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Volume 2010, Article ID 216760, 15 pagesdoi:10.1155/2010/216760 Research Article Global Existence, Uniqueness, and Type Wave Equation Caisheng Chen, Huaping Yao, and Ling Shao Department

Volume 2010, Article ID 216760, 15 pages

doi:10.1155/2010/216760

Research Article

Global Existence, Uniqueness, and

Type Wave Equation

Caisheng Chen, Huaping Yao, and Ling Shao

Department of Mathematics, Hohai University, Nanjing, Jiangsu, 210098, China

Correspondence should be addressed to Caisheng Chen,cshengchen@hhu.edu.cn

Received 10 May 2010; Accepted 13 July 2010

Academic Editor: Michel C Chipot

Copyrightq 2010 Caisheng Chen 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

We study the global existence and uniqueness of a solution to an initial boundary value problem

for the nonlinear wave equation with the p-Laplacian operator u tt −div|∇u| p−2∇u−Δu t gx, u 

f x Further, the asymptotic behavior of solution is established The nonlinear term g likes

g x, u  ax|u| α−1u − bx|u| β−1u with appropriate functions a x and bx, where α > β ≥ 1.

1 Introduction

This paper is concerned with the global existence, uniqueness, and asymptotic behavior of

solution for the nonlinear wave equation with the p-Laplacian operator

u tt− div|∇u| p−2∇u− Δut  gx, u  fx, in Ω × 0, ∞, 1.1

u x, 0  u0x, utx, 0  u1x, in Ω; u x, t  0, on ∂Ω × 0, ∞, 1.2

where 2 ≤ p < n and Ω is a boundary domain in R n with smooth boundary ∂Ω The assumptions on f, g, u0and u1will be made in the sequel

Recently, Ma and Soriano in1 investigated the global existence of solution ut for

the problem1.1-1.2 under the assumptions

p  n, g uu ≥ 0, g u ≤ Cβexp

β |u| n/ n−1

, u ∈ R. 1.3

Moreover, if f  0 and ugu ≥ Gu, then there exist positive constants c and γ such that

E t ≤ c exp−γt, t ≥ 0, if n  2, 1.4

E t ≤ c1  t −n/n−2 , t ≥ 0, if n ≥ 3, 1.5 where

E t  1

2utt2

2 1

n ∇ut n

n



ΩG x, utdx 1.6

with Gx, u u

0 f x, sds.

Gao and Ma in2 also considered the global existence of solution for 1.1-1.2 In

Theorem 3.1 of2 , the similar results to 1.4-1.5 for asymptotic behavior of solution were

obtained if the nonlinear function gx, u  gu satisfies

g u ≤ a|u| σ−1 b, ugu ≥ ρGu ≥ 0, in Ω × R, 1.7

where a, b > 0, ρ > 0, 1 < σ < np/n − p if 1 < p < n and 1 < σ < ∞ if n ≤ p.

More precisely, they obtained that the global existence of solution for1.1-1.2 if one

of the following assumptions was satisfied:

i 1 < σ < p, the initial data u0, u1 ∈ W 1,p

0 Ω × L2Ω;

ii p < σ, the initial data u0, u1 ∈ W 1,p

0 Ω × L2Ω is small

Similar consideration can be found in3 5 In 6 , Yang obtained the uniqueness of solution of the Laplacian wave equation1.1-1.2 for n  1 To the best of our knowledge,

there are few information on the uniqueness of solution of1.1-1.2 for n > 1 and p > 2.

In this paper, we are interested in the global existence, the uniqueness, the continuity and the asymptotic behavior of solution for1.1-1.2 The nonlinear term g in 1.1 likes

g x, u  ax|u| α−1u − bx|u| β−1u with α > β ≥ 1 and a, b ≥ 0 Obviously, the sign condition

ug u ≥ 0 fails to hold for this type of function.

For these purposes, we must establish the global existence of solution for1.1-1.2 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 ut.

