A nonlinear thermoelastic system with memory is considered, which is derived from a physical model with vibration in temperature environment. By some skillful and technical arguments, results of existence, uniqueness, and uniform decay on this generalized system are obtained.

1. Introduction

In this work we consider the following initial boundary value problem:
u′′-M(∥∇u∥2)Δu+∫0th(t-τ)Δu(τ)dτ-Δu′+|u|ρ-2u+β1θ=finQ,θ′-Δθ+β2u′=ginQ,u=θ=0onΣ,u(x,0)=u0(x),u′(x,0)=u1(x),θ(x,0)=θ0(x),x∈Ω,
where Q:=Ω×[0,T], Σ:=∂Ω×[0,T], ρ≥2, Ω is a bounded domain in Rn with C2 boundary, u′=du/dt, u′′=d2u/dt2, M(s) is C1 class function like 1+sγ, γ≥1 and β1, β2 are positive constants:
∥∇u∥2=∑i=1n∫Ω|∂u∂xi(x)|2dx,Δu=∑i=1n∂2u∂xi2,f, g is a known function and the function h(t) is positive and satisfies some conditions to be specified later.

However, (1.1) consists of a dynamical equation coupling a heat equation, which can be used to describe some physical process of thermoelastic material. Also, u(x,t) and θ(x,t) represent the displacement and temperature, respectively, at position x and time t. The coupling of the heat equation in the model of vibrations presents important aspects because it represents better than the reality, that is, allowing to influence the vibrations in a more adequate way. M(s) appearing in the dynamical part of system (1.1) is a nonlinear perturbation of Moeover, Kirchhoff-Carrier's model which describes small vibrations of a stretched string (dimension n = 1) when tension is assumed to have only a vertical component at each point of the string. Many researchers have investigated several types of problems involving the Kirchhoff equation among which we can cite the work in [1, 2]. Clark and Lima [3] studied the local existence for 0<T0<T of solutions to the mixed problem:
u′′-M(∫Ω|∇u|2dx)Δu+|u|ρu+θ=finQ,θ′-Δθ+u′=ginQ.

In this paper, we prove the global existence and uniqueness of weak solutions of (1.1) based on different definition of weak solution and estimate techniques from [3], we consider the Kirchhoff equation with the strong damping term Δu′ and so-called “memory” term ∫0th(t-τ)Δu(τ)dτ. Here we consider the memory effect in (1.1) because physically some materials could produce the viscosity of memory type [4]. Hence under appropriate assumptions on h(t), ρ, f, and g, and making use of Galerkin's approximations and compactness argument, we establish global existence and uniqueness. Meanwhile, by some suitable estimate techniques, we deal with the memory term and another nonlinear term appearing in the mixed problem of viscoelastic wave equation. In order to obtain the exponential decay of the energy, we make use of the perturbed energy method, see Komornik and Zuazua [5].

The rest of this paper is organized as follows: In Section 2 we give out assumptions and state the main result. In Section 3 we exploit Faedo-Galerkin's approximation, priori estimates, and compactness arguments to obtain the existence of solutions of a penalty problem. In Section 4, uniqueness is proved. In Section 5, the exponential decay of solution is obtained by using the perturbed energy method.

2. Assumptions and Main Results

Throughout this paper, we use the following notation:
(u,v)=∫Ωu(x)v(x)dx,∥u∥2=∫Ω|u(x)|2dx.

Now we state the main hypotheses in this paper.

(A.1) Assumption on Kernel h

Let h:R+→R+ be a nonnegative and bounded C2 function and suppose that there exist positive constants ξ1,ξ2,ξ3 such that
-ξ1h(t)≤h′(t)≤-ξ2h(t)∀t≥0,0≤h′′(t)≤ξ3h(t)∀t≥0.
Moreover, h verifies l:=1-∫0∞h(s)ds>0.

(A.2) Assumption on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M41"><mml:mi>ρ</mml:mi><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi>μ</mml:mi></mml:math></inline-formula>

Let ρ satisfies that
2≤ρ≤2n-2n-2ifn≥3,2≤ρ<∞ifn=1,2;μ is given by the Sobolev embedding inequality ∥u∥2≤μ∥∇u∥ for u∈H01(Ω), in the general case, we denote ∥u∥ρ≤C∥∇u∥.

