• Nem Talált Eredményt

In this paper we study decay properties of the solutions to the wave equation of p−Laplacian type with a weak nonlinear dissipative

N/A
N/A
Protected

Academic year: 2022

Ossza meg "In this paper we study decay properties of the solutions to the wave equation of p−Laplacian type with a weak nonlinear dissipative"

Copied!
8
0
0

Teljes szövegt

(1)

http://jipam.vu.edu.au/

Volume 7, Issue 1, Article 15, 2006

ENERGY DECAY OF SOLUTIONS OF A WAVE EQUATION OF p-LAPLACIAN TYPE WITH A WEAKLY NONLINEAR DISSIPATION

ABBÈS BENAISSA AND SALIMA MIMOUNI UNIVERSITÉDJILLALILIABÈS

FACULTÉ DESSCIENCES

DÉEPARTEMENT DEMATHÉMATIQUES

B. P. 89, SIDIBELABBÈS22000, ALGERIA.

benaissa_abbes@yahoo.com bbsalima@yahoo.fr

Received 03 November, 2005; accepted 15 November, 2005 Communicated by C. Bandle

ABSTRACT. In this paper we study decay properties of the solutions to the wave equation of p−Laplacian type with a weak nonlinear dissipative.

Key words and phrases: Wave equation ofp−Laplacian type, Decay rate.

2000 Mathematics Subject Classification. 35B40, 35L70.

1. INTRODUCTION

We consider the initial boundary problem for the nonlinear wave equation of p−Laplacian type with a weak nonlinear dissipation of the type

(P)





(|u0|l−2u0)0−∆pu+σ(t)g(u0) = 0inΩ×[0,+∞[, u= 0on∂Ω×[0,+∞[,

u(x,0) =u0(x), u0(x,0) =u1(x)inΩ.

where∆pu=div(|∇xu|p−2xu),p, l≥2,g :R→Ris a continuous non-decreasing function andσis a positive function.

When p = 2, l = 2 andσ ≡ 1, for the caseg(x) = δx (δ > 0), Ikehata and Suzuki [5]

investigated the dynamics, showing that for sufficiently small initial data(u0, u1), the trajectory (u(t), u0(t))tends to(0,0)inH01(Ω)×L2(Ω)ast→ +∞. Wheng(x) = δ|x|m−1x (m ≥1), Nakao [8] investigated the decay property of the problem(P). In [8] the author has proved the existence of global solutions to the problem(P).

For the problem(P)withσ ≡ 1, l = 2, when g(x) = δ|x|m−1x (m ≥ 1),Yao [1] proved that the energy decay rate isE(t) ≤ (1 +t)(mp−m−1)p for t ≥ 0 by using a general method

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

329-05

(2)

introduced by Nakao [8]. Unfortunately, this method does not seem to be applicable in the case of more general functionsσand is more complicated.

Our purpose in this paper is to give energy decay estimates of the solutions to the problem (P)for a weak nonlinear dissipation. We extend the results obtained by Yao and prove in some cases an exponential decay when p >2and the dissipative term is not necessarily superlinear near the origin.

We use a new method recently introduced by Martinez [7] (see also [2]) to study the decay rate of solutions to the wave equationu00−∆xu+g(u0) = 0inΩ×R+, whereΩis a bounded domain of Rn. This method is based on a new integral inequality that generalizes a result of Haraux [4].

Throughout this paper the functions considered are all real valued. We omit the space variable x of u(t, x), ut(t, x)and simply denote u(t, x), ut(t, x)by u(t), u0(t), respectively, when no confusion arises. Letl be a number with2≤ l ≤ ∞. We denote byk · kltheLlnorm overΩ.

In particular, theL2 norm is denoted by k · k2. (·)denotes the usual L2 inner product. We use familiar function spacesW01,p.

2. PRELIMINARIES ANDMAIN RESULTS

First assume that the solution exists in the class

(2.1) u∈C(R+, W01,p(Ω))∩C1(R+, Ll(Ω)).

λ(x), σ(t)andg satisfy the following hypotheses:

(H1)σ :R+→R+is a non increasing function of classC1onR+satisfying (2.2)

Z +∞

0

σ(τ)dτ = +∞.

