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volume 7, issue 1, article 15, 2006.

Received 03 November, 2005;

accepted 15 November, 2005.

Communicated by:C. Bandle

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Journal of Inequalities in Pure and Applied Mathematics

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

ABBÈS BENAISSA AND SALIMA MIMOUNI

Université Djillali Liabès Faculté des Sciences

Déepartement de Mathématiques B. P. 89, Sidi Bel Abbès 22000, ALGERIA.

EMail:benaissa_abbes@yahoo.com EMail:bbsalima@yahoo.fr

2000c Victoria University ISSN (electronic): 1443-5756 329-05

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Energy Decay of Solutions of a Wave Equation ofp−Laplacian Type with a Weakly Nonlinear

Dissipation

Abbès Benaissa and Salima Mimouni

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J. Ineq. Pure and Appl. Math. 7(1) Art. 15, 2006

Abstract

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

2000 Mathematics Subject Classification:35B40, 35L70.

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

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−2∇_xu), p, l ≥ 2, g : R → R is a continuous non- decreasing function andσis a positive function.

When p = 2, l = 2 and σ ≡ 1, for the case g(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) as t → +∞. When g(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)with σ ≡ 1, l = 2, wheng(x) = δ|x|^m−1x (m ≥ 1), Yao [1] proved that the energy decay rate isE(t)≤(1 +t)⁻^(mp−m−1)^p fort≥ 0 by using a general method introduced by Nakao [8]. Unfortunately, this method does not seem to be applicable in the case of more general functions σ and 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 whenp >2and the dissipative term is not necessarily superlinear near the origin.

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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) = 0in Ω×R⁺, where Ωis 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), u_t(t, x) and simply denote u(t, x), u_t(t, x)by u(t), u⁰(t), respectively, when no confusion arises. Let l be a number with 2≤l ≤ ∞. We denote byk · k_ltheL^lnorm overΩ. In particular, theL² norm is denoted by k · k₂. (·)denotes the usual L² inner product. We use familiar function spacesW₀^1,p.

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2. Preliminaries and Main Results

First assume that the solution exists in the class

(2.1) u∈C(R+, W₀^1,p(Ω))∩C¹(R+, L^l(Ω)).

λ(x), σ(t)andgsatisfy the following hypotheses:

(H1)σ:R+ →R+is a non increasing function of classC¹ onR+satisfying (2.2)

Z +∞

0

σ(τ)dτ = +∞.

(H2) Considerg :R→Ra non increasingC⁰ function such that g(v)v >0 for allv 6= 0.

and suppose that there existc_i >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)+p_n−p .

We define the energy associated to the solution given by (2.1) by the following formula

E(t) = l−1

l ku⁰k^l_l+1

pk∇xuk^p_p.

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

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Lemma 2.1 (Sobolev-Poincaré inequality). Let qbe a number with2 ≤ q <

+∞ (n = 1,2, . . . , p)or2 ≤ q ≤ _(n−p)^np (n ≥ p+ 1), then there is a constant c∗ =c(Ω, q)such that

kuk_q ≤c∗k∇uk_p for u∈W₀^1,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

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

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

E(t)≤cE(0)e^−ωt ∀t≥0, if q= 0,

wherecandω are 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 increasingC²function 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 <+∞,

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then we have

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

E(t)≤cE(0)e^−ωφ(t) ∀t≥0, if q = 0,

wherecandω are positive constants independent of the initial energyE(0).

Proof of Lemma2.3. Letf :R+→R+be defined byf(x) := E(φ⁻¹(x)). f is 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 Lemma2.2, we can deduce the desired results.

Our main result is the following

Theorem 2.4. Let (u₀, u₁) ∈ W₀^1,p×L^l(Ω) and suppose that(H1)and (H2) hold. Then the solutionu(x, t)of the problem(P)satisfies

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(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.

Examples 1) Ifσ(t) = 1

t^θ (0≤θ ≤1), by applying Theorem2.4we obtain E(t)≤C(E(0))e^1−ωt^1−θ ifθ∈[0,1[, l≥m+ 1, E(t)≤C(E(0))t⁻^mp−m−1^(1−θ)p 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

t^θlntln₂t . . .ln_kt, wherekis a positive integer and ( ln₁(t) = ln(t)

lnk+1(t) = ln(lnk(t)),

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by applying Theorem2.4, we obtain

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

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

3) Ifσ(t) = 1

t^θ(lnt)^γ, by applying Theorem2.4, we obtain

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

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

Proof of Theorem2.4.