(A.3) Assumption on Initial Condition, <italic>f</italic> and <italic>g </italic>

Assume that u0,u1,θ0∈H01(Ω)∩H2(Ω), and f,g∈Cloc1(0,∞;L2(Ω)). Next we define the energy E(t) with
E(t)=12(∥u′(t)∥2+∥θ(t)∥2+∥∇u(t)∥2+1γ+1∥∇u(t)∥2(γ+1)+2ρ∥u(t)∥ρρ).
The main result is as follow.

Theorem 2.1.

If assumptions (1)–(3) hold, then there exists a unique weak solution {u,θ} with u∈L∞(0,T;H01(Ω)), u′∈L∞(0,T;H01(Ω)), u′′∈L∞(0,T;L2(Ω)), θ∈L∞(0,T;H01(Ω)), and θ′∈L∞(0,T;L2(Ω)) such that
(u′′,w)+(∇u,∇w)+∥∇u∥2γ(∇u,∇w)-∫0th(t-τ)(∇u(τ),∇w)dτ+(∇u′,∇w)+β1(θ,w)+(|u|ρ-2u,w)-(f,w)=0,(θ′,w)+(∇θ,∇w)+β2(u′,w)-(g,w)=0∀w∈H01(Ω),u(0)=u0,u′(0)=u1,θ(0)=θ0.
Furthermore, if f=g=0, β1,β2 satisfy that 2/μ2≥β1+β2 and β1 small enough, we have the following decay estimate:
E(t)≤Cexp(-ξt),∀t≥t0,
where C and ξ are positive constants.

3. Existence of SolutionsProof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

We use Galerkin's approximation. Let w1,…,wm be a basis in H01(Ω) which is orthonormal in L2(Ω), and Vm the subspace of H01(Ω) generated by the first m of {wj}. For each m∈N, we seek the approximate solution:
um(t,x)=∑j=1mgm(t)wj(x),θm(t,x)=∑j=1mg̃m(t)wj(x),
of the following Cauchy problem:
(um′′,w)+(∇um,∇w)+∥∇um∥2γ(∇um,∇w)-∫0th(t-τ)(∇um(τ),∇w)dτ+(∇um′,∇w)+β1(θm,w)+(|um|ρ-2um,w)-(f,w)=0∀w∈Vm,(θm′,w)+(∇θm,∇w)+β2(um′,w)-(g,w)=0∀w∈Vm
satisfying the initial conditions
um(0)=u0m=∑j=1m(u0,wj)wj→u0stronglyinH01(Ω)∩H2(Ω),um′(0)=u1m=∑j=1m(u1,wj)wj→u1stronglyinH01(Ω)∩H2(Ω),θm(0)=θ0m=∑j=1m(θ0,wj)wj→θ0stronglyinH01(Ω)∩H2(Ω).
According to the ODE theory, we can solve the system (3.2)-(3.3) by Picard's iteration. Hence, this system has unique solution on interval [0,Tm] for each m. The following estimates allow us to extend the solution to the closed interval [0,T].

In the following proof, we will use ci, i=0,1,2,…, to denote various positive constants which may be different in different places and may be dependent on T in some cases.

The First Estimate

Taking w=um′(t) in (3.2) and w=θm(t) in (3.3), respectively, then adding the results and using assumption (1), we have
12ddt(∥um′(t)∥2+∥θm(t)∥2+∥∇um(t)∥2+1γ+1∥∇um(t)∥2(γ+1)+2ρ∥um(t)∥ρρ)+∥∇um′(t)∥2+∥∇θm(t)∥2=ddt[∫0th(t-τ)(∇um(τ),∇um(t))dτ]-(β1+β2)(um′,θm)+(f,um′(t))+(g,θm)-∫0th′(t-τ)(∇um(τ),∇um(t))dτ-h(0)∥∇um(t)∥2≤(β1+β2)2+12∥um′(t)∥2+∥θm∥2+ddt[∫0th(t-τ)(∇um(τ),∇um(t))dτ]+12∥f∥2+12∥g∥2+c1∥∇um(t)∥2+c2∫0t∥∇um(τ)∥2dτ.
Now integrating (3.5) over (0,t) for t<T, we have
12(∥um′(t)∥2+∥θm(t)∥2+∥∇um(t)∥2+1γ+1∥∇um(t)∥2(γ+1)+2ρ∥um(t)∥ρρ)+∫0t∥∇um′(τ)∥2dτ+∫0t∥∇θm(τ)∥2dτ≤(β1+β2)2+12∫0t∥um′(τ)∥2dτ+∫0th(t-τ)(∇um(τ),∇um(t))dτ+∫0t∥θm(τ)∥2dτ+c3∫0t∥∇um(τ)∥2dτ+c4.
Moreover, from assumption (1), we have
∫0th(t-τ)(∇um(τ),∇um(t))dτ≤c5(η)∫0t∥∇um(τ)∥2dτ+η∥∇um(t)∥2,
where η>0 is arbitrary.

