Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms

Miaomiao Chen; Wenjun Liu; Weican Zhou

doi:10.1515/anona-2016-0085

Open Access Published by De Gruyter October 11, 2016

Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms

Miaomiao Chen , Wenjun Liu and Weican Zhou

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2016-0085

Abstract

In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms:

{ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+β⁢θx=0,(x,t)∈(0,1)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢ψt⁢t⁢x+∫0tg⁢(t-s)⁢θx⁢x⁢(s)⁢ds+μ1⁢θt⁢(x,t)+μ2⁢θt⁢(x,t-τ)=0,(x,t)∈(0,1)×(0,∞),

together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ1,μ2 are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.

Keywords: Global existence; general energy decay; Timoshenko system; relaxation function

MSC 2010: 35B35; 35L55; 74D05; 93D15

1 Introduction

In this paper we investigate the existence and decay properties of solutions for the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms:

(1.1){ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+β⁢θx=0,(x,t)∈(0,1)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢ψt⁢t⁢x+∫0tg⁢(t-s)⁢θx⁢x⁢(s)⁢ds+μ1⁢θt⁢(x,t)+μ2⁢θt⁢(x,t-τ)=0,(x,t)∈(0,1)×(0,∞),

with the following initial datum and boundary conditions:

(1.2){φ⁢(x,0)=φ0,φt⁢(x,0)=φ1,ψ⁢(x,0)=ψ0,ψt⁢(x,0)=ψ1,x∈[0,1],θ⁢(x,0)=θ0,θt⁢(x,0)=θ1,x∈[0,1],φ⁢(0,t)=φ⁢(1,t)=ψ⁢(0,t)=ψ⁢(1,t)=θx⁢(0,t)=θx⁢(1,t)=0,t∈[0,∞),θt⁢(x,t-τ)=f0⁢(x,t-τ),(x,t)∈(0,1)×(0,τ),

where the coefficients ρ1,ρ2,ρ3,K,b,β,γ,δ,μ1 and μ2 are positive constants, and τ>0 represents the time delay.

System (1.1) arises in the theory of the transverse vibration of a beam, which was first introduced by Timoshenko. In 1921, Timoshenko [36] considered the following system of coupled hyperbolic equations:

(1.3){ρ⁢φt⁢t-K⁢(φx-ψ)x=0,(x,t)∈(0,L)×(0,∞),Iρ⁢ψt⁢t-(E⁢I⁢ψx)x-K⁢(φx-ψ)=0,(x,t)∈(0,L)×(0,∞),

where φ is the transverse displacement of the beam and ψ is the rotation angle of the filament of the beam. The coefficients ρ,Iρ,E,I and K are the density, the polar moment of inertia of a cross section, Young’s modulus of elasticity, the moment of inertia of a cross section, and the shear modulus, respectively.

Many mathematicians have studied system (1.3) and some results concerning the existence and asymptotic behavior of solutions have been established, see, for instance, [9, 21, 32, 33] and the references therein. Kim and Renardy [12] considered (1.3) together with two linear boundary conditions of the form

K⁢ψ⁢(L,t)-K⁢∂⁡φ∂⁡x⁢(L,t)=α⁢∂⁡φ∂⁡t⁢(L,t),t∈[0,∞),

E⁢I⁢∂⁡ψ∂⁡x⁢(L,t)=-β⁢∂⁡ψ∂⁡t⁢(L,t),t∈[0,∞),

and used the multiplier techniques to establish an exponential decay result for the energy of (1.3). They also provided numerical estimates for the eigenvalues of the operator associated with system (1.3). Soufyane and Wehbe [34] considered the system

(1.4){ρ⁢φt⁢t-K⁢(φx-ψ)x=0,(x,t)∈(0,L)×(0,∞),Iρ⁢ψt⁢t-(E⁢I⁢ψx)x-K⁢(φx-ψ)+b⁢(x)⁢ψt=0,(x,t)∈(0,L)×(0,∞),

where b is a positive and continuous function satisfying

b⁢(x)≥b0>0 for all ⁢x∈[a0,a1]⊂[0,L].

They proved that the uniform stability of (1.4) holds if and only if the wave speeds are equal (Kρ=E⁢IIρ); otherwise only the asymptotic stability can be proved. For more results related to system (1.3), we refer the readers to [3, 11, 29, 7, 26] and the references therein.

For the Timoshenko system of thermo-viscoelasticity of type III, Messaoudi and Said-Houari [24] considered the following one-dimensional linear Timoshenko system of thermoelastic type:

(1.5){ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+β⁢θx=0,(x,t)∈(0,1)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢ψt⁢t⁢x-κ⁢θt⁢x⁢x=0,(x,t)∈(0,1)×(0,∞).

They used the energy method to prove an exponential decay under the condition ρ1K=ρ2b. A similar result was also obtained by Rivera and Racke [25], and Messaoudi et al. [23]. Then, in [22], Messaoudi and Fareh also considered problem (1.5). By introducing the first and second-order energy functions, they proved a polynomial stability result under the condition ρ1K≠ρ2b.

The case of time delay in the Timoshenko system has been studied by some authors. Said-Houari and Laskri [30] considered the following Timoshenko system with a constant time delay in the feedback:

{ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+μ1⁢ψt+μ2⁢ψt⁢(x,t-τ)=0,(x,t)∈(0,1)×(0,∞).

They established an exponential decay result for the case of equal-speed wave propagation (ρ1K=ρ2b) under the assumption μ2<μ1. Then Kirane, Said-Houari and Anwar [14] considered the following Timoshenko system with a time-varying delay:

{ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+μ1⁢ψt+μ2⁢ψt⁢(x,t-τ⁢(t))=0,(x,t)∈(0,1)×(0,∞),

where τ⁢(t)>0 represents the time varying delay, 0<τ0≤τ⁢(t)≤τ¯ and μ1,μ2 are positive constants. Under the assumptions ρ1K=ρ2b and μ2<1-d⁢μ1, where d is a constant such that τ′⁢(t)≤d<1, they proved that the energy decays exponentially.