(H2) Considerg :R→Ra non increasingC0function such that g(v)v >0 for allv 6= 0.

and suppose that there existci >0;i= 1,2,3,4such that

(2.3) c1|v|m ≤ |g(v)| ≤c2|v|m1 if|v| ≤1, (2.4) c3|v|s ≤ |g(v)| ≤c4|v|rfor all|v| ≥1, wherem ≥1,l−1≤s≤r≤ n(p−1)+pn−p .

We define the energy associated to the solution given by (2.1) by the following formula E(t) = l−1

l ku0kll+1

pk∇xukpp.

We first state two well known lemmas, and then state and prove a lemma that will be needed later.

Lemma 2.1 (Sobolev-Poincaré inequality). Let q be a number with 2 ≤ q < +∞ (n = 1,2, . . . , p)or2≤q≤ (n−p)np (n≥p+ 1), then there is a constantc =c(Ω, q)such that

kukq ≤ck∇ukp for u∈W01,p(Ω).

Lemma 2.2 ([6]). LetE : R+ → R+ be a non-increasing function and assume that there are two constantsq≥0andA >0such that

Z +∞

S

Eq+1(t)dt ≤AE(S), 0≤S <+∞,

(3)

then we have

E(t)≤cE(0)(1 +t)−1q ∀t ≥0, if q >0 and

E(t)≤cE(0)e−ωt ∀t ≥0, if q= 0,

wherecandωare positive constants independent of the initial energyE(0).

Lemma 2.3 ([7]). Let E : R+ → R+ be a non increasing function and φ : R+ → R+ an increasingC2 function such that

φ(0) = 0 and φ(t)→+∞ as t→+∞.

Assume that there existq ≥0andA >0such that Z +∞

S

E(t)q+1(t)φ0(t)dt ≤AE(S), 0≤S <+∞, then we have

E(t)≤cE(0)(1 +φ(t))−1q ∀t ≥0, if q >0 and

E(t)≤cE(0)e−ωφ(t) ∀t≥0, if q= 0, wherecandωare positive constants independent of the initial energyE(0).

Proof of Lemma 2.3. Letf :R+ →R+be defined byf(x) :=E(φ−1(x)). fis non-increasing, f(0) =E(0)and if we setx:=φ(t)we obtain

Z φ(T) φ(S)

f(x)q+1dx= Z φ(T)

φ(S)

E φ−1(x)q+1

dx

= Z T

S

E(t)q+1φ0(t)dt

≤AE(S) =Af(φ(S)) 0≤S < T <+∞.

Settings :=φ(S)and lettingT →+∞, we deduce that Z +∞

s

f(x)q+1dx≤Af(s) 0≤s <+∞.

By Lemma 2.2, we can deduce the desired results.

Our main result is the following

Theorem 2.4. Let(u0, u1) ∈ W01,p×Ll(Ω) and suppose that (H1)and(H2)hold. Then the solutionu(x, t)of the problem(P)satisfies

(1) Ifl≥m+ 1, we have

E(t)≤C(E(0)) exp

1−ω Z t

0

σ(τ)dτ

∀t >0.

(2) Ifl < m+ 1, we have

E(t)≤ C(E(0)) Rt

0 σ(τ)dτ

!(mp−m−1)p

∀t >0.

(4)

Examples

1) Ifσ(t) = 1

tθ (0≤θ ≤1), by applying Theorem 2.4 we obtain E(t)≤C(E(0))e1−ωt1−θ ifθ ∈[0,1[, l≥m+ 1, E(t)≤C(E(0))t

(1−θ)p

mp−m−1 if0≤θ <1, l < m+ 1 and

E(t)≤C(E(0))(lnt)(mp−m−1)p ifθ = 1, l < m+ 1.

2) Ifσ(t) = 1

tθlntln2t . . .lnkt, wherek is a positive integer and ( ln1(t) = ln(t)

lnk+1(t) = ln(lnk(t)), by applying Theorem 2.4, we obtain

E(t)≤C(E(0))(lnk+1t)(mp−m−1)p ifθ = 1, l < m+ 1,

E(t)≤C(E(0))tmp−m−1(1−θ)p (lntln2t . . .lnkt)mp−m−1p if0≤θ <1, l < m+ 1.

