Let be bounded sequences and uniformly weakly convergent to . Then, for each , it follows that

Lemma 2.4 (see [17]). Let be an open set and satisfy Caratheodory condition and

If is bounded and convergent to in for all bounded , then for each , the following equality holds

3. Model Results

Let . Assume

(A1) there is a functional such that

(A2) functional is coercive, that is,

(A3) satisfies

for .

Theorem 3.1. Set , for each , then the following assertions hold.

(1)If satisfies (A1) and (A2), then (2.4) has a globally weak solution

(2)If satisfies (A1)–(A3), then (2.4) has a global weak solution

(3)Furthermore, if is symmetric sectorial operator, that is, , and satisfies

then .

Proof. Let be a common orthogonal basis of and , satisfying (2.3). Set

Clearly, , .

By using Galerkin method, there exits satisfying

for , and

for .

Firstly, we consider . Let in (3.9). Taking into account (2.2) and (3.1), it follows that

We get

Let . From (2.1) and (2.2), it is known that are orthogonal basis of . We find that in , and in . Since is imbedding, it follows that

From (3.2), (3.11), and (3.12), we obtain that,

Let

which implies that in is uniformly weakly convergent from that is compact imbedding.

If we have the following equality:

then is a weak solution of (2.4) in view of (3.8), (3.14).

We will show (3.15) as follows. It follows that from (2.5),

Taking into account (2.2), (2.5), and (3.9), we get that

From (2.1) and (3.14), we have

Then, we get

In view of (3.9), (3.14), we obtain for all

Since is dense in , for all , (3.20) holds for all . Thus, we have

From (3.14) and being compact imbedding, it follows that

Clearly,

Then, (3.14) follows from (3.19)–(3.21), which implies assertion (1).

Secondly, we consider . Let in (3.9). In view of (2.2) and (2.8), it follows that

From (3.3), we have

where .

By using Gronwall inequality, it follows that

which implies that for all ,

From (3.25) and (3.21), it follows that

Let

which implies that in is uniformly weakly convergent from that is compact imbedding.

The left proof is same as assertion (1).

Lastly, assume (3.6) hold. Let in (3.9). It follows that

From (3.26), the above inequality implies

We see that for all , is bounded. Thus .

4. Main Result

Now, we begin to consider the mixed problem (1.1). Set

We assume

where are constant and , is the best constant satisfying

Theorem 4.1. If the assumptions of (4.1)–(4.5) hold, for , then (1.1) is a strong solution

Proof. We introduce spatial sequences

where the inner products of and are defined by

where such that is an embedding.

Linear operators and are defined by

It is known that and satisfy (2.2), (2.3), and (2.5). Define by

We show that is -coercively weakly continuous. Let satisfying (2.7) and

We need to prove that

From (2.7) and Lemma 2.3, we obtain

From (4.3), we get

We have the deformation

From (4.14) and Lemma 2.4, we have

From (4.12), (4.15)–(4.17), it follows that

Since , we have

From (4.14), (4.19), (4.1), (4.4), and Lemma 2.3, we get (4.13).

Let , where is same as (4.1). We get

which implies Conditions (A1), (A2) of model results in Theorem 3.1.

We will show (3.3) as follows. It follows that

which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution

satisfying

Acknowledgments

The authors are grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This paper was funded by the National Natural Science Foundation of China (no. 11071177) and the NSF of Sichuan Science and Technology Department of China (no. 2010JY0057).

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