Recently, systems with viscoelastic damping have been investigated by many authors. It has been showed that the dissipation produced by the viscoelastic part is strong enough to produce the decay of the solution, see [1, 18, 31, 2, 4, 8, 15, 20, 35, 37, 39]. For example, Djebabla and Tatar [8] considered the Timoshenko system

{ρ1⁢φt⁢t-K⁢(φx+ψ)x=0,(x,t)∈(0,L)×(0,∞),ρ2⁢ψt⁢t-b⁢ψx⁢x+K⁢(φx+ψ)+β⁢θx=0,(x,t)∈(0,L)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢∫0tg⁢(t-s)⁢θx⁢x⁢(s)⁢ds+β⁢ψt⁢t⁢x=0,(x,t)∈(0,L)×(0,∞),

where ρ1,ρ2,ρ3,K,b,β,γ and δ are positive constants. They proved the exponential decay of solutions in the energy norm if and only if the coefficients satisfy b⁢ρ1K-ρ2=δ-K⁢ρ3ρ1=β and g decays uniformly.

Kirane and Said-Houari [13] examined the following system of viscoelastic wave equations with a linear damping and a delay term:

(1.6){ut⁢t-Δ⁢u+∫0tg⁢(t-s)⁢Δ⁢u⁢(x,s)⁢ds+μ1⁢ut⁢(x,t)+μ2⁢ut⁢(x,t-τ)=0,(x,t)∈Ω×(0,∞),u⁢(x,t)=0,(x,t)∈∂⁡Ω×(0,∞),u⁢(x,0)=u0⁢(x),ut⁢(x,0)=u1⁢(x),x∈Ω,ut⁢(x,t-τ)=f0⁢(x,t-τ),(x,t)∈Ω×(0,τ),

where Ω is a regular and bounded domain of ℝN(N≥1), μ1,μ2 are positive constants, τ>0 represents the time delay and u0,u1,f0 are given functions belonging to suitable spaces. They proved that the energy of problem (1.6) decreases exponentially as t tends to infinity provided that 0<μ2≤μ1 and g decays exponentially. After that, Liu [17] considered the viscoelastic wave equation with a linear damping and a time-varying delay term in the feedback. Under suitable assumptions, he established a general decay result. For more results about systems with viscoelastic damping and a delay term, we refer the readers to [38, 19] and the references therein.

Motivated by the above research, we consider the global existence and the general energy decay for problem (1.1). First, by using the Faedo–Galerkin approximations together with some energy estimates and under some restriction on the parameters μ1 and μ2, we prove the global existence of weak solutions. Then, under the hypothesis μ2≤μ1, we prove a general decay of the total energy of our problem by using an energy method. Our method of proof follows some ideas developed in [13] for the wave equation with a viscoelastic damping and a delay term, enabling us to obtain suitable Lyapunov functionals. We recall that for μ1=μ2, Nicaise and Pignotti showed in [27] that some instabilities may occur. Here, due to the presence of the viscoelastic damping, we can still establish a general energy decay result even if μ1=μ2.

The remaining part of this paper is organized as follows. In Section 2, we present some materials and recall some useful lemmas needed for our work and state our main results. In Section 3, we will prove the global existence of the solution. We will prove several technical lemmas and the general decay result under two cases, μ2<μ1 and μ2=μ1, in Section 4.

2 Preliminaries and main results

In this section, we present some assumptions and state the main results. We use the standard Lebesgue space L2⁢(0,1) and the Sobolev space H01⁢(0,1) with their usual scalar products and norms and define the space X as

X=[H01⁢(0,1)×L2⁢(0,1)]2×V×H,

where

V=H1⁢(0,1)∩H and H={θ∈L2⁢(0,1)∣θx⁢(0,t)=θx⁢(1,t)=0⁢ for all ⁢t∈[0,∞)}.

First, in order to exhibit the dissipative nature of system (1.1), as in [24], we introduce the new variables Φ=φt and Ψ=ψt. Then, as in [28], we introduce the function

z⁢(x,ρ,t)=θt⁢(x,t-ρ⁢τ),(x,ρ,t)∈(0,1)×(0,1)×(0,∞).

Then we have

τ⁢zt⁢(x,ρ,t)+zρ⁢(x,ρ,t)=0,(x,ρ,t)∈(0,1)×(0,1)×(0,∞).

Therefore, problem (1.1) is equivalent to

(2.1){ρ1⁢Φt⁢t-K⁢(Φx+Ψ)x=0,(x,t)∈(0,1)×(0,∞),ρ2⁢Ψt⁢t-b⁢Ψx⁢x+K⁢(Φx+Ψ)+β⁢θt⁢x=0,(x,t)∈(0,1)×(0,∞),ρ3⁢θt⁢t-δ⁢θx⁢x+γ⁢Ψt⁢x+∫0tg⁢(t-s)⁢θx⁢x⁢(s)⁢ds+μ1⁢θt⁢(x,t)+μ2⁢z⁢(x,1,t)=0,(x,t)∈(0,1)×(0,∞),τ⁢zt⁢(x,ρ,t)+zρ⁢(x,ρ,t)=0,(x,ρ,t)∈(0,1)×(0,1)×(0,∞)

with the following initial datum and boundary conditions:

(2.2){Φ⁢(x,0)=Φ0,Φt⁢(x,0)=Φ1,Ψ⁢(x,0)=Ψ0,Ψt⁢(x,0)=Ψ1,x∈[0,1],θ⁢(x,0)=θ0,θt⁢(x,0)=θ1,x∈[0,1],Φ⁢(0,t)=Φ⁢(1,t)=Ψ⁢(0,t)=Ψ⁢(1,t)=θx⁢(0,t)=θx⁢(1,t)=0,t∈[0,∞),z⁢(x,0,t)=θt⁢(x,t),(x,t)∈(0,1)×(0,∞),θt⁢(x,t-τ)=f0⁢(x,t-τ),(x,t)∈(0,1)×(0,τ),

where

Φ0=φ1,Φ1=Kρ1⁢(φ0⁢x+ψ0)x,

Ψ0=ψ1,Ψ1=bρ2⁢ψ0⁢x⁢x-Kρ2⁢(φ0⁢x+ψ0)-βρ2⁢θ0⁢x

Next, we denote by * the usual convolution, i.e.,

(g*h)⁢(t)=∫0tg⁢(t-s)⁢h⁢(s)⁢ds

and we define the binary operators ⋄ and □, respectively, by

(g⋄h)⁢(t)=∫0tg⁢(t-s)⁢(h⁢(t)-h⁢(s))⁢ds

and

(□⁡gh)⁢(t)=∫0tg⁢(t-s)⁢(h⁢(t)-h⁢(s))2⁢ds.

The following lemma was introduced in [13]. It will be used in Section 4 to prove the general energy decay result for problem (1.1)–(1.2).

Lemma 2.1 ([5, 6]).

For any function g∈C1⁢(R) and any h∈H1⁢(0,1), we have

(g*h)⁢(t)⁢ht⁢(t)=-12⁢g⁢(t)⁢|h⁢(t)|2+12⁢(□⁡g′h)⁢(t)-12⁢dd⁢t⁢{(□⁡gh)⁢(t)-(∫0tg⁢(s)⁢ds)⁢|h⁢(t)|2}.

The proof of this lemma follows by differentiating the term g⋄h.

Lemma 2.2 ([8]).

For any function g∈C⁢([0,∞),R+) and any h∈L2⁢(0,1), we have

[(g⋄h)⁢(t)]2≤(∫0tg⁢(s)⁢ds)⁢(□⁡gh)⁢(t),t≥0.

Now, we assume that the kernel g satisfies the following assumptions.

Hypothesis 1.

g:ℝ+→ℝ+ is a differential function such that

g⁢(0)>0,λ=δ-∫0∞g⁢(s)⁢ds=δ-g¯>0.

Hypothesis 2.

There exists a non-increasing differential function ζ:ℝ+→ℝ+ satisfying

g′⁢(t)≤-ζ⁢(t)⁢g⁢(t),t≥0 and ∫0+∞ζ⁢(t)⁢dt=+∞.

To state our decay result, we introduce the energy functional associated to problem (2.1), namely,

E⁢(t)=γ2⁢∫01{ρ1⁢Φt2+ρ2⁢Ψt2+(Φx+Ψ)2+b⁢Ψx2}⁢dx

(2.3)+β2⁢∫01{ρ3⁢θt2+(δ-∫0tg⁢(s)⁢ds)⁢θx2+(□⁡gθx)+ξ⁢∫01z2⁢(x,ρ,t)⁢dρ}⁢dx,

where ξ is a positive constant such that

(2.4)τ⁢μ2<ξ<τ⁢(2⁢μ1-μ2)if ⁢μ2<μ1,

(2.5)ξ=τ⁢μ2if ⁢μ2=μ1.

Our main results read as follows.

Theorem 2.3.

Assume that μ2≤μ1. Then, for any given (Φ0,Φ1,Ψ0,Ψ1,θ0,θ1)∈X, f0∈L2⁢((0,1)×(0,1)) and T>0, there exists a unique weak solution (Φ,Ψ,θ,z) of problem (2.1)–(2.2) on (0,T) such that

(Φ,Ψ,θ)∈C⁢([0,T],[H01⁢(0,1)]2×V)∩C1⁢([0,T],[L2⁢(0,1)]2×H).

Theorem 2.4.

Assume that μ2≤μ1 and that g satisfies Hypotheses 1 and 2. Assume further that the initial datum satisfy

(Φ0,Φ1,Ψ0,Ψ1,θ0,θ1)∈X,f0∈L2⁢((0,1)×(0,1)),

and the coefficients ρ1,ρ2,ρ3,K,b,γ,δ and β satisfy

(2.6)b⁢ρ1K-ρ2=γ,δ-K⁢ρ3ρ1=β.

Then, for any t0>0, there exist two positive constants A and ω, independent of the initial datum, such that

(2.7)E⁢(t)≤A⁢e-ω⁢∫t0tζ⁢(s)⁢ds,t≥t0.

Remark.

We note that the exponential and the polynomial decay estimates are only particular cases of (2.7). In fact, we obtain exponential decay for ζ≡a and polynomial decay for ζ=a⁢(1+t)-1, where a is a constant.

3 Proof of Theorem 2.3

In this section, we will use the Faedo–Galerkin approximations together with some energy estimates, to prove the existence of the unique solution of problem (2.1)–(2.2) as stated in Theorem 2.3. We divide the proof into two steps: we first construct Faedo–Galerkin approximations, and then thanks to certain energy estimates we pass to the limit.

Step 1: Faedo–Galerkin approximations.

As in [13] and [10], we construct approximations of the solution (Φ,Ψ,θ,z) by the Faedo–Galerkin method as follows. For every n≥1, let Wn=span⁢{ω1,…,ωn} be a Hilbert basis of the space H01⁢(0,1).

Now, we define for 1≤j≤n the sequence φ¯j⁢(x,ρ) as follows:

φ¯j⁢(x,0)=ωj⁢(x).