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

Lemma 2.5 (Energy identity). Let u(t, x)be a local solution to the problem (P)on[0,∞)as in Theorem2.4. Then we have

E(t) + Z

Ω

Z t 0

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

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Proof of the energy decay. From now on, we denote by cvarious positive constants which may be different at different occurrences. We multiply the first equation of(P)byE^qφ⁰u, whereφis a function satisfying all the hypotheses of Lemma2.3to obtain

0 = Z T

S

E^qφ⁰ Z

Ω

u((|u⁰|^l−2u⁰)_t−∆_pu+σ(t)g(u⁰))dx dt

=

E^qφ⁰ Z

Ω

uu⁰|u⁰|^l−2dx T

S

− Z T

S

(qE⁰E^q−1φ⁰+E^qφ⁰⁰) Z

Ω

uu⁰|u⁰|^l−2dxdt

− 3l−2 l

Z T S

E^qφ⁰ Z

Ω

|u⁰|²dxdt+ 2 Z T

S

E^qφ⁰ Z

Ω

l−1

l u⁰²+1 p|∇u|^p

dxdt +

Z T S

E^qφ⁰ Z

Ω

σ(t)ug(u⁰)dxdt+

1−2 p

Z T S

E^qφ⁰k∇uk^p_pdxdt.

We deduce that (2.5) 2

Z T S

E^q+1φ⁰dt≤ −

E^qφ⁰ Z

Ω

uu⁰|u⁰|^l−2dx T

S

+ Z T

S

Ω

uu⁰|u⁰|^l−2dxdt +3l−2

l Z T

S

E^qφ⁰ Z

Ω

|u⁰|^ldxdt− Z T

S

E^qφ⁰ Z

Ω

σ(t)ug(u⁰)dxdt.

SinceEis nonincreasing,φ⁰ is a bounded nonnegative function onR+(and we

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denote byµits maximum), using the Hölder inequality, we have

E(t)^qφ⁰ Z

Ω

uu⁰|u⁰|^l−2dx

≤cµE(S)^q+^l−1^l ⁺^p¹ ∀t≥S.

Z T S

Ω

uu⁰|u⁰|^l−2dxdt

≤cµ Z T

S

−E⁰(t)E(t)^q−¹^l⁺¹^pdt+c Z T

S

E(t)^q+^l−1^l ⁺¹^p(−φ⁰⁰(t))dt

≤cµE(S)^q+^l−1^l ⁺¹^p.

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−1^l ⁺¹^p +3l−2 l

Z T S

E^qφ⁰ Z

Ω

|u⁰|^ldxdt

− Z T

S

E^qφ⁰ Z

Ω

σ(t)ug(u⁰)dxdt

≤cE(S)^q+^l−1^l ⁺¹^p +3l−2 l

Z T S

E^qφ⁰ Z

Ω

|u⁰|^ldxdt

− Z T

S

E^qφ⁰ Z

|u⁰|≤1

σ(t)ug(u⁰)dxdt

− Z T

S

E^qφ⁰ Z

|u⁰|>1

σ(t)ug(u⁰)dxdt.

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Define

φ(t) = Z t

0

σ(s)ds.

It is clear thatφis a non decreasing function of classC²onR+. 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 Lemma2.3:

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

S

E^qφ⁰ Z

Ω

|u⁰|^ldxdt

≤C Z T

S

E^qφ⁰ Z

Ω

1

σ(t)u⁰ρ(t, u⁰)dx dt +C⁰

Z T S

E^qφ⁰ Z

Ω

1

σ(t)u⁰ρ(t, u⁰) _(m+1)^l

dx dt

≤C Z T

S

E^q φ⁰

σ(t)(−E⁰)dt+C⁰(Ω) Z T

S

E^q φ⁰ σ^m+1^l (t)

(−E⁰)^m+1^l dt

≤CE^q+1(S) +C⁰(Ω) Z T

S

E^qφ⁰^m+1−l^m+1 φ⁰

σ(t) _m+1^l

(−E⁰)^m+1^l dt.

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

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inequality, we obtain (2.8)

Z T S

E^qφ⁰ Z

Ω

|u⁰|^ldxdt

≤CE^q+1(S) +C⁰(Ω)m+−l m+ 1 ε

(m+1) (m+1−l)

Z T S

E^q^m+1−l^m+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

E^qφ⁰ Z

Ω

|u⁰|^ldxdt≤CE^q+1(S).