Hence letting η small enough and using Gronwall's inequality we obtain the first estimate:

∥um′(t)∥2+∥θm(t)∥2+∥∇um(t)∥2+∥∇um(t)∥2(γ+1)+∥um(t)∥ρρ+∫0t∥∇um′(τ)∥2dτ+∫0t∥∇θm(τ)∥2dτ≤L1,
where L1 is independent of m.

The Second Estimate

First we estimate the initial data um′′(0) in the L2-norm. Taking t=0 and w=um′′(0) in (3.2) we have
∥um′′(0)∥2≤(1+∥∇u0∥2γ)|(Δu0,um′′(0))|+β1(θ0,u′′(0))+(Δu1,u′′(0))+∥u0∥ρ-1ρ-1∥um′′(0)∥+∥f∥∥um′′(0)∥.
Hence, noticing the assumption on u0, u1, and θ0, we deduce
∥um′′(0)∥≤L2,
where L2 is independent of m.

Similarly, taking t=0 and w=θm′(0) in (3.3), we also deduce

∥θm′(0)∥≤L3,
where L3 is independent of m.

Differentiating (3.2) and (3.3), replacing w by um′′(t) and θm′(t) respectively, and then adding the results, we get

12ddt[∥um′′(t)∥2+∥θm′(t)∥2+∥∇um′(t)∥2]+∥∇um′′(t)∥2+∥∇θm′(t)∥2+h(0)∥∇um′(t)∥2≤-2γ∥∇um(t)∥2γ-2(∇um(t),∇um′(t))(∇um(t),∇um′′(t))-(∇um′(t),∇um′′(t))∥∇um(t)∥2γ-((ρ-1)|um(t)|ρ-2um′(t),um′′(t))+(β1+β2)|(θm′(t),um′′(t))|+(f′,um′′(t))+(g′,θm′(t))+ddt[∫0th′(t-τ)(∇um(τ),∇um′(t))dτ]-h′(0)(∇um(t),∇um′(t))-∫0th′′(t-τ)(∇um(τ),∇um′(t))dτ+h(0)ddt(∇um(t),∇um′(t)).
From the first estimate and Young's inequality, we have
2γ∥∇um(t)∥2γ-2(∇um(t),∇um′(t))(∇um(t),∇um′′(t))+(∇um′(t),∇um′′(t))∥∇um(t)∥2γ≤c1(η)∥∇um′(t)∥2+η∥∇um′′(t)∥2,
where η>0 is arbitrary.

Noticing 1/n+(n-2)/2n+1/2=1, assumption (2), and the first estimate, we have

((ρ-1)um(t)|ρ-2um′(t),um′′(t))≤(ρ-1)∥um(t)∥n(ρ-2)ρ-2∥um′(t)∥2n/(n-2)∥um′′(t)∥≤c2∥∇um(t)∥ρ-2∥∇um′(t)∥∥um′′(t)∥≤c3∥∇um′(t)∥2+c3∥um′′(t)∥2,h′(0)(∇um(t),∇u′m(t))≤h′(0)22∥∇um(t)∥2+12∥∇um′(t)∥2,
and by assumption (1), we have
∫0th′′(t-τ)(∇um(τ),∇um′(t))dτ≤c4∫0t∥∇um(τ)∥2dτ+12∥∇um′(t)∥2.
Therefore, combining (3.14)–(3.16), (3.10), (3.11) and integrating (3.12) over (0,t) we have
12[∥um′′(t)∥2+∥θm′(t)∥2+∥∇um′(t)∥2]+(1-η)∫0t∥∇um′′(τ)∥2dτ+∫0t∥∇θm′(τ)∥2dτ+h(0)∫0t∥∇um′(τ)∥2dτ≤(c1(η)+c3+1)∫0t∥∇um′(τ)∥2dτ+(β1+β2)2+12∫0t∥θm′(τ)∥2dτ+(c3+1)∫0t∥um′′(τ)∥2dτ+c5∫0t∥∇um(τ)∥2dτ+∫0th′(t-τ)(∇um(τ),∇um′(t))dτ+h(0)(∇um(t),∇um′(t))+c6.
Moreover, consider that
∫0th′(t-τ)(∇um(τ),∇um′(t))dτ≤c7(η)∫0t∥∇um(τ)∥2dτ+η∥∇um′(t)∥2,h(0)(∇um(t),∇um′(t))≤c8(η)∥∇um(t)∥2+η∥∇um′(t)∥2.

