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

Volume 2010, Article ID 216760,15pages doi:10.1155/2010/216760

Research Article

Global Existence, Uniqueness, and

Asymptotic Behavior of Solution for

p

-Laplacian

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,[email protected]

Received 10 May 2010; Accepted 13 July 2010

Academic Editor: Michel C. Chipot

Copyrightq2010 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 thep-Laplacian operatorutt−div|∇u|p−2∇u−Δutgx, u

fx. Further, the asymptotic behavior of solution is established. The nonlinear term g likes

gx, u ax|u|α−1u−bx|u|β−1uwith appropriate functionsaxandbx, whereα > β≥1.

1. Introduction

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

utt−div|∇u|p−2∇u−Δutgx, u fx, in Ω×0,∞, 1.1

ux,0 u0x, utx,0 u1x, inΩ; ux, t 0, on∂Ω×0,∞, 1.2

where 2 ≤ p < n and Ω is a boundary domain in Rn _{with smooth boundary ∂Ω. The}

assumptions onf, g, u0andu1will be made in the sequel.

Recently, Ma and Soriano in1 investigated the global existence of solutionutfor the problem1.1-1.2under the assumptions

pn, guu≥0, gu≤Cβexp

Moreover, iff 0 andugu≥Gu,then there exist positive constantscandγsuch that

Et≤cexp−γt, t≥0, ifn2, 1.4

Et≤c1t−n/n−2, t≥0, ifn≥3, 1.5

where

Et 1

2utt

2 2

1

n∇ut n n

ΩGx, utdx 1.6

withGx, u ₀ufx, sds.

Gao and Ma in2 also considered the global existence of solution for1.1-1.2. In Theorem 3.1 of2 , the similar results to1.4-1.5for asymptotic behavior of solution were obtained if the nonlinear functiongx, u gusatisfies

gu≤a|u|σ−1b, ugu≥ρGu≥0, in Ω×R, 1.7

wherea, b >0, ρ >0, 1< σ < np/n−pif 1< p < nand 1< σ <∞ifn≤p.

More precisely, they obtained that the global existence of solution for1.1-1.2if one of the following assumptions was satisfied:

i1< σ < p, the initial datau0, u1∈W₀1,pΩ×L2Ω;

iip < σ, the initial datau0, u1∈W₀1,pΩ×L2Ωis small.

Similar consideration can be found in3–5 . In6 , Yang obtained the uniqueness of solution of the Laplacian wave equation1.1-1.2forn 1. To the best of our knowledge, there are few information on the uniqueness of solution of1.1-1.2forn >1 andp >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 termg in1.1likes

gx, u ax|u|α−1_u−bx|u|β−1_{uwithα > β ≥1 anda, b ≥0. Obviously, the sign condition}

ugu≥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 solutionut. For the decay property, like1.5, we use the method recently introduced by Martinez8 to study the decay rate of solution to the wave equationutt−Δugut 0 inΩ×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

2. Assumptions and Main Results

We first give some notations and definitions. LetΩbe a bounded domain inRn_{with smooth}

boundary∂Ω. We denote the spaceLp_andW1,p

0 forLpΩandW 1,p

0 Ωand relevant norms

by · pand · 1,p, respectively. In general, · Xdenotes the norm of Banach spaceX. We

also denote by ·,·and ·,· the inner product of L2_{Ω and the duality pairing between}

W₀1,pΩand W−1,p_{Ω, respectively. As usual, we writeutinsteadux, t. Sometimes, let} utrepresent foruttand so on.

If T > 0 is given and X is a Banach space, we denote by Lp_{0, T;X the space of}

functions which areLp_{over0, Tand which take their values inX. In this space, we consider}

the norm

u_Lp_0,T;X

T

0

utp_Xdt

1/p

, 1≤p <∞,

u_L∞_0,T;Xess sup

0≤t≤T

ut_X.

2.1

Let us state our assumptions onfandg.

A1f ∈Lp

withpp/p−1, p >1.

