We will investigate
the decay estimate of the energy of the global
solutions to the p-Laplacian wave equation with
dissipation of the form
utt-div(∇xup-2∇xu)+σ(t)(ut-div(∇xutm-2∇xut))=0
under suitable assumptions on the positive
function
σ.
For this end we use the multiplier method
combined with nonlinear integral inequalities
given by Martinez; the proof is based on
the construction of a special weight function
that depends on the behavior of
σ.

1. Introduction

In this paper we are concerned with the energy decay rate of the p-Laplacian type wave equation of the form Putt-div∇xup-2∇xu+σtut-div∇xutm-2∇xut=0in Ω×0,+∞,u=0on Γ×0,+∞,ux,0=u0x,utx,0=u1xon Ω,where Ω is a bounded domain in Rn with smooth boundary Γ=∂Ω, p, m≥2 are real numbers, and σ is a positive function satisfying some conditions to be specified later.

Problem P can be considered as a system describing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Voigt model. The bibliography of works in this direction is so long. We mention, for instance, the works of Andrews [1], Andrews and Ball [2], Ang and Dinh [3], and Kawashima and Shibata [4].

Existence of global solution and the decay property of the energy for the wave equation related to the problem P have been investigated by many authors through various approaches (see [5–8]).

In [5], Benaissa and Messaoudi considered the following problem: (1)utt-Δu+a1+utm-2ut=-buup-2in Ω×0,+∞,u=0on Γ×0,+∞,ux,0=u0x,utx,0=u1xon Ω,where a, b>0. They showed that, for suitably chosen initial data, the problem has a global weak solution, which decays exponentially even if m>2. Further they proved the global existence by using the potential well theory introduced by Sattinger [9].

Similar results have been established by Ye [7, 10]. In these works the author used the Faedo-Galerkin approximation together with compactness criteria and difference inequality introduced by Nakao [11].

More related studies of the nonlinear p-Laplacian wave equation type with damping term can be found in the papers [6, 8, 12, 13].

In [14], with considering -div(∇xutm-2∇xut) instead of the damping term σtut-div∇xutm-2∇xut, we have obtained global existence result by using the argument in [15] combined with the concepts of so-called stable sets due to Sattinger [9]. We have also shown the asymptotic behavior of global solutions through the use of the integral inequality given by Komornik [16].

The purpose of this paper is to give an energy decay estimate of the solution of problem P. Our proof is based on the multiplier method combined with nonlinear integral inequalities given by Martinez [17].

The plan of the paper is as follows. In the next section we present some assumptions, technical lemmas, and main result. Then in Section 3 we are devoted to the proof of decay estimate.

For simplicity of notation, we denote by ·p the Lebesgue space Lp(Ω) norm. In particular ·2 denotes L2(Ω) and (·,·) the inner product of L2(Ω). We also write equivalent norm ∇·p instead of W01,p(Ω) norm ·W01,p(Ω). As usual, we write, respectively, u(t) and ut(t) instead of u(x,t) and ut(x,t). Furthermore, throughout this paper the functions considered are all real valued.

2. Preliminaries and Main Result

First assume that σ:R+→R+ is a nonincreasing positive function of class C1 on R+, satisfying(2)∫0+∞στdτ=+∞.Let us define the energy equality associated with the solution of the problem P by the following formula:(3)Et=12ut22+1p∇upp,for u∈W01,p(Ω) and t≥0.

We state, without proof, a global existence result for the problem P. For more details we refer the reader to [6].

Theorem 1.

Let 2≤m≤p and assume that (u0,u1)∈W01,p(Ω)×L2(Ω). Then, for any T>0, the problem P has a unique strong solution u(t) on Ω×[0,T] in the class (4)ut∈L∞0,T,W01,pΩ∩W1,∞0,T,L2Ω∩W1,m0,T,W01,mΩ.

We now present the following well-known lemmas which will be needed later.

Lemma 2 (energy identity).

Let u(x,t) be a global solution to the problem P on 0,∞. Then one has (5)Et+∫Ω∫0tσsutut-div∇xutm-2∇xutdsdx=E0,for all t∈0,∞.

Lemma 3 (Sobolev-Poincaré inequality).

Let p∈N∗ and r∈R with 2≤r<+∞(n=1,2,…,p) or 2≤r≤np/n-p(n≥p+1). Then there is a constant c∗=c∗(Ω,p,r) such that (6)ur≤c∗∇upfor u∈W01,pΩ.The case p=r=2 gives the known Poincaré’s inequality.

Before stating our main result, we introduce the following lemma which plays an important part in studying the decay estimate of energy associated with the solution of the problem P.

Lemma 4 (see [<xref ref-type="bibr" rid="B9">17</xref>]).

Let E:R+→R+ be a nonincreasing function and ϕ:R+→R+ an increasing C2 function such that (7)ϕ0=0,ϕt⟶+∞as t⟶+∞.Assume that there exist q≥0 and γ>0 such that(8)∫S+∞Etq+1ϕ′tdt≤γ-1E0qES,0≤S<+∞.Then one has(9)Et≤E01+q1+qγϕt1/q∀t≥0 if q>0,Et≤E0exp1-γϕt∀t≥0 if q=0.

Now we are in position to state and prove our main result.

Theorem 5.

Let (u0,u1)∈W01,p(Ω)×L2(Ω) and p≥m≥2. Suppose that (2) holds. Then the solution u(x,t) of the problem P satisfies the following energy decay estimates.