3) Ifσ(t) = 1

tθ(lnt)γ, by applying Theorem 2.4, we obtain

E(t)≤C(E(0))tmp−m−1(1−θ)p (lnt)mp−m−1γp if0≤θ <1, l < m+ 1, E(t)≤C(E(0))(lnt)

(1−γ)p

mp−m−1 ifθ = 1,0≤γ <1, l < m+ 1, E(t)≤C(E(0))(ln2t)mp−m−1p ifθ = 1, γ = 1, l < m+ 1.

Proof of Theorem 2.4.

First we have the following energy identity to the problem(P)

Lemma 2.5 (Energy identity). Letu(t, x)be a local solution to the problem(P)on[0,∞)as in Theorem 2.4. Then we have

E(t) + Z

Z t 0

σ(s)u0(s)g(u0(s))ds dx=E(0) for allt∈[0,∞).

Proof of the energy decay. From now on, we denote bycvarious positive constants which may be different at different occurrences. We multiply the first equation of(P)byEqφ0u, where φ is a function satisfying all the hypotheses of Lemma 2.3 to obtain

0 = Z T

S

Eqφ0 Z

u((|u0|l−2u0)t−∆pu+σ(t)g(u0))dx dt

=

Eqφ0 Z

uu0|u0|l−2dx T

S

− Z T

S

(qE0Eq−1φ0 +Eqφ00) Z

uu0|u0|l−2dxdt

− 3l−2 l

Z T S

Eqφ0 Z

|u0|2dxdt+ 2 Z T

S

Eqφ0 Z

l−1

l u02+1 p|∇u|p

dxdt +

Z T S

Eqφ0 Z

σ(t)ug(u0)dxdt+

1−2 p

Z T S

Eqφ0k∇ukppdxdt.

(5)

We deduce that (2.5) 2

Z T S

Eq+1φ0dt≤ −

Eqφ0 Z

uu0|u0|l−2dx T

S

+ Z T

S

(qE0Eq−1φ0+Eqφ00) Z

uu0|u0|l−2dxdt + 3l−2

l Z T

S

Eqφ0 Z

|u0|ldxdt− Z T

S

Eqφ0 Z

σ(t)ug(u0)dxdt.

SinceEis nonincreasing,φ0 is a bounded nonnegative function onR+(and we denote byµits maximum), using the Hölder inequality, we have

E(t)qφ0 Z

uu0|u0|l−2dx

≤cµE(S)q+l−1l +1p ∀t≥S.

Z T S

(qE0Eq−1φ0+Eqφ00) Z

uu0|u0|l−2dxdt

≤cµ Z T

S

−E0(t)E(t)q−1l+1p dt+c Z T

S

E(t)q+l−1l +1p(−φ00(t))dt

≤cµE(S)q+l−1l +1p.

Using these estimates we conclude from the above inequality that 2

Z T S

E(t)1+qφ0(t)dt (2.6)

≤cE(S)q+l−1l +1p +3l−2 l

Z T S

Eqφ0 Z

|u0|ldxdt− Z T

S

Eqφ0 Z

σ(t)ug(u0)dxdt

≤cE(S)q+l−1l +1p +3l−2 l

Z T S

Eqφ0 Z

|u0|ldxdt

− Z T

S

Eqφ0 Z

|u0|≤1

σ(t)ug(u0)dxdt− Z T

S

Eqφ0 Z

|u0|>1

σ(t)ug(u0)dxdt.

Define

φ(t) = Z t

0

σ(s)ds.

It is clear thatφis a non decreasing function of classC2 onR+. The hypothesis (2.2) ensures that

(2.7) φ(t)→+∞ast→+∞.

Now, we estimate the terms of the right-hand side of (2.6) in order to apply the results of Lemma 2.3:

Using the Hölder inequality, we get forl < m+ 1 Z T

S

Eqφ0 Z

|u0|ldxdt

≤C Z T

S

Eqφ0 Z

1

σ(t)u0ρ(t, u0)dx dt+C0 Z T

S

Eqφ0 Z

1

σ(t)u0ρ(t, u0) (m+1)l

dx dt

(6)

≤C Z T

S

Eq φ0

σ(t)(−E0)dt+C0(Ω) Z T

S

Eq φ0 σm+1l (t)

(−E0)m+1l dt

≤CEq+1(S) +C0(Ω) Z T

S

Eqφ0m+1−lm+1 φ0

σ(t) m+1l

(−E0)m+1l dt.

