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Global Existence, Uniqueness, and Asymptotic Behavior of Solution for -Laplacian Type Wave Equation

Journal of Inequalities and Applications20102010:216760

https://doi.org/10.1155/2010/216760

Received: 10 May 2010

Accepted: 13 July 2010

Published: 2 August 2010

Abstract

We study the global existence and uniqueness of a solution to an initial boundary value problem for the nonlinear wave equation with the -Laplacian operator . Further, the asymptotic behavior of solution is established. The nonlinear term likes with appropriate functions and , where .

Keywords

Laplacian OperatorGlobal ExistenceSobolev InequalityApproximate EquationNonlinear Wave Equation

1. Introduction

This paper is concerned with the global existence, uniqueness, and asymptotic behavior of solution for the nonlinear wave equation with the -Laplacian operator
(1.1)
(1.2)

where and is a boundary domain in with smooth boundary . The assumptions on and will be made in the sequel.

Recently, Ma and Soriano in [1] investigated the global existence of solution for the problem (1.1)-(1.2) under the assumptions
(1.3)
Moreover, if and then there exist positive constants and such that
(1.4)
(1.5)
where
(1.6)

with

Gao and Ma in [2] also considered the global existence of solution for (1.1)-(1.2). In Theorem of [2], the similar results to (1.4)-(1.5) for asymptotic behavior of solution were obtained if the nonlinear function satisfies
(1.7)

where if and if .

More precisely, they obtained that the global existence of solution for (1.1)-(1.2) if one of the following assumptions was satisfied:

(i) , the initial data ;

(ii) , the initial data is small.

Similar consideration can be found in [35]. In [6], Yang obtained the uniqueness of solution of the Laplacian wave equation (1.1)-(1.2) for . To the best of our knowledge, there are few information on the uniqueness of solution of (1.1)-(1.2) for and .

In this paper, we are interested in the global existence, the uniqueness, the continuity and the asymptotic behavior of solution for (1.1)-(1.2). The nonlinear term in (1.1) likes with and . Obviously, the sign condition fails to hold for this type of function.

For these purposes, we must establish the global existence of solution for (1.1)-(1.2). Several methods have been used to study the existence of solutions to nonlinear wave equation. Notable among them is the variational approach through the use of Faedo-Galerkin approximation combined with the method of compactness and monotonicity, see [7]. To prove the uniqueness, we need to derive the various estimates for assumed solution . For the decay property, like (1.5), we use the method recently introduced by Martinez [8] to study the decay rate of solution to the wave equation in , where is a bounded domain of .

This paper is organized as follows. In Section 2, some assumptions and the main results are stated. In Section 3, we use Faedo-Galerkin approximation together with a combination of the compactness and the monotonicity methods to prove the global existence of solution to problem (1.1)-(1.2). Further, we establish the uniqueness of solution by some a priori estimate to assumed solutions. The proof of asymptotic behavior of solution is given in Section 4.

2. Assumptions and Main Results

We first give some notations and definitions. Let be a bounded domain in with smooth boundary . We denote the space and for and and relevant norms by and , respectively. In general, denotes the norm of Banach space . We also denote by and the inner product of and the duality pairing between and , respectively. As usual, we write instead . Sometimes, let represent for and so on.

If is given and is a Banach space, we denote by the space of functions which are over and which take their values in . In this space, we consider the norm
(2.1)

Let us state our assumptions on and .

() with .

()Let and satisfy

(2.2)
and growth condition
(2.3)

with some and the nonnegative functions , , , where , .

A typical function is with the appropriate nonnegative functions and , where .

Definition 2.1 (see [7]).

A measurable function on is said to be a (weak) solution of (1.1)-(1.2) if all , , , , and satisfies (1.2) with and the integral identity
(2.4)

for all .

Now we are in a position to state our results.

Theorem 2.2.

Assume - hold and . Then the problem (1.1)-(1.2) admits a solution satisfying
(2.5)
and the following estimates
(2.6)
where
(2.7)

with .

Further, if and , the solution satisfying (2.5)-(2.6) is unique.

Theorem 2.3.

Let be a solution of (1.1)-(1.2) with . In addition, let and
(2.8)
Then there exists , such that
(2.9)

The following theorem shows that the asymptotic estimate (2.9) can be also derived if assumption (2.8) fails to hold.

Theorem 2.4.

Let be a solution of (1.1)-(1.2) with . In addition, let and
(2.10)
with . Then there exists and , such that the solution satisfies
(2.11)

3. Proof of Theorem 2.2

In this section, we assume that all assumptions in Theorem 2.2 are satisfied. We first prove the global existence of a solution to problem (1.1)-(1.2) with the Faedo-Galerkin method as in [1, 2, 7, 9].