A2Letgx, u∈C1Ω×Rand satisfy

ugx, u h1x|u| ≥k0Gx, u h1x|u|≥0, inΩ×R 2.2

and growth condition

_{gx, u≤k1}

|u|αh2x

, gux, u≤k1

|u|α−1h3x

, inΩ×R 2.3

with somek0, k1 > 0 and the nonnegative functions h1x ∈ Lp

,h2 ∈ L2 ∩Lα1/α, h3 ∈

L2_∩Lα1/α−1_{, where 1≤α≤np/n−p−1,Gx, u} u

0 gx, sds.

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

nonnegative functionsaxandbx, whereα > β≥1.

Definition 2.1see7 . A measurable functionu ux, tonΩ×R is said to be aweak solution of1.1-1.2if allT >0,u∈L∞0, T;W₀1,p,ut∈L20, T;W₀1,2,utt∈L20, T;W−1,p, andusatisfies1.2withu0, u1∈W01,pand the integral identity

Ω

uttφ|∇u|p−2∇u· ∇φ∇ut· ∇φgφ−fφdx0 2.4

for allφ∈C∞₀ Ω.

Theorem 2.2. AssumeA1-A2hold andu0, u1∈W₀1,p×L2. Then the problem1.1-1.2admits

a solutionutsatisfying

u∈C0,∞;, W₀1,2∩L∞0,∞;, W₀1,p,

ut∈L20,∞;, W₀1,2, utt∈L2_loc0,∞;, W−1,p,

2.5

and the following estimates

∇utt22∇ut

p p

_t

0

∇uts22ds≤C1AB, ∀t≥0, 2.6

where

Au0pp∇u0αp1u122, BH1H2H3F, 2.7

withFf_pp, Hihip

p, i1,2, H3h3 λ1

λ1, λ1n/2.

Further, if 1≤ α≤np/n−pand 2 ≤p ≤4, the solution satisfying2.5-2.6is unique.

Theorem 2.3. Letube a solution of 1.1-1.2withf0. In addition, let2< p < nand

gx, uu≥pGx, u≥0, inΩ×R. 2.8

Then there existsC0C0u0, u1, such that

∇utt2₂∇utpp

ΩGx, ux, tdx≤C01t

−p/p−2_{, ∀t≥0. 2.9}

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

Theorem 2.4. Letube a solution of 1.1-1.2withf0. In addition, let2< p < nand

gx, u λ|u|α−1u− |u|β−1u, inΩ×R 2.10

withp < β1<2p, β < α < np/n−p. Then there existsC0C0u0, u1>0andλ2λ2α, β>0,

such thatλ > λ2,the solutionutsatisfies

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 problem1.1-1.2with the Faedo-Galerkin method as in1,2,7,9 .

Let r be an integer for which the embedding Hr

0Ω W

r,2

0 Ω → W 1,p

0 Ω is

continuous. Letwjj1,2, . . .be eigenfunctions of the spectral problem

wj, v

Hr 0λj

wj, v

, ∀v∈Hr

0Ω, 3.1

where ·,·_Hr

0 represents the inner product in H

r

0Ω. Then the family {w1, w2, . . . , wm, . . .}

yields a basis for bothH₀rΩandL2Ω. For each integerm, letVmspan{w1, w2, . . . , wm}.

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

umt m

j1

Tjmtwj, 3.2

whereTjmtare the solution of the nonlinear ODE system in the variantt:

u_m, wj−Δpum, wj−Δu_m, wjg, wjf, wj, j1,2, . . . m 3.3

with thep-Laplacian operatorΔpudiv|∇u|p−2∇u and the initial conditions

um0 u0m, um 0 u1m, 3.4

whereu0mandu1mare chosen inVmso that

u0m−→u0 inW01,p, u1m−→u1 inL2. 3.5

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

0, tm.We claim that for anyT >0, such a solution can be extended to the whole interval0, T

by using the first a priori estimate below. We denote byCkthe constant which is independent ofmand the initial datau0andu1.