For the decay property, like1.5, we use the method recently introduced by Martinez 8 to

study the decay rate of solution to the wave equation utt − Δu  gut  0 in Ω × R, whereΩ

is a bounded domain ofRn

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 problem1.1-1.2 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 LetΩ be a bounded domain in Rnwith smooth

boundary ∂Ω We denote the space L p and W01,p for L p Ω and W 1,p

0 Ω and relevant norms

by · pand · 1,p, respectively In general, · Xdenotes the norm of Banach space X We

also denote by ·, · and ·, · the inner product of L2Ω and the duality pairing between

W01,p Ω and W −1,p Ω, respectively As usual, we write ut instead ux, t Sometimes, let

u t represent for utt and so on.

If T > 0 is given and X is a Banach space, we denote by L p 0, T; X the space of functions which are L pover0, T and which take their values in X In this space, we consider

the norm

u L p 0,T;X

T

0

ut p

X dt

1/p , 1≤ p < ∞,

u L∞0,T;X ess sup

0≤t≤Tut X

2.1

Let us state our assumptions on f and g.

A1 f ∈ L p with p  p/p − 1, p > 1.

A2 Let gx, u ∈ C1Ω × R and satisfy

ug x, u  h1x|u| ≥ k0Gx, u  h1x|u| ≥ 0, in Ω × R 2.2 and growth condition

g x, u ≤ k1



|u| α  h2x, g ux, u ≤ k1



|u| α−1 h3x, inΩ × R 2.3

with some k0, k1 > 0 and the nonnegative functions h1x ∈ L p , h2 ∈ L2 ∩ L α1/α , h3 ∈

L2∩ L α1/α−1, where 1≤ α ≤ np/n − p − 1, Gx, u u

0 g x, sds.

A typical function g is gx, u  ax|u| α−1u − bx|u| β−1u with the appropriate

nonnegative functions ax and bx, where α > β ≥ 1.

Definition 2.1see 7  A measurable function u  ux, t on Ω × R is said to be aweak solution of1.1-1.2 if all T > 0, u ∈ L∞0, T; W 1,p

0 , ut ∈ L20, T; W 1,2

0 , utt ∈ L20, T; W −1,p,

and u satisfies1.2 with u0, u1 ∈ W 1,p

0 and the integral identity



Ω



u tt φ  |∇u| p−2∇u · ∇φ  ∇ut · ∇φ  gφ − fφdx 0 2.4

for all φ ∈ C∞

0 Ω

Now we are in a position to state our results

Theorem 2.2 Assume A1-A2 hold and u0, u1 ∈ W01,p ×L2 Then the problem1.1-1.2 admits

a solution u t satisfying

u ∈ C0, ∞; , W 1,2

0



∩ L∞

0, ∞; , W01,p,

u t ∈ L2

0, ∞; , W 1,2

0



, u tt ∈ L2

loc



0, ∞; , W −1,p

,

2.5

and the following estimates

∇utt2

2 ∇ut p

p

t

0∇uts2

2ds ≤ C1A  B, ∀t ≥ 0, 2.6

where

A  u0p

p  ∇u0α1

p  u12

2, B  H1 H2 H3 F, 2.7

with F  f p

p , H i  hi p

p , i  1, 2, H3 h3λ1

λ1, λ1 n/2.

Further, if 1≤ α ≤ n  p/n − p and 2 ≤ p ≤ 4, the solution satisfying 2.5-2.6 is unique

Theorem 2.3 Let u be a solution of 1.1-1.2 with f  0 In addition, let 2 < p < n and

g x, uu ≥ pGx, u ≥ 0, in Ω × R. 2.8

Then there exists C0 C0u0, u1, such that

∇utt2

2 ∇ut p

p



ΩG x, ux, tdx ≤ C01  t −p/p−2 , ∀t ≥ 0. 2.9

The following theorem shows that the asymptotic estimate2.9 can be also derived if assumption2.8 fails to hold

