- Research Article
- Open access
- Published:
Global Existence, Uniqueness, and Asymptotic Behavior of Solution for
-Laplacian Type Wave Equation
Journal of Inequalities and Applications volume 2010, Article number: 216760 (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
.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ3_HTML.gif)
Moreover, if and
then there exist positive constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ7_HTML.gif)
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 [3–5]. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ8_HTML.gif)
Let us state our assumptions on and
.
() with
.
()Let and satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ9_HTML.gif)
and growth condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ12_HTML.gif)
and the following estimates
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ13_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ14_HTML.gif)
with .
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_IEq104_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ15_HTML.gif)
Then there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ16_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ17_HTML.gif)
with . Then there exists
and
, such that
the solution
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ20_HTML.gif)
where are the solution of the nonlinear ODE system in the variant
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ21_HTML.gif)
with the -Laplacian operator
and the initial conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ22_HTML.gif)
where and
are chosen in
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ23_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ24_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ25_HTML.gif)
By (2.2) and Young inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ26_HTML.gif)
Let be so small that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ27_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ28_HTML.gif)
for some .
Thus, it follows from (3.6) and (3.10) that, for any and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ29_HTML.gif)
By assumption , we obtain that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ30_HTML.gif)
Then it follows (3.5) and (3.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ31_HTML.gif)
Hence, for any and
, we have from (3.11) and (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ32_HTML.gif)
With this estimate we can extend the approximate solution to the interval
and we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ33_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ34_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ35_HTML.gif)
Now we recall that operator is bounded, monotone, and hemicontinuous from
to
with
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ37_HTML.gif)
From (3.15)-(3.16), going to a subsequence if necessary, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ38_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ40_HTML.gif)
and in view of (3.18), there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ41_HTML.gif)
By applying the Lions-Aubin compactness Lemma in [7], we get, from (3.15) and (3.16),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ42_HTML.gif)
and a.e. in
.
Since the embedding is compact, we get, from (3.18) and (3.19),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ43_HTML.gif)
Using the growth condition (2.3) and (3.25), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ44_HTML.gif)
is bounded and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ45_HTML.gif)
Therefore, from [7, Chapter , Lemma
], we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ46_HTML.gif)
With these convergences, we can pass to the limit in the approximate equation and then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ47_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ48_HTML.gif)
Multiplying (3.30) by and integrating over
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ49_HTML.gif)
Now setting , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ50_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ51_HTML.gif)
Then, by the estimates (2.6) and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ52_HTML.gif)
with .
For the term of the right side to (3.31), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ53_HTML.gif)
with ,
.
By the assumption and
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ54_HTML.gif)
By the estimate (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ55_HTML.gif)
Therefore, there exists , depending
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ56_HTML.gif)
Since ,
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ57_HTML.gif)
Then (3.35) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ58_HTML.gif)
Therefore, it follows from (3.31), (3.34), and (3.40) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ59_HTML.gif)
The integral inequality (3.41) shows that there exists , such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ60_HTML.gif)
Consequently, .
Repeating the above procedure, we conduce that on
and
on
. This ends the proof of uniqueness.
Next, we prove that . Let
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ61_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ62_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ63_HTML.gif)
4.1. The Proof of Theorem 2.3
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ64_HTML.gif)
Then, we have from (1.1) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ65_HTML.gif)
This shows that is nonincreasing in
.
Multiplying (1.1) by with
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ66_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ67_HTML.gif)
Then we havefrom (4.5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ68_HTML.gif)
Since ,
. Further, by (4.4), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ69_HTML.gif)
with .
This gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ70_HTML.gif)
where the fact that is nonincreasing is used.
Similarly, we derive the following estimates
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ71_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ72_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ73_HTML.gif)
Then we get from (4.9)–(4.12) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ74_HTML.gif)
for any , letting
, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ75_HTML.gif)
By Lemma 4.1, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ76_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ77_HTML.gif)
Let be a solution for (1.1)-(1.2) in Theorem 2.2. By (2.10),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ78_HTML.gif)
Obviously, there exists , such that
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ79_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ80_HTML.gif)
On the other hand, we have, from (4.18) and (4.19),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ81_HTML.gif)
It shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ82_HTML.gif)
Then, by (4.9) and (4.11)–(4.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ83_HTML.gif)
The applications of Lemma 4.1 and (4.19) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F216760/MediaObjects/13660_2010_Article_2088_Equ84_HTML.gif)
This ends the proof of Theorem 2.4.
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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).
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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). https://doi.org/10.1155/2010/216760
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DOI: https://doi.org/10.1155/2010/216760