Multiplying3.3byT_jm tand summing the resulting equations overj, we get after integration by parts

E_mt ∇u_mt2₂0, ∀t≥0, 3.6

where

Emt

1 2u

mt

2 2

1

p∇umt p p

ΩGx, umdx−

By2.2and Young inequality, we have

ΩGx, umdx≥ −

Ωh1x|um|dx≥ −ε∇um

p

p−Cεh1p

p,

Ωfxumdx≥ −ε∇um

p

p−Cεfp

p.

3.8

Letε >0 be so small that 2p−1_−4ε≥p−1_{. Then}

Emt≥ 1

2u

mt

2 2

1

2p∇umt p

p−C1H1F, 3.9

or

u_mt₂2∇umtpp≤C1Emt H1F1 3.10

for someC1>0.

Thus, it follows from3.6and3.10that, for anym1,2, . . . ,andt≥0

u_mt2₂∇umtpp

_t

0

∇ums22ds≤C2Em0 H1F1. 3.11

By assumptionA2, we obtain thatα1≤np/n−pand

ΩGx, umdx

≤k1

umα_α1₁

Ω|h2||um|dx

≤C2

∇umα_p1umpph2p

p

≤C2

∇umα_p1∇umppH2

.

3.12

Then it follows3.5and3.6that

Emt≤Em0

1 2u

1m

2 2

1

p∇u0m p p

ΩGx, u0mdx−

Ωfxu0mdx

≤C2

u122∇u0pp∇u0αpH1H2F

≤C2AB.

3.13

Hence, for anyt≥0 andm1,2, . . ., we have from3.11and3.13that

u_mt2₂∇umtpp

_t

0

With this estimate we can extend the approximate solutionumtto the interval0, T

and we have that

{umt} is bounded inL∞

0, T;W₀1,p, 3.15

{u_mt} is bounded inL∞0, T;L2_{, 3.16}

{u_mt}is bounded inL2_{0, T;W}1,2 0

. 3.17

Now we recall that operator−Δpu −div|∇u|p−2∇uis bounded, monotone, and

hemicontinuous fromW₀1,ptoW−1,p_{withp≥2. Then we have}

−Δpumtis boundedL∞0, T;W−1,p. 3.18

By the standard projection argument as in 1 , we can get from the approximate equation3.3and the estimates3.15–3.17that

u_mtis bounded inL2_{0, T; H}−r_{Ω. 3.19}

From3.15-3.16, going to a subsequence if necessary, there existsusuch that

um uweakly star inL∞0, T;W₀1,p, 3.20

u_m uweakly star inL∞0, T;L2, 3.21

u_m u weakly inL20, T;L2, 3.22

and in view of3.18, there existsχtsuch that

−Δpumt χtweakly star inL∞

0, T;W−1,p_{. 3.23}

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

3.16,

um−→ustrongly inL2

0, T;L2_{, 3.24}

Since the embeddingW₀1,2→L2_{is compact, we get, from3.18and3.19,}

u_m−→u strongly inL20, T;L2. 3.25

Using the growth condition2.3and3.25, we see that

_T

0

Ω

gx, umx, tα1/αdx dt 3.26

is bounded and

gx, um−→gx, u a.e. inΩ×T. 3.27

Therefore, from7, Chapter 1, Lemma 1.3 , we infer that

gx, um gx, uweakly inLα1/α

0, T;Lα1/α. 3.28

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

d dt

ut, vχt, v∇u,∇vg, vf, v, ∀v∈W₀1,p. 3.29

Obviously,usatisfies the estimates2.5-2.6. Finally, using the standard monotonic-ity argument as done in 1, 7 , we get that χt −Δput. This completes the proof of

existence of solutionut.

To prove the uniqueness, we assume thatutandvtare two solutions which satisfy

2.5-2.6 and u0 v0, ut0 vt0. Setting Ut utt, Vt vtt, andWt Ut−Vt. We see from1.1and1.2that

Wt−ΔW−div

|∇u|p−2∇u− |∇v|p−2∇vgx, v−gx, u. 3.30

Multiplying3.30byWand integrating overΩ, we have

1 2

d

dtWt

2

2∇Wt22

Ω

|∇u|p−2∇u− |∇v|p−2∇v∇Wdx

Ω

gx, v−gx, uWdx,

Wt2₂2

_t

0

∇Ws2₂ds2

_t

0

Ω

|∇u|p−2∇u− |∇v|p−2∇v∇W dx dτ

2

_t

0

Ω

gx, v−gx, uW dx ds

Now settingUu 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 estimates2.6and 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