Hence, from (3.17), (3.18), the first estimate, letting η small enough and using Gronwall's inequality, we get the second estimate:

∥um′′(t)∥2+∥θm′(t)∥2+∥∇um′(t)∥2+∫0T∥∇um′′(τ)∥2dτ+∫0T∥∇θm′(τ)∥2dτ≤L4,∀0≤t≤T,
where L4 is independent of m.

The Third Estimate

Taking w=θm′(t) in (3.3), we have
∥θm′∥2+12ddt∥∇θm∥2≤β2|(um′(t),θm′(t))|+|(g(t),θm′(t))|.
Hence we easily get ∥∇θ∥2≤L5,∀0≤t≤T, and L5 is independent of m.

The Fourth Estimate

Let m1≥m2 be two natural numbers and consider ym:=um1-um2, zm:=θm1-θm2. From the system (3.2), we have
(ym′′,w)+(∇ym,∇w)+(∥∇um1∥2γ∇ym,∇w)+((∥∇um1∥2γ-∥∇um2∥2γ)∇um2,∇w)-∫0th(t-τ)(∇ym(τ),∇w)dτ+β1(zm,w)+(∇ym′,∇w)+(|um1(t)|ρ-2um1(t)-|um2(t)|ρ-2um2(t),w)=0,

Taking w=ym′ in (3.21), we have

12ddt[∥ym′(t)∥2+∥∇ym(t)∥2+∥∇um1(t)∥2γ∥∇ym(t)∥2]+∥∇ym′(t)∥2≤((∥∇um2(t)∥2γ-∥∇um1(t)∥2γ)∇um2(t),∇ym′(t))-(|um1(t)|ρ-2um1(t)-|um2(t)|ρ-2um2(t),ym′(t))+12∥∇ym(t)∥2ddt∥∇um1(t)∥2γ+β1|(zm(t),ym′(t))|+∫0th(t-τ)(∇ym(τ),∇ym′(t))dτ.
Noticing that
∫0th(t-τ)(∇ym(τ),∇ym′(t))dτ=-h(0)∥∇ym(t)∥2-∫0th′(t-τ)(∇ym(τ),∇ym(t))dτ+ddt(∫0th(t-τ)(∇ym(τ),∇ym(t))dτ),
hence, using assumption (2.2) and integrating (3.22) over (0,t), we get
12[∥ym′(t)∥2+∥∇ym(t)∥2+∥∇um1(t)∥2γ∥∇ym(t)∥2]+h(0)∫0t∥∇ym(τ)∥2dτ+∫0t∥∇ym′(τ)∥2dτ≤∫0t|∥∇um1(τ)∥2γ-∥∇um2(τ)∥2γ|∥∇um2(τ)∥∥∇ym′(τ)∥dτ+∫0t∥|um1(τ)|ρ-2um1(τ)-|um2(τ)|ρ-2um2(τ)∥∥ym′(τ)∥dτ+∫0tβ1|(zm(τ),ym′(τ))|dτ+12∫0t∥∇ym(τ)∥2ddτ∥∇um1(τ)∥2γdτ+c1∫0th(t-τ)|(∇ym(τ),∇ym(t))|dτ+c2(∥y1m∥2+∥∇y0m∥2).
Notice that
c1∫0th(t-τ)|(∇ym(τ),∇ym(t))|dτ≤c3(η)∫0t∥∇ym(τ)∥2dτ+η∥∇ym(t)∥2.
where η>0 is arbitrary:
|∥∇um1(τ)∥2γ-∥∇um2(τ)∥2γ|≤c4(∥∇um1(τ)∥2γ-1+∥∇um2(τ)∥2γ-1)∥∇ym(τ)∥,ddτ∥∇um1(τ)∥2γ≤c5∥∇um1(τ)∥2γ-1∥∇u′m1(τ)∥.
Moreover, by mean value theorem and assumption (2), we have
∥|um1(τ)|ρ-2um1(τ)-|um2(τ)|ρ-2um2(τ)∥≤c6(∥∇um1(τ)∥ρ-2+∥∇um2(τ)∥ρ-2)∥∇ym(τ)∥.
Therefore, by (3.25)–(3.27), letting η>0 small enough, by the first estimate, and using the Gronwall's lemma of integral form (see [6]) in (3.24) we obtain that
∥ym′(t)∥2+∥∇ym(t)∥2+∫0T∥∇ym′(τ)∥2dτ≤c7(T)(∥y1m∥2+∥∇y0m∥2+∫0T∥zm(τ)∥2dτ).

