Asymptotic Behavior of Solution for p-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≤C_βexp

β|u|^n/n−1

, u∈R. 1.3

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

2utt²₂ 1

n∇utⁿ_n

ΩGx, utdx 1.6

withGx, u _u

0 fx, 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∈W₀^1,pΩ×L²Ω;

iip < σ, the initial datau0, u1∈W₀^1,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−1u−bx|u|^β−1uwithα > 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 Section 4.

2. Assumptions and Main Results

We first give some notations and definitions. LetΩbe a bounded domain inRⁿwith smooth boundary∂Ω. We denote the spaceL^pandW₀^1,pforL^pΩandW₀^1,pΩand relevant norms by · pand · 1,p, respectively. In general, · Xdenotes the norm of Banach spaceX. We also denote by ·,·and ·,· the inner product of L²Ω and the duality pairing between W₀^1,pΩand W^−1,pΩ, 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 L^p0, T;X the space of functions which areL^pover0, Tand which take their values inX. In this space, we consider the norm

u_Lp0,T;X ^T

0

ut^p_Xdt _1/p

, 1≤p <∞, u_L^∞_0,T;Xess sup

0≤t≤Tut_X.

2.1

Let us state our assumptions onfandg.

A1f ∈L^pwithpp/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 ∈ L^p,h₂ ∈ L² ∩L^α1/α, h₃ ∈ L²∩L^α1/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−1u − bx|u|^β−1u 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;W₀^1,p,ut∈L²0, T;W₀^1,2,utt∈L²0, T;W^−1,p, andusatisfies1.2withu0, u₁∈W₀^1,pand the integral identity

Ω

uttφ|∇u|^p−2∇u· ∇φ∇ut· ∇φgφ−fφ

dx0 2.4

for allφ∈C^∞₀ Ω.

Now we are in a position to state our results.

Theorem 2.2. AssumeA1-A2hold andu0, u1∈W₀^1,p×L². Then the problem1.1-1.2admits a solutionutsatisfying

u∈C

0,∞;, W₀^1,2

∩L^∞

0,∞;, W₀^1,p , ut∈L²

0,∞;, W₀^1,2

, utt∈L²_loc

0,∞;, W^−1,p ,

2.5

and the following estimates

∇utt²₂∇ut^pp _t

0

∇uts²₂ds≤C₁AB, ∀t≥0, 2.6

where

Au0^pp∇u0^α1_p u1²₂, BH1H2H3F, 2.7

withFf^p_p, H_ihi^p_p, i1,2, H₃h3^λ_λ¹

1, 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, let 2< p < nand

gx, uu≥pGx, u≥0, inΩ×R. 2.8

Then there existsC0C0u0, u1, such that

∇utt²₂∇ut^p_p

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, let 2< 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>0 andλ2λ2α, β>0, such thatλ > λ₂,the solutionutsatisfies

∇utt²₂∇ut^pput^α1_α1≤C01t^−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 problem1.1-1.2with the Faedo-Galerkin method as in1,2,7,9 .

Let r be an integer for which the embedding H₀^rΩ W₀^r,2Ω → W₀^1,pΩ is continuous. Letw_jj1,2, . . .be eigenfunctions of the spectral problem

w_j, v

H^r₀λ_j w_j, v

, ∀v∈H₀^rΩ, 3.1

where ·,·_H^r

0 represents the inner product in H₀^rΩ. Then the family {w1, w₂, . . . , w_m, . . .}

yields a basis for bothH₀^rΩandL²Ω. 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, wj

g, wj

f, wj

, j1,2, . . . m 3.3

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

um0 u0m, u_m0 u1m, 3.4

whereu_0mandu_1mare chosen inV_mso that

u_0m−→u₀ inW₀^1,p, u_1m−→u₁ inL². 3.5 As it is well known, the system3.3-3.4has a local solutionu_mton 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_mt²

20, ∀t≥0, 3.6

where

E_mt 1

2u_mt²

21

p∇umt^pp

ΩGx, umdx−

Ωfxumdx. 3.7

By2.2and Young inequality, we have

ΩGx, umdx≥ −

Ωh1x|um|dx≥ −ε∇um^pp−Cεh1^p_p,

Ωfxumdx≥ −ε∇um^pp−C_εf^p

p.

3.8

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

Emt≥ 1

2u_mt²

2 1

2p∇umt^pp−C1H1F, 3.9

or

u_mt²

2∇umt^pp≤C₁Emt H₁F₁ 3.10

for someC1>0.