∇uτpp−2∇vτpp−2

∇Ws₂∇Wτ₂ds dτ

≤C1ABp−2/p

_t

0

_τ

0

∇Ws₂∇Wτ₂ds dτ

≤C1ABp−2/p

t

0

∇Ws₂ds

2

≤C2t

t

0

∇Ws2₂ds

3.34

withC2C1ABp−2/p.

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

G1

_t

0

Ω

gx, v−gx, u|W|dx dτ

_t 0 Ω ₁ 0 d

dgx, Ud

|W|dx dτ

≤ _t 0 Ω ₁ 0

gux, Uuτ−vτWτd dxdτ

≤ t 0 τ 0 1 0

_{gux, U}

λ1duss−vssλ2Wτλ2d ds dτ

3.35

By the assumptionA2and 1≤α≤np/n−p, we see that

_{gux, U}λ1

λ1 ≤k1

Ω

|uτ|α−1|vτ|α−1|h3|

n/2

dx

≤C3

Ω

|uτ|nα−1/2|vτ|nα−1/2|h3|n/2

dx

≤C3

∇uτ_pnα−1/2∇vτn_pα−1/2H3

.

3.36

By the estimate2.6, we have

∇ut_p, vt_p≤C2AB1/p, ∀t≥0. 3.37

Therefore, there existsC4 >0, dependingu0, v0, f, hisuch that

gux, U_λ1≤C4, ∀t≥0. 3.38

Sinceu, v∈W₀1,p⊂W₀1,2,ut, vt∈W01,2, we get

uss−vssλ2≤C0∇uss−vss2C0∇Ws2,

Wτ₂ ≤C0∇Wτ2.

3.39

Then3.35becomes

G1≤C4

_t

0

_τ

0

Ws_λ2Wτ_λ2dsdτ ≤C4

t

0

∇Ws₂ds

2

≤C4t

_t

0

∇Ws2₂ds.

3.40

Therefore, it follows from3.31,3.34, and3.40that

Wt2₂2

_t

0

∇Ws2₂ds≤C2C4t

_t

0

∇Ws2₂. 3.41

The integral inequality3.41shows that there existsT1>0, such that

Wt 0, 0≤t≤T1. 3.42

Repeating the above procedure, we conduce thatut vtonT1,2T1 ,2T1,3T1 , . . .

andut vton0,∞. This ends the proof of uniqueness.

Next, we prove thatu∈C0,∞;W₀1,2. Lett > s≥0, we have

∇ut−us2₂

Ω

_t

s

∇uττdτ

2

dx≤

Ω

_t

s

|∇uττ|2ds dxt−s

t−s

t

s

∇uττ2₂dτ −→0, ast−→s.

3.43

This shows thatut∈C0,∞;W₀1,2. 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 4.1see10 . LetE : R → R be a nonincreasing function and assume that there are constantsq≥0andγ >0, such that

_∞

S

Eq1tdt≤γ−1Eq0ES, ∀S≥0. 4.1

Then, we have

Et≤E0

_1q

1qγt

1/q

, ∀t≥0, ifq >0,

Et≤E0e1−γt, ∀t≥0, ifq0.

4.2

4.1. The Proof of

Theorem 2.3

Let

Et 1

2utt

2 2

1

p∇ut p p

ΩGx, udx, t≥0. 4.3

Then, we have from1.1that

Et ∇utt₂20, ∀t≥0. 4.4

Multiplying1.1byEq_{tutwithq p−2/p >0, we get}

_T

S Eqt

Ωu

utt−Δpu−Δutgx, u

dx dt0, ∀T > S≥0. 4.5

Note that

_T

S

Eqtu, uttdt Eqtu, ut|TS−

_T

S

qEq−1tEtu, ut Eqtutt22

dt

−

T

S

Eqtu,Δpudt

T

S

Eqt∇utppdt,

−

T

S

Eqtu,Δutdt

T

S

Eqt∇u,∇utdt.