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

S

E^qφ⁰ Z

|u⁰|≤1

σ(t)ug(u⁰)dxdt (2.10)

≤ε₁ Z T

S

E^qφ⁰ Z

|u⁰|≤1

kuk^p_pdt+C(ε₁) Z T

S

E^qφ⁰ Z

|u⁰|≤1

(σg(u⁰))^p−1^p dx

≤cε₁ Z T

S

E^q+1φ⁰dt+C(ε₁) Z T

S

E^qφ⁰ Z

|u⁰|≤1

(σg(u⁰))^p−1^p dx.

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

S

E^qφ⁰ Z

|u⁰|≤1

(σg(u⁰))^p−1^p dx dt (2.11)

≤ Z T

S

E^qφ⁰ Z

|u⁰|≤1

(u⁰g(u⁰))^(m+1)(p−1)^p dx dt

≤ Z T

S

E^qφ⁰ 1 σ^(m+1)(p−1)^p

Z

|u⁰|≤1

(σu⁰g(u⁰))^(m+1)(p−1)^p dx dt

≤C(Ω) Z T

S

E^qφ⁰ 1

σ^(m+1)(p−1)^p (−E⁰)^(m+1)(p−1)^p dt.

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

Z T S

E^qφ⁰ Z

|u⁰|≤1

(σg(u⁰))^p−1^p 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

E^q

(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), we choseqsuch that

q (m+ 1)(p−1)

(m+ 1)(p−1)−p =q+ 1.

thus we findq = ^mp−m−1_p and thusq_m+1−l^m+1 =q+ 1 +αwithα = (m+1)(p l−p−l) p(m+1−l) .

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

Z T S

E^qφ⁰ Z

|u⁰|≥1

σ(t)ug(u⁰)dxdt

≤ Z T

S

E^qφ⁰σ(t) Z

Ω

|u|^r+1dx

_(r+1)¹ Z

|u⁰|>1

|g(u⁰)|^r+1^r dx _r+1^r

dt

≤c Z T

S

E^q+¹^pφ⁰σ^(r+1)¹ (t) Z

|u⁰|>1

σu⁰g(u⁰)dx _r+1^r

dt

≤c Z T

S

E^q+¹^pφ⁰σ^(r+1)¹ (t)(−E⁰)^r+1^r dt.

Applying Young’s inequality, we obtain Z T

S

E^qφ⁰ Z

|u⁰|≥1

σ(t)ug(u⁰)dxdt (2.13)

≤ε₃ Z T

S

(E^q+¹^pφ⁰σ^(r+1)¹ (t))^r+1dt+c(ε₃) Z T

S

(−E⁰)dt

≤ε₃µ^r+1E(p−1)(mr−1)

p (0)

Z T S

E^q+1φ⁰dt+c(ε₃)E(S).

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

Choosingε, ε₁, ε₂ andε₃ small enough, we deduce from (2.6), (2.8), (2.10),

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(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−1^l ⁺¹^p +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) for l ≥ m+ 1

Z T S

E(t)^1+qφ⁰(t)dt≤CE(S)^q+1+C⁰E(S)^q+^l−1^l ⁺¹^p+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 Lemma2.3.

Remark 1. We obtain the same results for the following problem











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

u(x,0) = u₀(x), u⁰(x,0) =u₁(x)inΩ,

where Φis a positive function such that Φ ∈ L^∞(Ω), in this case (u₀, u₁) ∈ W_0,Φ^1,p×L^l_Φ, where

W_0,Φ^1,p(Ω) =

u∈W₀^1,p(Ω), Z

Ω

e^Φ(x)|∇_xu|^pdx <∞

, L^l_Φ(Ω) =

u∈L^l(Ω), Z

Ω

e^Φ(x)|u|^ldx <∞

.

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

E(t) = l−1

l ke^Φ(x)/lu⁰k^l_l+ 1

pke^Φ(x)/p∇xuk^p_p.

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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. BENAISSA ANDS. MOKEDDEM, Global existence and energy decay of solutions to the Cauchy problem for a wave equation with a weakly non- linear dissipation, Abstr. Appl. Anal., 11 (2004), 935–955.

[3] Y. EBIHARA, M. NAKAO AND T. 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: Re- search Notes in Mathematics, Pitman, 1985, p. 161–179.

[5] R. IKEHATAANDT. SUZUKI, Stable and unstable sets for evolution equa- tions of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475–

491.

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

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

[8] M. NAKAO, A difference inequality and its applications to nonlinear evo- lution equations, J. Math. Soc. Japan, 30 (1978), 747–762.

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