Then we may extend φ¯j⁢(x,0) by φ¯j⁢(x,ρ) over L2⁢((0,1)×[0,1]) and denote Vn=span⁢{φ¯1,…,φ¯n}. We choose sequences (Φ0n,Ψ0n,θ0n) and (Φ1n,Ψ1n,θ1n) in Wn and a sequence (z0n) in Vn such that

(Φ0n,Φ1n,Ψ0n,Ψ1n,θ0n,θ1n)→(Φ0,Φ1,Ψ0,Ψ1,θ0,θ1) strongly in X

and z0n→f0 strongly in L2⁢((0,1)×(0,1)).

We define now the approximations

(Φn⁢(x,t),Ψn⁢(x,t),θn⁢(x,t))=∑j=1n(fjn⁢(t),yjn⁢(t),hjn⁢(t))⁢ωj⁢(x)

and

zn⁢(x,ρ,t)=∑j=1nljn⁢(t)⁢φ¯j⁢(x,ρ),

where (Φn⁢(t),Ψn⁢(t),θn⁢(t),zn⁢(t)) satisfies the following problem:

(3.1){ρ1⁢∫01Φt⁢tn⁢ωj⁢dx+K⁢∫01(Φxn+Ψn)⁢ωx⁢j⁢dx=0,ρ2⁢∫01Ψt⁢tn⁢ωj⁢dx+b⁢∫01Ψxn⁢ωx⁢j⁢dx+K⁢∫01(Φxn+Ψn)⁢ωj⁢dx-β⁢∫01θtn⁢ωx⁢j⁢dx=0,ρ3⁢∫01θt⁢tn⁢ωj⁢dx+δ⁢∫01θxn⁢ωx⁢j⁢dx-γ⁢∫01Ψtn⁢ωx⁢j⁢dx-∫0tg⁢(t-s)⁢∫01θxn⁢(s)⁢ωx⁢j⁢dx⁢ds+∫01(μ1⁢θtn⁢(x,t)+μ2⁢zn⁢(x,1,t))⁢ωj⁢dx=0,(Φn⁢(0),Ψn⁢(0),θn⁢(0))=(Φ0n,Ψ0n,θ0n),(Φtn⁢(0),Ψtn⁢(0),θtn⁢(0))=(Φ1n,Ψ1n,θ1n),zn⁢(x,0,t)=θtn⁢(x,t)

and

(3.2)∫01(τ⁢ztn⁢(x,ρ,t)+zρn⁢(x,ρ,t))⁢φ¯j⁢dx=0 for 1≤j≤n,zn⁢(x,ρ,0)=z0n.

According to the standard theory of ordinary differential equations, the finite dimensional problem (3.1)–(3.2) has a solution (fjn⁢(t),yjn⁢(t),hjn⁢(t),ljn⁢(t))j=1,…,n, defined on [0,tn). Then the a priori estimates that follow imply that in fact tn=T.

Step 2: Energy estimates.

Multiplying, in (2.1), the first equation by γ⁢(fjn)′, the second equation by γ⁢(yjn)′ and the third equation by β⁢(hjn)′, and then integrating over (0,1) using integration by parts and Lemma 2.1, we get, for every n≥1,

γ2⁢∫01{ρ1⁢(Φtn)2+ρ2⁢(Ψtn)2+K⁢(Φxn+Ψn)2+b⁢(Ψxn)2}⁢dx

+β2⁢∫01{ρ3⁢(θtn)2+(δ-∫0tg⁢(s)⁢ds)⁢(θxn)2+(□⁡gθxn)}⁢dx-β2⁢∫0t(□⁡g′θxn)⁢(s)⁢ds

+β⁢μ1⁢∫0t∫01(θtn)2⁢(s)⁢dx⁢ds+β⁢μ2⁢∫0t∫01θtn⁢(x,s)⁢zn⁢(x,1,s)⁢dx⁢ds+β2⁢∫0tg⁢(s)⁢∫01(θxn)2⁢(s)⁢dx⁢ds

(3.3)=γ2⁢∫01{ρ1⁢(Φ1n)2+ρ2⁢(Ψ1n)2+[K⁢(Φxn+Ψn)2+b⁢(Ψxn)2]⁢(x,0)}⁢dx+β2⁢∫01{ρ3⁢(θ1n)2+δ⁢(θxn)2⁢(x,0)}⁢dx.

Let ξ>0 to be chosen later. Multiplying the fourth equation in (2.1) by ξτ⁢(ljn)′⁢(t) and integrating over (0,t)×(0,1), we obtain

(3.4)ξ2⁢∫01∫01(zn)2⁢(x,ρ,t)⁢dρ⁢dx+ξτ⁢∫0t∫01∫01zρn⁢zn⁢(x,ρ,s)⁢dρ⁢dx⁢ds=ξ2⁢∫01∫01(z0n)2⁢(x,ρ,0)⁢dρ⁢dx.

Now, to handle the last term in the left-hand side of (3.4), we remark that

∫0t∫01∫01zρn⁢zn⁢(x,ρ,t)⁢dρ⁢dx⁢ds=12⁢∫0t∫01∫01∂∂⁡ρ⁢(zn)2⁢(x,ρ,s)⁢dρ⁢dx⁢ds

(3.5)=12⁢∫0t∫01((zn)2⁢(x,1,s)-(zn)2⁢(x,0,s))⁢dx⁢ds.