Passage to the Limit

From above estimates, we deduce that there exist functions u,θ and subsequences of {um}, {θm} which we still denote by {um}, {θm} satisfying
um→uinL∞(0,T;H01(Ω))weak*,um′→u′inL∞(0,T;H01(Ω))weak*,um′′→u′′inL∞(0,T;L2(Ω))weak*,um′′→u′′inL2(0,T;H01(Ω))weakly,θm→θinL∞(0,T;H01(Ω))weak*,θm′→θ′inL2(0,T;H01(Ω))weakly,θm′→θ′inL∞(0,T;L2(Ω))weak*.
Moreover, according to the compactness of Aubin-Lions, we have
um→ustronglyinL2(0,T;L2(Ω)),θm→θstronglyinL2(0,T;L2(Ω)).
Hence combing (3.31) and the fourth estimate (3.28), we deduce that
um→ustronglyinC0(0,T;H01(Ω)).

Thus we can pass the limit in system (3.2)-(3.3). Let m→∞, we prove that {u,θ} is a weak solution of the system (1.1).

4. Uniqueness of the Solution

The proof of uniqueness of solution is similar to the fourth estimate, but for integrity, we still give the detailed proof.

Let (u1,θ1) and (u2,θ2) be two solutions of couple system (1.1) under the conditions of Theorem 2.1, then we have (u,θ):=(u1-u2,θ1-θ2) verifying
(u′′,w)+(∇u,∇w)+(∥∇u1∥2γ∇u,∇w)+((∥∇u1∥2γ-∥∇u2∥2γ)∇u2,∇w)-∫0th(t-τ)(∇u(τ),∇w)dτ+β1(θ,w)+(∇u′,∇w)+(|u1(t)|ρ-2u1(t)-|u2(t)|ρ-2u2(t),w)=0,(θ′,w)+(∇θ,∇w)+β2(u′,w)=0∀w∈H01(Ω),u(0)=u′(0)=θ(0)=0.

Taking w=u′ in (4.1) and w=θ in (4.2), respectively, and adding the results, we have
12ddt[∥u′(t)∥2+∥θ(t)∥2+∥∇u(t)∥2+∥∇u1(t)∥2γ∥∇u(t)∥2]+∥∇u′(t)∥2+∥∇θ(t)∥2≤((∥∇u2(t)∥2γ-∥∇u1(t)∥2γ)∇u2(t),∇u′(t))-(|u1(t)|ρ-2u1(t)-|u2(t)|ρ-2u2(t),u′(t))+12∥∇u(t)∥2ddt∥∇u1(t)∥2γ+(β1+β2)|(θ(t),u′(t))|+∫0th(t-τ)(∇u(τ),∇u′(t))dτ.
Noticing that
∫0th(t-τ)(∇u(τ),∇u′(t))dτ=-h(0)∥∇u(t)∥2-∫0th′(t-τ)(∇u(τ),∇u(t))dτ+ddt(∫0th(t-τ)(∇u(τ),∇u(t))dτ),
hence, using assumption (2.2) and integrating (4.3) over (0,t), we get
12[∥u′(t)∥2+∥θ(t)∥2+∥∇u(t)∥2+∥∇u1(t)∥2γ∥∇u(t)∥2]+h(0)∫0t∥∇u(τ)∥2dτ+∫0t∥∇u′(τ)∥2dτ+∫0t∥∇θ(τ)∥2dτ≤∫0t|∥∇u1(τ)∥2γ-∥∇u2(τ)∥2γ|∥∇u2(τ)∥∥∇u′(τ)∥dτ+∫0t∥|u1(τ)|ρ-2u1(τ)-|u2(τ)|ρ-2u2(τ)∥∥u′(τ)∥dτ+∫0t(β1+β2)|(θ(τ),u′(τ))|dτ+12∫0t∥∇u(τ)∥2ddτ∥∇u1(τ)∥2γdτ+c1∫0th(t-τ)|(∇u(τ),∇u(t))|dτ.
Notice that
c1∫0th(t-τ)|(∇u(τ),∇u(t))|dτ≤c2(η)∫0t∥∇u(τ)∥2dτ+η∥∇u(t)∥2,
where η>0 is arbitrary:
|∥∇u1(τ)∥2γ-∥∇u2(τ)∥2γ|≤c3(∥∇u1(τ)∥2γ-1+∥∇u2(τ)∥2γ-1)∥∇u(τ)∥,ddτ∥∇u1(τ)∥2γ≤c4∥∇u1(τ)∥2γ-1∥∇u′1(τ)∥.
Moreover, by mean value theorem and assumption (2), we have
∥|u1(τ)|ρ-2u1(τ)-|u2(τ)|ρ-2u2(τ)∥≤c5(∥∇u1(τ)∥ρ-2+∥∇u2(τ)∥ρ-2)∥∇u(τ)∥.
Therefore, by (4.6)–(4.8), Cauchy inequality, Young's inequality, and using Gronwall's lemma in (4.5), we get
∥u′(t)∥2+∥θ(t)∥2+∥∇u(t)∥2+∫0T∥∇u′(τ)∥2dτ+∫0T∥∇θ(τ)∥2dτ=0.
Thus, we have proved the uniqueness consequence.