Let be an integer for which the embedding is continuous. Let   be eigenfunctions of the spectral problem
(3.1)
where represents the inner product in . Then the family yields a basis for both and . For each integer , let  span . We look for an approximate solution to problem (1.1)-(1.2) in the form
(3.2)
where are the solution of the nonlinear ODE system in the variant :
(3.3)
with the -Laplacian operator and the initial conditions
(3.4)
where and are chosen in so that
(3.5)

As it is well known, the system (3.3)-(3.4) has a local solution on some interval We claim that for any , such a solution can be extended to the whole interval by using the first a priori estimate below. We denote by the constant which is independent of and the initial data and .

Multiplying (3.3) by and summing the resulting equations over , we get after integration by parts
(3.6)
where
(3.7)
By (2.2) and Young inequality, we have
(3.8)
Let be so small that . Then
(3.9)
or
(3.10)

for some .

Thus, it follows from (3.6) and (3.10) that, for any and
(3.11)
By assumption , we obtain that and
(3.12)
Then it follows (3.5) and (3.6) that
(3.13)
Hence, for any and , we have from (3.11) and (3.13) that
(3.14)
With this estimate we can extend the approximate solution to the interval and we have that
(3.15)
(3.16)
(3.17)
Now we recall that operator is bounded, monotone, and hemicontinuous from to with . Then we have
(3.18)
By the standard projection argument as in [1], we can get from the approximate equation (3.3) and the estimates (3.15)–(3.17) that
(3.19)
From (3.15)-(3.16), going to a subsequence if necessary, there exists such that
(3.20)
(3.21)
(3.22)
and in view of (3.18), there exists such that
(3.23)
By applying the Lions-Aubin compactness Lemma in [7], we get, from (3.15) and (3.16),
(3.24)

and a.e. in .

Since the embedding is compact, we get, from (3.18) and (3.19),
(3.25)
Using the growth condition (2.3) and (3.25), we see that
(3.26)
is bounded and
(3.27)
Therefore, from [7, Chapter , Lemma ], we infer that
(3.28)
With these convergences, we can pass to the limit in the approximate equation and then
(3.29)

Obviously, satisfies the estimates (2.5)-(2.6). Finally, using the standard monotonicity argument as done in [1, 7], we get that . This completes the proof of existence of solution .

To prove the uniqueness, we assume that and are two solutions which satisfy (2.5)-(2.6) and . Setting , and . We see from (1.1) and (1.2) that
(3.30)
Multiplying (3.30) by and integrating over , we have
(3.31)
Now setting , then
(3.32)
Note that
(3.33)
Then, by the estimates (2.6) and , we have
(3.34)

with .

For the term of the right side to (3.31), we have
(3.35)

with , .

By the assumption and , we see that
(3.36)
By the estimate (2.6), we have
(3.37)
Therefore, there exists , depending such that
(3.38)
Since , , we get
(3.39)
Then (3.35) becomes
(3.40)
Therefore, it follows from (3.31), (3.34), and (3.40) that
(3.41)
The integral inequality (3.41) shows that there exists , such that
(3.42)

Consequently, .

Repeating the above procedure, we conduce that on and on . This ends the proof of uniqueness.

Next, we prove that . Let , we have
(3.43)

This shows that . We complete the proof of Theorem 2.2.

4. Proof of Theorem 2.3

Let us first state a well-known lemma that will be needed later.

Lemma 4.1 (see [10]).

Let be a nonincreasing function and assume that there are constants and , such that
(4.1)
Then, we have
(4.2)

4.1. The Proof of Theorem 2.3

Let
(4.3)
Then, we have from (1.1) that
(4.4)

This shows that is nonincreasing in .

Multiplying (1.1) by with , we get
(4.5)
Note that
(4.6)
Then we havefrom (4.5) that
(4.7)
Since , . Further, by (4.4), we see that
(4.8)

with .

This gives
(4.9)

where the fact that is nonincreasing is used.

Similarly, we derive the following estimates
(4.10)
(4.11)
(4.12)
Then we get from (4.9)–(4.12) that
(4.13)
for any , letting , we find that
(4.14)
By Lemma 4.1, we obtain that
(4.15)

This is (2.9) and we complete the proof of Theorem 2.3.

4.2. The Proof of Theorem 2.4

By Sobolev inequality, we know that there exists such that
(4.16)
Let be a solution for (1.1)-(1.2) in Theorem 2.2. By (2.10),
(4.17)
Obviously, there exists , such that ,
(4.18)
This implies that
(4.19)
On the other hand, we have, from (4.18) and (4.19),
(4.20)
It shows that
(4.21)
Then, by (4.9) and (4.11)–(4.14), we have
(4.22)
The applications of Lemma 4.1 and (4.19) yields that
(4.23)

This ends the proof of Theorem 2.4.

Declarations

Acknowledgments

The authors express their sincere gratitude to the anonymous referees for a number of valuable comments and suggestions. The work was supported by the Science Funds of Hohai University (Grant Nos. 2008430211 and 2008408306) and the Fundamental Research Funds for the Central Universities (Grant No. 2010B17914).

Authors’ Affiliations

(1)
Department of Mathematics, Hohai University, Nanjing, China

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Copyright

© Caisheng Chen et al. 2010

This article is published under license to BioMed Central Ltd. 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.

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