Now, fix an arbitrarily small ε > 0 (to be chosen later). By applying Young’s inequality, we obtain

(2.8) Z T

S

Eqφ0 Z

|u0|ldxdt

≤CEq+1(S) +C0(Ω)m+−l m+ 1 ε

(m+1) (m+1−l)

Z T S

Eqm+1−lm+1 φ0dt +C0(Ω) l

m+ 1 1 ε(m+1)l

E(S).

Ifl ≥m+ 1, we easily obtain from (2.3) and (2.4) (2.9)

Z T S

Eqφ0 Z

|u0|ldxdt≤CEq+1(S).

Next, we estimate the third term of the right-hand of (2.6). We get forl < m+ 1 Z T

S

Eqφ0 Z

|u0|≤1

σ(t)ug(u0)dxdt (2.10)

≤ε1 Z T

S

Eqφ0 Z

|u0|≤1

kukppdt+C(ε1) Z T

S

Eqφ0 Z

|u0|≤1

(σg(u0))p−1p dx

≤cε1 Z T

S

Eq+1φ0dt+C(ε1) Z T

S

Eqφ0 Z

|u0|≤1

(σg(u0))p−1p dx.

We now estimate the last term of the above inequality to get Z T

S

Eqφ0 Z

|u0|≤1

(σg(u0))p−1p dx dt (2.11)

≤ Z T

S

Eqφ0 Z

|u0|≤1

(u0g(u0))(m+1)(p−1)p dx dt

≤ Z T

S

Eqφ0 1 σ(m+1)(p−1)p

Z

|u0|≤1

(σu0g(u0))(m+1)(p−1)p dx dt

≤C(Ω) Z T

S

Eqφ0 1

σ(m+1)(p−1)p (−E0)

p

(m+1)(p−1)dt.

Setε2 >0; due to Young’s inequality, we obtain (2.12)

Z T S

Eqφ0 Z

|u0|≤1

(σg(u0))p−1p dxdt

≤C(Ω)(m+ 1)(p−1)−p (m+ 1)(p−1) ε

(m+1)(p−1) (m+1)(p−1)−p

2

Z T S

Eq

(m+1)(p−1) (m+1)(p−1)−pφ0dt + C(Ω)p

(m+ 1)(p−1) 1 ε

(m+1)(p−1) p

2

E(S),

(7)

we choseqsuch that

q (m+ 1)(p−1)

(m+ 1)(p−1)−p =q+ 1.

thus we findq = mp−m−1p and thusqm+1−lm+1 =q+ 1 +αwithα= (m+1)(p l−p−l) p(m+1−l) .

Using the Hölder inequality, the Sobolev imbedding and the condition (2.4), we obtain Z T

S

Eqφ0 Z

|u0|≥1

σ(t)ug(u0)dxdt

≤ Z T

S

Eqφ0σ(t) Z

|u|r+1dx

(r+1)1 Z

|u0|>1

|g(u0)|r+1r dx r+1r

dt

≤c Z T

S

Eq+1pφ0σ(r+1)1 (t) Z

|u0|>1

σu0g(u0)dx r+1r

dt

≤c Z T

S

Eq+1pφ0σ(r+1)1 (t)(−E0)r+1r dt.

Applying Young’s inequality, we obtain Z T

S

Eqφ0 Z

|u0|≥1

σ(t)ug(u0)dxdt (2.13)

≤ε3 Z T

S

(Eq+1pφ0σ(r+1)1 (t))r+1dt+c(ε3) Z T

S

(−E0)dt

≤ε3µr+1E

(p−1)(mr−1)

p (0)

Z T S

Eq+1φ0dt+c(ε3)E(S).

Ifl ≥m+ 1, the last inequality is also valid in the domain{|u0|<1}and withminstead ofr.