5. Existence of Solutions

In this section, we follow the additional assumptions appeared in Theorem 2.1. We introduce the energy
e(t):=12(∥u′(t)∥2+∥θ(t)∥2+(1-∫0th(s)ds)∥∇u(t)∥2+(h□∇u)(t)+1γ+1∥∇u(t)∥2(γ+1)+2ρ∥u(t)∥ρρ),
where we define
(h□y)(t)=∫0th(t-τ)∥y(t)-y(τ)∥22dτ.

Remark 5.1.

Taking w=u′(t) in (2.6) and w=θ(t) in (2.7), respectively, then adding the results we have
12ddt(∥u′(t)∥2+∥θ(t)∥2+∥∇u(t)∥2+1γ+1∥∇u(t)∥2(γ+1)+2ρ∥u(t)∥ρρ)+∥∇u′(t)∥2+∥∇θ(t)∥2+(β1+β2)(θ(t),u′(t))=∫0th(t-τ)(∇u(τ),∇u′(t))dτ.
Noticing
∫0th(t-τ)(∇u(τ),∇u′(t))dτ=12(h′□∇u)(t)-12(h□∇u)′(t)+12(∫0th(s)ds∥∇u(t)∥2)′-12h(t)∥∇u(t)∥2,
and combining the assumptions on β1,β2 appeared in Theorem 2.1, we deduce
e′(t)≤-M1∥∇u′(t)∥2-M1∥∇θ(t)∥2+12(h′□∇u)(t)-12h(t)∥∇u(t)∥2≤-M1∥∇u′(t)∥2-M1∥∇θ(t)∥2-ξ22(h□∇u)(t)≤0,
where we denote M1=1-(β1+β2)(μ2/2)≥0. Thus, we have the energy e(t) is uniformly bounded (by e(0) and is decreasing in t.

Remark 5.2.

Furthermore, from the assumption (1), we have
E(t)≤12(∥u′(t)∥2+∥θ(t)∥2+1l(1-∫0th(s)ds)∥∇u(t)∥2+1γ+1∥∇u(t)∥2(γ+1)+2ρ∥u(t)∥ρρ)≤l-1e(t).

For every ε>0, we define the perturbed energy by setting
eε(t)=e(t)+εψ(t),whereψ(t)=(u′(t),u(t)).

Lemma 5.3.

There exists M2>0 such that
|eε(t)-e(t)|≤εM2e(t),∀t≥0.

Proof.

From (5.7), we obtain
|ψ(t)|≤μ∥u′(t)∥∥∇u(t)∥≤μ2∥u′(t)∥2+μ2∥∇u(t)∥2≤μle(t),
hence we have
|eε(t)-e(t)|≤εM2e(t),∀t≥0,
where M2=μ/l.

Lemma 5.4.

There exists M3>0 and ε¯ such that for ε∈(0,ε¯],e′ε(t)≤-εM3e(t).

Proof.