4.6

Then we have from4.5that

p

T

S

Eq1tdt −Eqtu, ut|T_Sq

T

S

Eq−1tEtu, utdt

1p 2

T

S

Eqtutt22dt−

_T

S

Eqt∇u,∇utdt

T

S Eqt

Ω

pGu−ugudx dt.

4.7

Since_ΩGx, udx≥0,Et≥0. Further, by4.4, we see that

∇utt2≤

−Et1/2, ∇ut_p≤pE1/pt, ∀t≥0,

|Eqtu, ut| ≤Eqtut2utt2≤C0Eqt∇utp∇utt2≤C0Etμ1

4.8

withμ1q1/21/p.

This gives

Eqtu, ut|TS≤C1Eμ1S, ∀T > S≥0, 4.9

Similarly, we derive the following estimates

T

S

Eqtutt2₂dt≤C1

T

S

Eqt∇utt2₂dt

C1

T

S

Eq_t−Etdt≤C

1Eq1S,

4.10

q

_T

S

Eq−1tEtu, utdt≤C1

_T

S

Eq−1tEtut₂utt2dt

≤C1

T

S

Eμ1−1_tEtdt≤_C

1Eμ1S,

4.11

_T

S

|Eqt∇u,∇ut|dt≤

_T

S

Eqt∇ut₂∇utt2dt

≤C1

T

S

Eq1/pt−Et1/2dt

≤

T

S

Eq1_tdtC 1

T

S

Eq2/p−1_t−Etdt

≤

_T

S

Eq1tdtC1Eq2/pS.

4.12

Then we get from4.9–4.12that

T

S

Eq1tdt≤C1

Eμ1_{S E}q1_{S E}q2/p_S

≤C1ES

Eμ1_{S E}q_{S E}q2/p−1_S

≤C1ESEq0

E1/p−1/20 1E2/p−10

≡γ−1Eq0ES,

4.13

for anyT > S≥0, lettingT → ∞, we find that

_∞

S

Eq1tdt≤γ−1ESEq0, ∀S≥0. 4.14

ByLemma 4.1, we obtain that

Et 1

2utt

2 2

1

p∇ut p p

ΩGx, udx≤E0

_1q

1qγt

1/q

≤C2E01t−p/p−2.

4.15

4.2. The Proof of

Theorem 2.4

By Sobolev inequality, we know that there existsλ0>0 such that

λ0upp≤ ∇upp, ∀u∈W₀1,pΩ. 4.16

Letube a solution for1.1-1.2inTheorem 2.2. By2.10,

Gu λ α1|u|

α1₋ 1

β1|u|

β1_{. 4.17}

Obviously, there existsλ2>0, such thatλ > λ2,

λ0

2p|u|

p_Gu≥ 1

2α1|u|

α1_{, ∀u∈R. 4.18}

This implies that

λ0

2pu p p

ΩGudx≥ 1 2α1u

α1

α1,

Et≥ 1

2utt

2 2

1

2p∇ut p p

1

2α1 ut

α1

α1.

4.19

On the other hand, we have, from4.18and4.19,

pGu−ugu β1−p β1 |u|

β1₋λ

α1−p α1 |u|

α1

≤ β1−p β1 |u|

β1_β1−p λ

α1|u|

α1_−Gu

≤β1−pλ0 p|u|

p_Gu

.

4.20

It shows that

_T

S Eqt

Ω

pGu−gudxdt≤β1−p T

S

Eq1tdt. 4.21

Then, by4.9and4.11–4.14, we have

2p−β−1

T

S

Eq1tdt≤C0

Eq1/p2S Eq1S Eq2/pS

≤γ−1ESEq0.

The applications ofLemma 4.1and4.19yields that

utt2₂∇ut2₂ut_αα1₁≤C01t−p/p−2, ∀t≥0. 4.23

This ends the proof ofTheorem 2.4.

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 UniversityGrant Nos. 2008430211 and 2008408306and the Fundamental Research Funds for the Central UniversitiesGrant No. 2010B17914.

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