If p=m, then there exists a positive constant ω such that(10)Et≤cE0exp1-ω∫0tστdτ∀t>0.

If p>m, then there exists a positive constant c(E(0)) depending continuously on E(0) such that(11)Et≤cE0∫0tστdτpm-1/p-m∀t>0.

3. Proof of Main Result

From now on, we denote by c various positive constants depending on the known constants and they may be different at each appearance.

Multiplying by Eqϕ′(t)u on both sides of the first equation of P and integrating over Ω×[T,S], where ϕ is a function satisfying all the hypotheses of Lemma 4 and 0≤S≤T≤+∞, we obtain that (12)0=∫STEqϕ′∫Ωuutt-div∇xup-2∇xu+σtut-div∇xutm-2∇xutdxdt.By an integration by parts we see that (13)0=Eqϕ′∫ΩuutST-∫STqE′Eq-1ϕ′+Eqϕ′′·∫Ωuutdxdt-∫STEqϕ′∫Ωut2dxdt+∫STEqϕ′∫Ω∇updxdt+∫STEqϕ′σt·∫Ωuut-div∇xutm-2∇xutdxdt.Hence from the definition of energy and a simple argument we can obtain(14)p∫STEq+1ϕ′dt=-Eqϕ′∫ΩuutST+∫STqE′Eq-1ϕ′+Eqϕ′′∫Ωuutdxdt+p2+1·∫STEqϕ′∫Ωut2dxdt-∫STEqϕ′σ·∫Ωuut-div∇xutm-2∇xutdxdt.Now we must estimate both sides of (14) to arrive at a similar inequality as (8).

Define (15)ϕt=∫0tστdτ.It is clear that ϕ is a nondecreasing function of class C2 on R+ and hypothesis (2) ensures that(16)ϕt⟶+∞as t⟶+∞.Since E is nonincreasing, ϕ′ is a bounded nonnegative function on R+ (we denote by μ its maximum) and, using the definition of energy, Cauchy-Schwartz inequality, and Sobolev-Poincaré inequality, we have(17)-Eqϕ′∫ΩuutdxST≤cμESq+1/2+1/p,where the above estimate follows from the fact that(18)∫Ωuutdx≤u2ut2≤c∇uput2≤cEt1/pEt1/2.Again, exploiting Cauchy-Schwartz inequality, Sobolev-Poincaré inequality, the definition of energy, and (18), we obtain(19)∫STqE′Eq-1ϕ′+Eqϕ′′∫Ωuutdxdt≤cμ∫ST-E′Eq-1/2+1/pdt+∫STcEq+1/2+1/p-ϕ′′dt≤cμESq+1/2+1/p,where the fact that E(t) is nonincreasing is used.

Furthermore, by using Lemma 2, we have(20)1+p2∫STEqϕ′∫Ωut2dxdt≤1+p2∫STEqϕ′∫Ωut2+∇utmdxdt≤1+p2∫STEqϕ′-E′tσtdt≤cEq+1S.We then estimate the last term in (14) as follows: (21)∫STEqϕ′σt∫Ωuut-div∇xutm-2∇xutdxdt≤∫STEqϕ′σt∫Ωuutdxdt+∫STEqϕ′σt∫Ωudiv∇xutm-2∇xutdxdt,which implies that (22)∫STEqϕ′σt∫Ωudiv∇xutm-2∇xutdxdt≤∫STEqϕ′σt∫Ω∇u∇utm-1dxdt.Using Hölder’s and Sobolev-Poincaré’s inequalities, we see that (23)∫STEqϕ′tσt∫Ωudiv∇xutm-2∇xutdxdt≤∫STEqϕ′σt∇up∇utpm-1/p-1m-1dt.We also have (24)∇utpm-1/p-1m-1≤cΩp-m/pm∇utmm-1,∇utmm-1≤-E′tσtm-1/m.This gives (25)∫STEqϕ′tσt∫Ωudiv∇xutm-2∇xut≤c∫STEq+1/pϕ′σt-E′tσtm-1/mdt=c∫STEq+1/pϕ′σt1/m-E′tm-1/mdt.Further, by Young inequality, we have for ε>0(26)∫STEqϕ′tσt∫Ωudiv∇xutm-2∇xut≤1mεm∫STEq+1/pmϕ′tmσtdt+m-1m1εm/m-1ES=1mεm∫STEq+1/pmϕ′tσtmdt+m-1m1εm/m-1ES.We choose q such that (27)mq+1p=q+1.Thus q=(p-m)/p(m-1).

Combining estimates (17)–(26), (14) becomes(28)∫STEq+1ϕ′tdt≤cESq+1/2+1/p+c′ESq+1+c′′ES≤cE0q-1/2+1/p+c′E0q+c′′E0qE0qES,where c, c′, and c′′ are different positive constants independent of E(0).

Letting T→+∞, this yields the following estimate: (29)∫S+∞Eq+1ϕ′tdt≤cE0q-1/2+1/p+c′E0q+c′′E0qE0qES,∀S≥0,and we conclude from Lemma 4 that (30)Et≤cE0q-1/2+1/p+c′E0q+c′′1/q·1+qq1/q∫0tσtdt.It is clear that, for p=m, we have q=0 and the energy E(t) associated with the solution of the problem P satisfies the decay property in (10).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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