Thus, it follows from3.6and3.10that, for anym1,2, . . . ,andt≥0 u_mt²

2∇umt^pp _t

0

∇ums²₂ds≤C₂Em0 H₁F₁. 3.11

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

ΩGx, umdx ≤k1

um^α1_α1

Ω|h2||um|dx

≤C2

∇um^α1_p um^pph2^p_p

≤C2

∇um^α1_p ∇um^ppH2

.

3.12

Then it follows3.5and3.6that E_mt≤E_m0 1

2u_1m²

2 1

p∇u0m^pp

ΩGx, u0mdx−

Ωfxu0mdx

≤C₂

u1²₂∇u0^pp∇u0^α_pH₁H₂F

≤C2AB.

3.13

Hence, for anyt≥0 andm1,2, . . ., we have from3.11and3.13that u_mt²

2∇umt^pp _t

0

∇u_ms²

2ds≤C₂AB, ∀t≥0. 3.14

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

{umt} is bounded inL^∞

0, T;W₀^1,p

, 3.15

{u_mt} is bounded inL^∞

0, T;L²

, 3.16

{u_mt}is bounded inL²

0, T;W₀^1,2

. 3.17

Now we recall that operator−Δpu −div|∇u|^p−2∇uis bounded, monotone, and hemicontinuous fromW₀^1,ptoW^−1,pwithp≥2. Then we have

−Δpumt

is 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_mt

is bounded inL²

0, T; H^−rΩ

. 3.19

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

um uweakly star inL^∞

0, T;W₀^1,p

, 3.20

u_m uweakly star inL^∞

0, T;L²

, 3.21

u_m u weakly inL²

0, T;L²

, 3.22

and in view of3.18, there existsχtsuch that

−Δpu_mt 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,

u_m−→ustrongly inL²

0, T;L²

, 3.24

andu_m → ua.e. inΩ×0, T.

Since the embeddingW₀^1,2→L²is compact, we get, from3.18and3.19, u_m−→u strongly inL²

0, T;L²

. 3.25

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

0

Ω

gx, umx, t^α1/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 inL^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

ut, v

χt, v

∇u,∇v

g, v

f, v

, ∀v∈W₀^1,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 v_t0. Setting Ut u_tt, Vt v_tt, andWt Ut−Vt. We see from1.1and1.2that

W_t−ΔW−div

|∇u|^p−2∇u− |∇v|^p−2∇v

gx, v−gx, u. 3.30

Multiplying3.30byWand integrating overΩ, we have 1

2 d

dtWt²₂∇Wt²₂

Ω

|∇u|^p−2∇u− |∇v|^p−2∇v

∇Wdx

Ω

gx, v−gx, u Wdx,

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, u

W dx ds

3.31

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∇U

d

|∇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

≤C_{1 t}

0

_τ

0

∇uτ^p−2p ∇vτ^p−2p

∇Ws₂∇Wτ₂ds dτ

≤C₁AB^p−2/p _t

0

_τ

0

∇Ws₂∇Wτ₂ds dτ

≤C1AB^{p−2/p t}

0

∇Ws₂ds ₂

≤C2t _t

0

∇Ws²₂ds

3.34

withC2C1AB^p−2/p.

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

G_{1 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

g_ux, Uuτ−vτWτd dxdτ

≤ _t

0

_τ

0

₁

0

gux, U

λ1duss−vss_λ2Wτ_λ2d ds dτ

3.35

withλ₁ n/2,λ₂2n/n−2.

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^nα−1/2_p ∇vτ^nα−1/2_p H3

.

3.36

By the estimate2.6, we have

∇ut_p, vt_p≤C2AB^1/p, ∀t≥0. 3.37

Therefore, there existsC₄ >0, dependingu₀, v₀, f, h_isuch that g_ux, U

λ1≤C₄, ∀t≥0. 3.38

Sinceu, v∈W₀^1,p⊂W₀^1,2,u_t, v_t∈W₀^1,2, we get

uss−v_ss_λ2≤C₀∇uss−v_ss₂C₀∇Ws₂, Wτ₂ ≤C0∇Wτ₂.

3.39

Then3.35becomes

G₁≤C_{4 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

Consequently,ut−vt u0−v0 0, 0≤t≤T₁.

Repeating the above procedure, we conduce thatut vtonT1,2T1 ,2T1,3T1 , . . . andut vton0,∞. This ends the proof of uniqueness.