Summing up the identities (3.3) and (3.4), and taking into account (3.5), we get

ℰn⁢(t)+β⁢(μ1-ξ2⁢τ)⁢∫0t∫01(θtn)2⁢dx⁢ds+ξ⁢β2⁢τ⁢∫0t∫01(zn)2⁢(x,1,s)⁢dx⁢ds+β⁢μ2⁢∫0t∫01zn⁢(x,1,s)⁢θtn⁢(x,s)⁢dx⁢ds

+β2⁢∫0tg⁢(s)⁢∫01(θxn)2⁢(s)⁢dx⁢ds-β2⁢∫0t(□⁡g′θxn)⁢(s)⁢ds

(3.6)=ℰn⁢(0),

where

ℰn⁢(t)=γ2⁢∫01{ρ1⁢(Φtn)2+ρ2⁢(Ψtn)2+K⁢(Φxn+Ψn)2+b⁢(Ψxn)2}⁢dx

+β2⁢∫01{ρ3⁢(θtn)2+(δ-∫0tg⁢(s)⁢ds)⁢(θxn)2+(□⁡gθxn)}⁢dx+β⁢ξ2⁢∫01∫01(zn)2⁢(x,ρ,t)⁢dρ⁢dx.

At this point, we have to distinguish the following two cases. Case 1: We suppose that μ2<μ1. Let us choose ξ satisfies inequality (2.4). Using Young’s inequality, (3.6) leads to

ℰn⁢(t)+β⁢(μ1-ξ2⁢τ-μ22)⁢∫0t∫01(θtn)2⁢(s)⁢dx⁢ds+β⁢(ξ2⁢τ-μ22)⁢∫0t∫01(zn)2⁢(x,1,s)⁢dx⁢ds

+β2⁢∫0tg⁢(s)⁢∫01(θxn)2⁢(s)⁢dx⁢ds-β2⁢∫0t(□⁡g′θxn)⁢(s)⁢ds

(3.7)≤ℰn⁢(0).

Consequently, we can find two positive constants c1 and c2 such that

ℰn⁢(t)+c1⁢∫0t∫01(θtn)2⁢(s)⁢dx⁢ds+c2⁢∫0t∫01(zn)2⁢(x,1,s)⁢dx⁢ds

+β2⁢∫0tg⁢(s)⁢∫01(θxn)2⁢(s)⁢dx⁢ds-β2⁢∫0t(□⁡g′θxn)⁢(s)⁢ds

≤ℰn⁢(0).

Case 2: We suppose that μ2=μ1=μ and choose ξ so that it satisfies inequality (2.5). Then inequality (3.7) takes the form

ℰn⁢(t)+β2⁢∫0tg⁢(s)⁢∫01(θxn)2⁢(s)⁢dx⁢ds-β2⁢∫0t(□⁡g′θxn)⁢(s)⁢ds≤ℰn⁢(0).

Now, in both cases, since the sequences (Φ0n)n∈ℕ, (Φ1n)n∈ℕ, (Ψ0n)n∈ℕ, (Ψ1n)n∈ℕ, (θ0n)n∈ℕ, (θ1n)n∈ℕ and (z0n)n∈ℕ converge, using Hypotheses 1 and 2, we can find a positive constant C independent of n such that

(3.8)ℰn⁢(t)≤C.

Therefore, from (3.8) and the Aubin–Lions theorem [16], we can pass to the limit in (3.1)–(3.2). The rest of the proof is routine.

4 Proof of Theorem 2.4

In this section, under the hypothesis μ2≤μ1, we show that the energy of the solution of problem (2.1)–(2.2) decreases generally as t tends to infinity by using the energy method and constructing suitable Lyapunov functionals. We will separately discuss the two cases which are the case μ2<μ1 and the case μ2=μ1, since the proofs are slightly different.

4.1 The case μ2<μ1

Our goal now is to prove that the above energy E⁢(t) is a non-increasing functional along the trajectories. More precisely, we have the following result.

Lemma 4.1.

Suppose that Hypotheses 1 and 2 hold and let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). Then we have

E′⁢(t)≤-β2⁢g⁢(t)⁢∫01θx2⁢dx+β2⁢∫01(□⁡g′θx)⁢dx-β⁢(μ1-ξ2⁢τ-μ22)⁢∫01θt2⁢dx

(4.1)-β⁢(ξ2⁢τ-μ22)⁢∫01z2⁢(x,1,t)⁢dx≤0,t≥0.

Proof.

Multiplying, in (2.1), the first equation by γ⁢Φt, the second equation by γ⁢Ψt and the third equation by β⁢θt, and then integrating over (0,1) using integration by parts, we find

γ2⁢dd⁢t⁢[∫01{ρ1⁢Φt2+ρ2⁢Ψt2+K⁢(Φx+Ψ)2+b⁢Ψx2}⁢dx]+β2⁢dd⁢t⁢[∫01{ρ3⁢θt2+(δ-∫0tg⁢(τ)⁢dτ)⁢θx2+(□⁡gθx)}⁢dx]

=-β⁢μ1⁢∫01θt2⁢dx-β⁢μ2⁢∫01θt⁢z⁢(x,1,t)⁢dx-β2⁢g⁢(t)⁢∫01θx2⁢dx+β2⁢∫01(□⁡g′θx)⁢dx.

Now, multiplying the fourth equation in (2.1) by ξτ⁢z, and then integrating the result over (0,1)×(0,1) with respect to ρ and x, respectively, we obtain

ξ2⁢dd⁢t⁢∫01∫01z2⁢(x,ρ,t)⁢dρ⁢dx=-ξτ⁢∫01∫01z⁢zρ⁢(x,ρ,t)⁢dρ⁢dx=-ξ2⁢τ⁢∫01∫01∂∂⁡ρ⁢z2⁢(x,ρ,t)⁢dρ⁢dx

(4.2)=ξ2⁢τ⁢∫01[z2⁢(x,0,t)-z2⁢(x,1,t)]⁢dx=ξ2⁢τ⁢∫01(θt2-z2⁢(x,1,t))⁢dx.