Theorem 2.4 Let u be a solution of 1.1-1.2 with f  0 In addition, let 2 < p < n and

g x, u  λ|u| α−1u − |u| β−1u, inΩ × R 2.10

with p < β 1 < 2p, β < α < np/n−p Then there exists C0 C0u0, u1 > 0 and λ2 λ2α, β > 0,

such that λ > λ2, the solution u t satisfies

∇utt2

2 ∇ut p

p  ut α1

α1≤ C01  t −p/p−2 , ∀t ≥ 0. 2.11

3 Proof of Theorem 2.2

In this section, we assume that all assumptions inTheorem 2.2are satisfied We first prove the global existence of a solution to problem1.1-1.2 with the Faedo-Galerkin method as

in1,2,7,9

Let r be an integer for which the embedding H r

0Ω  W r,2

0

1,p

0 Ω is

continuous Let w jj  1, 2,  be eigenfunctions of the spectral problem



w j , v

H r

0 λjw j , v

, ∀v ∈ H r

0Ω, 3.1

where ·, · H r

0 represents the inner product in H r

0Ω Then the family {w1, w2, , w m , }

yields a basis for both H0r Ω and L2Ω For each integer m, let Vm  span{w1, w2, , w m}.

We look for an approximate solution to problem1.1-1.2 in the form

u mt m

j1

where Tjmt are the solution of the nonlinear ODE system in the variant t:



u m , w j



−Δpu m , w j



−Δu m , w j



g, w j



f, w j



, j  1, 2, m 3.3

with the p-Laplacian operatorΔpu  div|∇u| p−2∇u and the initial conditions

u m0  u 0m , u m 0  u 1m , 3.4

where u 0m and u 1m are chosen in V mso that

u 0m −→ u0 in W01,p , u 1m −→ u1 in L2. 3.5

As it is well known, the system3.3-3.4 has a local solution umt on some interval

0, tm We claim that for any T > 0, such a solution can be extended to the whole interval 0, T

by using the first a priori estimate below We denote by Ckthe constant which is independent

of m and the initial data u0and u1

Multiplying3.3 by T jm t and summing the resulting equations over j, we get after

integration by parts

E m t  ∇um t 2

2 0, ∀t ≥ 0, 3.6 where

E mt  1

2 u m t 2

21

p ∇umt p

p



ΩG x, umdx −



Ωf xum dx. 3.7

By2.2 and Young inequality, we have



ΩG x, umdx ≥ −



Ωh1x|um|dx ≥ −ε∇um p

p − Cεh1p

p ,



Ωf xum dx ≥ −ε∇um p

p − Cε f p

p

3.8

Let ε > 0 be so small that 2p−1− 4ε ≥ p−1 Then

E mt ≥ 1

2 u m t 2

2 1

2p ∇umt p

p − C1H1 F, 3.9

or

u m t 2

2 ∇umt p

p ≤ C1Emt  H1 F1 3.10

for some C1> 0.

Thus, it follows from3.6 and 3.10 that, for any m  1, 2, , and t ≥ 0

u m t 2

2 ∇umt p

p

t

0

∇ums2

2ds ≤ C2Em0  H1 F1. 3.11

By assumptionA2, we obtain that α  1 ≤ np/n − p and





ΩG x, umdx

 ≤ k1 um α1

α1



Ω|h2||um|dx



≤ C2



∇um α1

p  um p

p  h2p

p



≤ C2



∇um α1

p  ∇um p

p  H2



.

3.12

Then it follows3.5 and 3.6 that

E mt ≤ Em0  1

2 u 1m 2

2 1

p ∇u 0mp

p



ΩG x, u 0m  dx −



Ωf xu 0m dx

≤ C2



u12

2 ∇u0p

p  ∇u0α

p  H1 H2 F

≤ C2A  B.