Next, we prove thatu∈C0,∞;W₀^1,2. 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,∞;W₀^1,2. 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≥0 andγ >0, such that

_∞

S

E^q1tdt≤γ⁻¹E^q0ES, ∀S≥0. 4.1

Then, we have

Et≤E0 1q

1qγt 1/q

, ∀t≥0, ifq >0, Et≤E0e^1−γt, ∀t≥0, ifq0.

4.2

4.1. The Proof ofTheorem 2.3 Let

Et 1

2utt²₂ 1

p∇ut^p_p

ΩGx, udx, t≥0. 4.3

Then, we have from1.1that

Et ∇utt²₂0, ∀t≥0. 4.4

This shows thatEtis nonincreasing in0,∞.

Multiplying1.1byE^qtutwithq p−2/p >0, we get _T

S

E^qt

Ωu

u_tt−Δpu−Δutgx, u

dx dt0, ∀T > S≥0. 4.5

Note that _T

S

E^qtu, uttdt E^qtu, ut|^T_S− _T

S

qE^q−1tEtu, ut E^qtutt²₂ dt

− _T

S

E^qt u,Δpu

dt _T

S

E^qt∇ut^ppdt,

− _T

S

E^qtu,Δutdt _T

S

E^qt∇u,∇utdt.

4.6

Then we have from4.5that

p _T

S

E^q1tdt −E^qtu, ut|^T_Sq _T

S

E^q−1tEtu, utdt

1p

2

T S

E^qtutt²₂dt− _T

S

E^qt∇u,∇utdt

_T

S

E^qt

Ω

pGu−ugudx dt.

4.7

Since

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

∇utt₂≤

−Et_1/2

, ∇ut_p≤pE^1/pt, ∀t≥0,

|E^qtu, ut| ≤E^qtut₂utt₂≤C₀E^qt∇ut_p∇utt₂≤C₀Et^μ1 4.8

withμ1q1/21/p.

This gives

E^qtu, ut|^T_S≤C₁E^μ1S, ∀T > S≥0, 4.9

where the fact thatEtis nonincreasing is used.

Similarly, we derive the following estimates _T

S

E^qtutt²₂dt≤C1

_T

S

E^qt∇utt²₂dt

C_{1 T}

S

E^qt

−Et

dt≤C₁E^q1S,

4.10

q _T

S

E^q−1tEtu, utdt≤C_{1 T}

S

E^q−1tEtut₂utt₂dt

≤C1

_T

S

E^μ1−1tEtdt≤C1E^μ1S,

4.11

_T

S

|E^qt∇u,∇ut|dt≤ _T

S

E^qt∇ut₂∇utt₂dt

≤C1

_T

S

E^q1/pt

−Et_1/2 dt

≤ _T

S

E^q1tdtC_{1 T}

S

E^q2/p−1t

−Et dt

≤ _T

S

E^q1tdtC₁E^q2/pS.

4.12

Then we get from4.9–4.12that _T

S

E^q1tdt≤C1

E^μ1S E^q1S E^q2/pS

≤C1ES

E^μ1S E^qS E^q2/p−1S

≤C₁ESE^q0

E^1/p−1/20 1E^2/p−10

≡γ⁻¹E^q0ES,

4.13

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

S

E^q1tdt≤γ⁻¹ESE^q0, ∀S≥0. 4.14

ByLemma 4.1, we obtain that Et 1

2utt²₂1

p∇ut^pp

ΩGx, udx≤E0 1q

1qγt 1/q

≤C₂E01t^−p/p−2. 4.15 This is2.9and we complete the proof ofTheorem 2.3.

4.2. The Proof ofTheorem 2.4

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

λ₀u^pp≤ ∇u^pp, ∀u∈W₀^1,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|^pGu≥ 1

2α1|u|^α1, ∀u∈R. 4.18

This implies that λ₀ 2pu^pp

ΩGudx≥ 1

2α1u^α1_α1, Et≥ 1

2utt²₂ 1

2p∇ut^pp 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|^pGu

.

4.20

It shows that _T

S

E^qt

Ω

pGu−gu

dxdt≤

β1−p

T S

E^q1tdt. 4.21

Then, by4.9and4.11–4.14, we have 2p−β−1

T S

E^q1tdt≤C0

E^q1/p2S E^q1S E^q2/pS

≤γ⁻¹ESE^q0.

4.22

The applications ofLemma 4.1and4.19yields that

utt²₂∇ut²₂ut^α1_α1≤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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