By Young’s inequality, we have

(4.3)-μ2⁢∫01θt⁢z⁢(x,1,t)⁢dx≤μ22⁢∫01θt2⁢dx+μ22⁢∫01z2⁢(x,1,t)⁢dx.

Then, exploiting (4.2) and (4.3), our conclusion follows. ∎

Now we are going to construct a Lyapunov functional ℒ equivalent to E. For this, we define several functionals which allow us to obtain the needed estimates. We introduce the multiplier w given by the solution of the Dirichlet problem

-wx⁢x=Ψx,w⁢(0)=w⁢(1)=0,

and define the functional

I1⁢(t)=∫01(ρ2⁢Ψt⁢Ψ+ρ1⁢Φt⁢w+β⁢θx⁢Ψ)⁢dx,t≥0.

The derivative of this functional will provide us the term

-∫01Ψx2⁢dx.

Lemma 4.2.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). For any ε1>0, we have

(4.4)I1′⁢(t)≤-b⁢∫01Ψx2⁢dx+(3⁢ρ22+ρ12⁢Cp4⁢ε1)⁢∫01Ψt2⁢dx+ε1⁢∫01Φt2⁢dx+β22⁢ρ2⁢∫01θx2⁢dx,

where Cp is the Poincaré constant.

Proof.

By differentiating I1 with respect to t and using equations (2.1) and (2.2), we conclude that

I1′⁢(t)=-b⁢∫01Ψx2⁢dx+ρ2⁢∫01Ψt2⁢dx-K⁢∫01Ψ2⁢dx+K⁢∫01wx2⁢dx+ρ1⁢∫01Φt⁢wt⁢dx+β⁢∫01θx⁢Ψt⁢dx.

By exploiting the inequalities

∫01wx2⁢dx≤∫01Ψ2⁢dx≤Cp⁢∫01Ψx2⁢dx,∫01wt2⁢dx≤Cp⁢∫01wt⁢x2⁢dx≤Cp⁢∫01Ψt2⁢dx

and Young’s inequality, we find that

I1′⁢(t)≤-b⁢∫01Ψx2⁢dx+3⁢ρ22⁢∫01Ψt2⁢dx+β22⁢ρ2⁢∫01θx2⁢dx+ε1⁢∫01Φt2⁢dx+ρ124⁢ε1⁢∫01wt2⁢dx.

Thus,

I1′⁢(t)≤-b⁢∫01Ψx2⁢dx+(3⁢ρ22+ρ12⁢Cp4⁢ε1)⁢∫01Ψt2⁢dx+ε1⁢∫01Φt2⁢dx+β22⁢ρ2⁢∫01θx2⁢dx,

which is exactly (4.4). ∎

In order to obtain the negative term of ∫01θx2⁢dx, we define the functional

I2⁢(t)=∫01(ρ3⁢θt⁢θ+γ⁢Ψx⁢θ+μ12⁢θ2)⁢dx.

Lemma 4.3.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). For any ε2>0, we have

I2′⁢(t)≤-λ2⁢∫01θx2⁢dx+(ρ3+γ24⁢ε2)⁢∫01θt2⁢dx+ε2⁢∫01Ψx2⁢dx

(4.5)+1λ⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx+μ22⁢Cpλ⁢∫01z2⁢(x,1,t)⁢dx.

Proof.

A simple differentiation leads to

I2′⁢(t)=ρ3⁢∫01θt2⁢dx+ρ3⁢∫01θt⁢t⁢θ⁢dx+γ⁢∫01Ψt⁢x⁢θ⁢dx+γ⁢∫01Ψx⁢θt⁢dx+μ1⁢∫01θt⁢θ⁢dx.

By using the third equation in (2.1) and (2.2), we arrive at

I2′⁢(t)=-(δ-∫0tg⁢(s)⁢ds)⁢∫01θx2⁢dx+ρ3⁢∫01θt2⁢dx+γ⁢∫01Ψx⁢θt⁢dx-∫01(g⋄θx)⁢θx⁢dx-μ2⁢∫01z⁢(x,1,t)⁢θ⁢dx.

The last three terms can be estimated, using Young’s inequality and Lemma 2.2, as follows:

γ⁢∫01Ψx⁢θt⁢dx≤ε2⁢∫01Ψx2⁢dx+γ24⁢ε2⁢∫01θt2⁢dx,

-μ2⁢∫01z⁢(x,1,t)⁢θ⁢dx≤α⁢∫01θx2⁢dx+μ22⁢Cp4⁢α⁢∫01z2⁢(x,1,t)⁢dx,

-∫01(g⋄θx)⁢θx⁢dx≤α⁢∫01θx2⁢dx+14⁢α⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx

for α>0. We deduce that

I2′⁢(t)≤-(δ-∫0tg⁢(s)⁢ds-2⁢α)⁢∫01θx2⁢dx+(ρ3+γ24⁢ε2)⁢∫01θt2⁢dx

+ε2⁢∫01Ψx2⁢dx+14⁢α⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx+μ22⁢Cp4⁢α⁢∫01z2⁢(x,1,t)⁢dx.

The choice of α=λ4 gives the result. ∎

In order to get the negative terms of ∫01Φt2⁢dx and ∫01Ψt2⁢dx, we define the functional

I3⁢(t)=-ρ1⁢∫01Φt⁢Φ⁢dx-ρ2⁢∫01Ψt⁢Ψ⁢dx.

Lemma 4.4.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). Then we have

(4.6)I3′⁢(t)≤-ρ1⁢∫01Φt2⁢dx-ρ2⁢∫01Ψt2⁢dx+3⁢b2⁢∫01Ψx2⁢dx+K⁢∫01(Φx+Ψ)2⁢dx+β22⁢b⁢∫01θt2⁢dx.