Choosingε, ε1, ε2andε3small enough, we deduce from (2.6), (2.8), (2.10), (2.12) and (2.13) forl < m+ 1

Z T S

E(t)1+qφ0(t)dt ≤CE(S)q+1+C0E(S)q+l−1l +p1 +C00E(S) +C000E(0)

(p l−p−l)(m+1)

p l E(S) +C0000E(0)

(m r−1)(p−1) p r E(S), whereC, C0, C00, C000, C000 are different positive constants independent ofE(0).

Choosingε3 small enough, we deduce from (2.6), (2.9) and (2.13) forl≥m+ 1 Z T

S

E(t)1+qφ0(t)dt≤CE(S)q+1+C0E(S)q+l−1l +1p +C00E(0)(m

2−1)(p−1) p m E(S),

where C, C0, C00 are different positive constants independent of E(0), we may thus complete

the proof by applying Lemma 2.3.

Remark 2.6. We obtain the same results for the following problem





(|u0|l−2u0)0−e−Φ(x)div(eΦ(x)|∇xu|p−2xu) +σ(t)g(u0) = 0inΩ×[0,+∞[, u= 0on∂Ω×[0,+∞[,

u(x,0) =u0(x), u0(x,0) =u1(x)inΩ,

(8)

whereΦis a positive function such thatΦ∈L(Ω), in this case(u0, u1)∈W0,Φ1,p×LlΦ, where W0,Φ1,p(Ω) =

u∈W01,p(Ω), Z

eΦ(x)|∇xu|pdx <∞

, LlΦ(Ω) =

u∈Ll(Ω), Z

eΦ(x)|u|ldx <∞

. Thus the energy associated to the solution is given by the following formula

E(t) = l−1

l keΦ(x)/lu0kll+1

pkeΦ(x)/pxukpp. REFERENCES

[1] YAO-JUN YE, On the decay of solutions for some nonlinear dissipative hyperbolic equations, Acta Math. Appl. Sin. Engl. Ser., 20(1) (2004), 93–100.

[2] A. BENAISSAANDS. MOKEDDEM, Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly nonlinear dissipation, Abstr. Appl. Anal., 11 (2004), 935–

955.

[3] Y. EBIHARA, M. NAKAOANDT. NAMBU, On the existence of global classical solution of initial boundary value problem foru00−∆u−u3 =f, Pacific J. of Math., 60 (1975), 63–70.

[4] A. HARAUX, Two remarks on dissipative hyperbolic problems, in: Research Notes in Mathematics, Pitman, 1985, p. 161–179.

[5] R. IKEHATA ANDT. SUZUKI, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475–491.

[6] V. KOMORNIK, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wi- ley, Paris, 1994.

[7] P. MARTINEZ, A new method to decay rate estimates for dissipative systems, ESAIM Control Op- tim. Calc. Var., 4 (1999), 419–444.

[8] M. NAKAO, A difference inequality and its applications to nonlinear evolution equations, J. Math.

Soc. Japan, 30 (1978), 747–762.

[9] M. NAKAO, On solutions of the wave equations with a sublinear dissipative term, J. Diff. Equat., 69 (1987), 204–215.

Hivatkozások

KAPCSOLÓDÓ DOKUMENTUMOK

In this paper, we prove the existence of nontrivial nonnegative classical time periodic solutions to the viscous diffusion equation with strongly nonlinear periodic sources..

In this paper we analyze the existence of bounded solutions for a nonlinear second-order neutral difference equation, which is more general than other equations of this type

We prove the existence of weak solutions to the Dirichlet boundary value problem for equations involving the p ( x ) -Laplacian-like operator in the principal part, with reaction

In this article, we establish the existence of weak solutions for a nonlinear transmission problem involving nonlocal coefficients of p ( x ) -Kirchhoff type in two dif- ferent

Our aim in this paper is to find decay mild solutions of the nonlocal Cauchy problem for a class of second order evolution equations with memory.. (1.1) is a nonlinear wave

Existence of solutions for a Kirchhoff type problem involving the fractional p-Laplacian operator.. Wenjing Chen B and

In this paper, we study the oscillation and asymptotic properties of solutions of certain nonlinear third order differential equations with delay.. In particular, we extend results

For instance, the authors of [13] studied the wave equation with a linear internal damping term with constant delay (τ = const in the problem (P) and determined suitable