By using the problem (1.1), we obtain
ψ′(t)=∥u′(t)∥2+(u′′(t),u(t))=∥u′(t)∥2-∥∇u(t)∥2-∥∇u(t)∥2γ+2+∫0th(t-τ)(∇u(τ),∇u(t))dτ-(∇u′(t),∇u(t))-β1(θ(t),u(t))-(|u|ρ-2u,u).
Notice that
∫0th(t-τ)(∇u(τ),∇u(t))dτ=∫0th(t-τ)(∇u(τ)-∇u(t),∇u(t))dτ+∥∇u(t)∥2∫0th(t-τ)dτ≤14η∫0th(t-τ)∥∇u(τ)-∇u(t)∥2dτ+(1+η)∥∇u(t)∥2∫0th(t-τ)dτ=14η(h□∇u)(t)+(1+η)∥∇u(t)∥2∫0th(t-τ)dτ≤14η(h∇u)(t)+(1+η)(1-l)∥∇u(t)∥2,β1(θ(t),u(t))≤β1μ22∥∇θ(t)∥2+β1μ22∥∇u(t)∥2,(∇u′(t),∇u(t))≤η∥∇u(t)∥2+14η∥∇u′(t)∥2,
where η>0 is arbitrary.

Hence, from (5.12)–(5.15), we have

ψ′(t)≤∥u′(t)∥2+(-l+η(2-l)+β1μ22)∥∇u(t)∥2-∥∇u(t)∥2γ+2+14η(h□∇u)(t)+14η∥∇u′(t)∥2+β1μ22∥∇θ(t)∥2-∥u(t)∥ρρ.
Therefore, from (5.5) and (5.16), we get
e′ε(t)=e′(t)+εψ′(t)≤-M1∥∇u′(t)∥2-M1∥∇θ(t)∥2-ξ22(h□∇u)(t)+εμ2∥∇u′(t)∥2+ε(-l+η(2-l)+β1μ22)∥∇u(t)∥2-ε∥∇u(t)∥2γ+2+ε4η(h□∇u)(t)+ε4η∥∇u′(t)∥2+εβ1μ22∥∇θ(t)∥2-ε∥u(t)∥ρρ≤-(M1-εμ2-ε4η)∥∇u′(t)∥2-(M1-εβ1μ22)∥∇θ(t)∥2-(ξ22-ε4η)(h□∇u)(t)-ε(l-η(2-l)-β1μ22)∥∇u(t)∥2-ε∥∇u(t)∥2γ+2-ε∥u(t)∥ρρ.
Taking β1 and η small enough, we have l-η(2-l)-β1μ2/2≥0. Moreover if we denote
ε̃=min{M1μ2+1/4η,2M1β1μ2,2ηξ2},
and choosing ε∈(0,ε̃], we obtain
e′ε(t)≤-εM3e(t)
for some constant M3>0.

Proof of Decay

Let us define ε̂=min{1/2M2,ε̃} and consider ε∈(0,ε̂]. From Lemma 5.3, we have
(1-M2ε)e(t)≤eε(t)≤(1+M2ε)e(t),
and so
12e(t)≤eε(t)≤32e(t).
From (5.21), we get
-εM3e(t)≤-ε23M3eε(t).
Hence from (5.22) and Lemma 5.4, we obtain
e′ε(t)≤-ε23M3eε(t).
that is,
ddt(eε(t)exp{2ε3M3t})≤0.
Integrating last inequality over [0,t], we get
eε(t)≤eε(0)exp{-2ε3M3t}.
From (5.21) and (5.25), we have
e(t)≤3e(0)exp{-2ε3M3t}.

Hence, from (5.6) and (5.26), we obtain

E(t)≤l-1e(t)≤3e(0)l-1exp{-2ε3M3t},t≥t0,∀ε∈(0,ε̂],
that is,
E(t)≤Cexp(-ξt),∀t≥t0,
where C=3e(0)l-1 and ξ=(2ε/3)M3.

Therefore, we have proved the exponential decay of solution.

Acknowledgment

This work is supported by NSFC of Yunnan Province (07Y40422, 2007A196M) and the National Natural Science Foundation of China under Grant 10471072.

MatosM. P.PereiraD. C.On a hyperbolic equation with strong dampingMedeirosL. A.MirandaM. M.On a nonlinear wave equation with dampingClarkM. R.LimaO. A.On a mixed problem for a coupled nonlinear systemDuvautG.LionsJ.-L.KomornikV.ZuazuaE.A direct method for the boundary stabilization of the wave equationShowalterR. E.