Proof.

A differentiation of I3, taking into account (2.1)–(2.2), gives

I3′⁢(t)=-ρ1⁢∫01Φt2⁢dx-ρ2⁢∫01Ψt2⁢dx+b⁢∫01Ψx2⁢dx+K⁢∫01(Φx+Ψ)2⁢dx-β⁢∫01θt⁢Ψx⁢dx.

Using Young’s and Poincaré’s inequalities for the last term, we obtain (4.6). ∎

In order to get the negative term of ∫01(Φx+Ψ)2⁢dx, we define the functional

I4⁢(t)=ρ2⁢∫01Ψt⁢(Φx+Ψ)⁢dx+(ρ2+γ)⁢∫01Ψx⁢Φt⁢dx+ρ3⁢∫01θt⁢Φt⁢dx

+(K⁢ρ3ρ1+β)⁢∫01θx⁢Φx⁢dx-∫01(g*θx)⁢Φx⁢dx.

Lemma 4.5.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2) and assume that (2.6) holds. Then, for any ε4>0, we get

I4′⁢(t)≤ε4⁢(1+ρ3⁢K2ρ12⁢b)⁢[Φx2⁢(1)+Φx2⁢(0)]+b24⁢ε4⁢[Ψx2⁢(1)+Ψx2⁢(0)]+b⁢ρ34⁢ε4⁢[θt2⁢(1)+θt2⁢(0)]

-K⁢∫01(Φx+Ψ)2⁢dx+ε4⁢∫01Ψx2⁢dx+δ2+μ124⁢ε4⁢∫01θt2⁢dx+2⁢ε4⁢∫01Φt2⁢dx

(4.7)+ρ2⁢∫01Ψt2⁢dx+ε4⁢∫01Φx2⁢dx+g2⁢(0)2⁢ε4⁢∫01θx2⁢dx-g⁢(0)2⁢ε4⁢∫01(□⁡g′θx)⁢dx+μ224⁢ε4⁢∫01z2⁢(x,1,t)⁢dx.

Proof.

By differentiating the functional I4, using the first three equations in (2.1), we obtain

I4′⁢(t)=∫01[b⁢Ψx⁢x-K⁢(Φx+Ψ)-β⁢θt⁢x]⁢(Φx+Ψ)⁢dx+ρ2⁢∫01Ψt⁢(Φx+Ψ)t⁢dx

+(ρ2+γ)⁢∫01Ψt⁢x⁢Φt⁢dx+(ρ2+γ)⁢∫01Ψx⁢Φt⁢t⁢dx+K⁢ρ3ρ1⁢∫01θt⁢(Φx+Ψ)x⁢dx

+∫01(δ⁢θx⁢x-γ⁢Ψt⁢x-μ1⁢θt-μ2⁢z⁢(x,1,t)-g*θx⁢x)⁢Φt⁢dx+(K⁢ρ3ρ1+β)⁢∫01θt⁢x⁢Φx⁢dx

+(K⁢ρ3ρ1+β)⁢∫01θx⁢Φt⁢x⁢dx-∫01(g*θx)t⁢Φx⁢dx-∫01(g*θx)⁢Φt⁢x⁢dx

=b⁢[Φx⁢Ψx]x=0x=1-b⁢ρ1K⁢∫01Ψx⁢Φt⁢t⁢dx-K⁢∫01(Φx+Ψ)2⁢dx-β⁢∫01θt⁢x⁢Φx⁢dx+β⁢∫01θt⁢Ψx⁢dx

+ρ2⁢∫01Ψt⁢Φt⁢x⁢dx+ρ2⁢∫01Ψt2⁢dx+(ρ2+γ)⁢∫01Ψt⁢x⁢Φt⁢dx+(ρ2+γ)⁢∫01Ψx⁢Φt⁢t⁢dx

+K⁢ρ3ρ1⁢∫01θt⁢Ψx⁢dx+K⁢ρ3ρ1⁢[Φx⁢θt]x=0x=1-K⁢ρ3ρ1⁢∫01θt⁢x⁢Φx⁢dx-δ⁢∫01θx⁢Φt⁢x⁢dx-γ⁢∫01Ψt⁢x⁢Φt⁢dx

-∫01(μ1⁢θt+μ2⁢z⁢(x,1,t))⁢Φt⁢dx+∫01(g*θx)⁢Φt⁢x⁢dx+(K⁢ρ3ρ1+β)⁢∫01θt⁢x⁢Φx⁢dx

+(K⁢ρ3ρ1+β)⁢∫01θx⁢Φt⁢x⁢dx-∫01(g*θx)t⁢Φx⁢dx-∫01(g*θx)⁢Φt⁢x⁢dx

=b⁢[Φx⁢Ψx]x=0x=1+K⁢ρ3ρ1⁢[Φx⁢θt]x=0x=1-K⁢∫01(Φx+Ψ)2⁢dx+(β+K⁢ρ3ρ1)⁢∫01θt⁢Ψx⁢dx+ρ2⁢∫01Ψt2⁢dx

-∫01(μ1⁢θt+μ2⁢z⁢(x,1,t))⁢Φt⁢dx-∫01(g*θx)t⁢Φx⁢dx+(ρ2+γ-b⁢ρ1K)⁢∫01Ψx⁢Φt⁢t⁢dx

+(K⁢ρ3ρ1+β-δ)⁢∫01θx⁢Φt⁢x⁢dx.

By using (2.6), we have

I4′⁢(t)=b⁢[Φx⁢Ψx]x=0x=1+K⁢ρ3ρ1⁢[Φx⁢θt]x=0x=1-K⁢∫01(Φx+Ψ)2⁢dx+δ⁢∫01θt⁢Ψx⁢dx

(4.8)+ρ2⁢∫01Ψt2⁢dx-∫01(μ1⁢θt+μ2⁢z⁢(x,1,t))⁢Φt⁢dx-∫01(g*θx)t⁢Φx⁢dx.

