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


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.


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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).

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Correspondence to Caisheng Chen.

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Chen, C., Yao, H. & Shao, L. Global Existence, Uniqueness, and Asymptotic Behavior of Solution for -Laplacian Type Wave Equation. J Inequal Appl 2010, 216760 (2010).

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