Open Access

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

Journal of Inequalities and Applications20102010:216760

Received: 10 May 2010

Accepted: 13 July 2010

Published: 2 August 2010


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 .

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

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
Moreover, if and then there exist positive constants and such that


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

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

Let us state our assumptions on and .

() with .

()Let and satisfy

and growth condition

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

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
and the following estimates

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
Then there exists , such that

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
with . Then there exists and , such that the solution satisfies

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
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
where are the solution of the nonlinear ODE system in the variant :
with the -Laplacian operator and the initial conditions
where and are chosen in so that

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
By (2.2) and Young inequality, we have
Let be so small that . Then

for some .

Thus, it follows from (3.6) and (3.10) that, for any and
By assumption , we obtain that and
Then it follows (3.5) and (3.6) that
Hence, for any and , we have from (3.11) and (3.13) that
With this estimate we can extend the approximate solution to the interval and we have that
Now we recall that operator is bounded, monotone, and hemicontinuous from to with . Then we have
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
From (3.15)-(3.16), going to a subsequence if necessary, there exists such that
and in view of (3.18), there exists such that
By applying the Lions-Aubin compactness Lemma in [7], we get, from (3.15) and (3.16),

and a.e. in .

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

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
Multiplying (3.30) by and integrating over , we have
Now setting , then
Note that
Then, by the estimates (2.6) and , we have

with .

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

with , .

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

Consequently, .

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

Next, we prove that . Let , we have

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
Then, we have

4.1. The Proof of Theorem 2.3

Then, we have from (1.1) that

This shows that is nonincreasing in .

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

with .

This gives

where the fact that is nonincreasing is used.

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

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

This ends the proof of Theorem 2.4.



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

Department of Mathematics, Hohai University, Nanjing, China


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© Caisheng Chen et al. 2010

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