3.13

Hence, for any t ≥ 0 and m  1, 2, , we have from 3.11 and 3.13 that

u m t 2

2 ∇umt p

p

t

0

∇um s 2

2ds ≤ C2A  B, ∀t ≥ 0. 3.14

With this estimate we can extend the approximate solution umt to the interval 0, T

and we have that

{umt} is bounded in L∞

0, T; W01,p

{u m t} is bounded in L∞

0, T; L2

{u m t} is bounded in L2

0, T; W01,2

Now we recall that operator−Δpu  − div|∇u| p−2∇u is bounded, monotone, and hemicontinuous from W01,p to W −1,p with p≥ 2 Then we have



−Δpu mtis bounded L∞

0, T; W −1,p

By the standard projection argument as in 1 , we can get from the approximate equation3.3 and the estimates 3.15–3.17 that



u m tis bounded in L2

0, T; H −rΩ. 3.19

From3.15-3.16, going to a subsequence if necessary, there exists u such that

u m u weakly star in L∞

0, T; W01,p

u m u weakly star in L∞

0, T; L2

u m u weakly in L2

0, T; L2

and in view of3.18, there exists χt such that

−Δpu mt χt weakly star in L∞

0, T; W −1,p

. 3.23

By applying the Lions-Aubin compactness Lemma in 7 , we get, from 3.15 and

3.16,

u m −→ u strongly in L2

0, T; L2

and u m → u a.e in Ω × 0, T.

Since the embedding W01,2 → L2is compact, we get, from3.18 and 3.19,

u m −→ u strongly in L2

0, T; L2

Using the growth condition2.3 and 3.25, we see that

T

0



Ω

g x, umx, tα1/α dx dt 3.26

is bounded and

g x, um −→ gx, u a.e in Ω × T. 3.27 Therefore, from7, Chapter 1, Lemma 1.3 , we infer that

g x, um gx, u weakly in L α1/α

0, T; L α1/α

. 3.28

With these convergences, we can pass to the limit in the approximate equation and then

d

dt



u t, vχ t, v∇u , ∇vg, v

f, v

, ∀v ∈ W 1,p

0 . 3.29

Obviously, u satisfies the estimates2.5-2.6 Finally, using the standard monotonic-ity argument as done in 1, 7 , we get that χt  −Δp u t This completes the proof of existence of solution ut.

To prove the uniqueness, we assume that ut and vt are two solutions which satisfy

2.5-2.6 and u0  v0, ut0  vt0 Setting Ut  utt, V t  vtt, and Wt 

U t − V t We see from 1.1 and 1.2 that

W t − ΔW − div|∇u| p−2∇u − |∇v| p−2∇v gx, v − gx, u. 3.30 Multiplying3.30 by W and integrating over Ω, we have

1

2

d

dt Wt2

2 ∇Wt2

2



Ω



|∇u| p−2∇u − |∇v| p−2∇v∇Wdx 



Ω



g x, v − gx, uWdx,

Wt2

2 2

t

0

∇Ws2

2ds 2

t

0



Ω



|∇u| p−2∇u − |∇v| p−2∇v∇W dx dτ

 2

t

0



Ω



g x, v − gx, uW dx ds

3.31

Now setting U  u  1 − v, 0 ≤  ≤ 1, then

t

0



Ω



|∇u| p−2∇u − |∇v| p−2∇v∇Wdx dτ

≤

t

0



Ω







1

0

d d



|∇U| p−2∇Ud





|∇W |dx dτ

≤p− 1 t

0



Ω

1

0

|∇U| p−2|∇uτ − vτ||∇W|d dx dτ ≡ I.

3.32

Note that

|∇Uτ| ≤ |∇uτ|  |∇vτ|,

|∇uτ − vτ| ≤

τ

0

|∇uss − vss|ds 

τ

0

|∇Ws|ds. 3.33

Then, by the estimates2.6 and 2 ≤ p ≤ 4, we have

I ≤ C1

t

0



Ω

τ

0



|∇uτ| p−2 |∇vτ| p−2

|∇Ws||∇Wτ|dx ds dτ

≤ C1

t

0

τ

0



∇uτ p−2

p  ∇vτ p−2

p



∇Ws2∇Wτ2ds dτ

≤ C1A  B p−2/p

t

0

τ

0

∇Ws2∇Wτ2ds dτ

≤ C1A  B p−2/p

t

0

∇Ws2ds

2

≤ C2t

t

0

∇Ws2

2ds

3.34

with C2 C1A  B p−2/p

For the term of the right side to3.31, we have

G1

t

0



Ω

g x, v − gx, u|W|dxdτ t

0



Ω







1

0

d d g x, Ud



|W |dx dτ

≤

t

0



Ω

1

0

g ux, Uuτ − vτWτd dxdτ

≤

t

0

τ

0

1

0

g ux, U

λ1d uss − vss λ2Wτ λ2d ds dτ

3.35

with λ1  n/2, λ2 2n/n − 2.

By the assumptionA2 and 1 ≤ α ≤ n  p/n − p, we see that

g ux, U λ1

λ1 ≤ k1



Ω



|uτ| α−1 |vτ| α−1 |h3|n/2 dx

≤ C3



Ω



|uτ| n α−1/2  |vτ| n α−1/2  |h3|n/2

dx

≤ C3



∇uτ n α−1/2

p  ∇vτ n α−1/2

p  H3



.

3.36

By the estimate2.6, we have

∇ut p , vt p ≤ C2A  B 1/p , ∀t ≥ 0. 3.37

Therefore, there exists C4 > 0, depending u0, v0, f, h isuch that

g ux, U

λ1≤ C4, ∀t ≥ 0. 3.38

Since u, v ∈ W 1,p

0 ⊂ W 1,2

0 , u t , v t ∈ W 1,2

0 , we get

uss − vss λ2≤ C0∇uss − vss2 C0∇Ws2,

Wτ2 ≤ C0∇Wτ2.

3.39

Then3.35 becomes

G1≤ C4

t

0

τ

0

Ws λ2Wτ λ2dsdτ ≤ C4

t

0

∇Ws2ds

2

≤ C4t

t

0

∇Ws2

2ds.

3.40 Therefore, it follows from3.31, 3.34, and 3.40 that

Wt2

2 2

t

0

∇Ws2

2ds ≤ C2 C4t

t

0

∇Ws2

2. 3.41 The integral inequality3.41 shows that there exists T1> 0, such that

W t  0, 0≤ t ≤ T1. 3.42

Consequently, ut − vt  u0 − v0  0, 0 ≤ t ≤ T1

Repeating the above procedure, we conduce that ut  vt on T1, 2T1 , 2T1, 3T1 , and ut  vt on 0, ∞ This ends the proof of uniqueness.

Next, we prove that u ∈ C0, ∞; W 1,2

0  Let t > s ≥ 0, we have

∇ut − us2

2



Ω







t

s

∇uττdτ





2

dx≤



Ω

t

s

|∇uττ|2ds dx t − s

 t − s

t

s

∇uτ τ2

2dτ −→ 0, as t −→ s.

3.43

This shows that ut ∈ C0, ∞; W 1,2

0  We complete the proof ofTheorem 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 E : R → R be a nonincreasing function and assume that there are constants q ≥ 0 and γ > 0, such that

∞

S

E q1tdt ≤ γ−1E q 0ES, ∀S ≥ 0. 4.1

Then, we have

E t ≤ E0 1 q

1 qγt

1/q , ∀t ≥ 0, if q > 0,

E t ≤ E0e1−γt, ∀t ≥ 0, if q  0.

4.2

Let

E t  1

2utt2

2 1

p ∇ut p p



ΩG x, udx, t ≥ 0. 4.3 Then, we have from1.1 that

E t  ∇utt2

2 0, ∀t ≥ 0. 4.4

This shows that Et is nonincreasing in 0, ∞.

With this estimate we can extend the approximate solution umt to the interval 0, T

and we...

Ωf xum dx. 3.7

By2.2 and Young inequality, we have



ΩG... − 2.

By the assumptionA2 and ≤ α ≤ n  p/n − p, we see