Now we estimate the terms in the right-hand side of (4.8). Applying Young’s and Poincaré’s inequalities and Lemma 2.2, for any ε4>0, we obtain

δ⁢∫01θt⁢Ψx⁢dx≤ε4⁢∫01Ψx2⁢dx+δ24⁢ε4⁢∫01θt2⁢dx,

-μ1⁢∫01θt⁢Φt⁢dx≤ε4⁢∫01Φt2⁢dx+μ124⁢ε4⁢∫01θt2⁢dx,

-μ2⁢∫01z⁢(x,1,t)⁢Φt⁢dx≤ε4⁢∫01Φt2⁢dx+μ224⁢ε4⁢∫01z2⁢(x,1,t)⁢dx,

b⁢[Φx⁢Ψx]x=0x=1≤ε4⁢[Φx2⁢(1)+Φx2⁢(0)]+b24⁢ε4⁢[Ψx2⁢(1)+Ψx2⁢(0)],

K⁢ρ3ρ1⁢[Φx⁢θt]x=0x=1≤ε4⁢ρ3⁢K2ρ12⁢b⁢[Φx2⁢(1)+Φx2⁢(0)]+b⁢ρ34⁢ε4⁢[θt2⁢(1)+θt2⁢(0)]

and

-∫01(g*θx)t⁢Φx⁢dx=-∫01(g⁢(0)⁢θx+g′*θx)⁢Φx⁢dx

=-∫01(g⁢(t)⁢θx-g′⋄θx)⁢Φx⁢dx

≤-g⁢(0)2⁢ε4⁢∫01(□⁡g′θx)⁢dx+ε42⁢∫01Φx2⁢dx+ε42⁢∫01Φx2⁢dx+g2⁢(0)2⁢ε4⁢∫01θx2⁢dx.

Combining all the above estimates, we get the desired results. ∎

In order to absorb the boundary terms appearing in (4.7), we exploit, as in [25], the function

q⁢(x)=2-4⁢x,x∈[0,1].

We will also introduce the functionals J1 and J2 defined by

J1⁢(t)=ρ1⁢∫01Φt⁢q⁢Φx⁢dx

and

J2⁢(t)=γ⁢ρ2⁢b⁢∫01Ψt⁢q⁢Ψx⁢dx+β⁢bδ⁢∫01(ρ3⁢θt+γ⁢Ψx)⁢q⁢(δ⁢θx-(g*θx))⁢dx.

Lemma 4.6.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). For any ε4>0, we have

(4.9)J1′⁢(t)≤-K⁢[Φx2⁢(1)+Φx2⁢(0)]+2⁢ρ1⁢∫01Φt2⁢dx+3⁢K⁢∫01Φx2⁢dx+K⁢∫01Ψx2⁢dx

and

J2′⁢(t)≤-b2⁢γ⁢[Ψx2⁢(1)+Ψx2⁢(0)]-β⁢b⁢ρ3⁢[θt2⁢(1)+θt2⁢(0)]+ε42⁢K2⁢∫01(Φx+Ψ)2⁢dx+2⁢γ⁢ρ2⁢b⁢∫01Ψt2⁢dx

+(2⁢b2⁢γ+γ2⁢b24⁢ε42+ε4)⁢∫01Ψx2⁢dx+(2⁢β⁢b⁢ρ3+5⁢ε42)⁢∫01θt2⁢dx+C1⁢(ε4)⁢∫01θx2⁢dx

+3⁢ε42⁢∫01z2⁢(x,1,t)⁢dx+2⁢β⁢b⁢∫0tg⁢(s)⁢dsδ2⁢ε4⁢[4⁢δ⁢ε4+β⁢b⁢μ12+β⁢b⁢μ22]⁢∫01(□⁡gθx)⁢dx

(4.10)-2⁢g⁢(0)⁢β2⁢b2⁢(γ2+ρ32)ε4⁢δ2⁢∫01(□⁡g′θx)⁢dx,

where

C1⁢(ε4)=4⁢β⁢bδ⁢[2⁢(∫0tg⁢(s)⁢ds)2+δ2]+2⁢β2⁢b2ε4⁢[g2⁢(0)⁢(γ2+ρ32)δ2+(μ12+μ22)⁢(1+(∫0tg⁢(s)⁢ds)2δ2)].

Proof.

A direct differentiation of J1 yields

J1′⁢(t)=K⁢∫01(Φx+Ψ)x⁢q⁢Φx⁢dx+ρ1⁢∫01Φt⁢q⁢Φt⁢x⁢dx

=K⁢∫01Φx⁢x⁢q⁢Φx⁢dx+K⁢∫01Ψx⁢q⁢Φx⁢dx-ρ12⁢∫01qx⁢Φt2⁢dx

=K2⁢[q⁢Φx2]x=0x=1-K2⁢∫01qx⁢Φx2⁢dx+K⁢∫01Ψx⁢q⁢Φx⁢dx+2⁢ρ1⁢∫01Φt2⁢dx

=-K⁢[Φx2⁢(1)+Φx2⁢(0)]+2⁢K⁢∫01Φx2⁢dx+2⁢ρ1⁢∫01Φt2⁢dx+K⁢∫01Ψx⁢q⁢Φx⁢dx.

The Young’s inequality applied to the last term gives the result.

Differentiating J2⁢(t) along solutions of (2.1), we find

J2′⁢(t)=γ⁢b⁢∫01[b⁢Ψx⁢x-K⁢(Φx+Ψ)-β⁢θt⁢x]⁢q⁢Ψx⁢dx+γ⁢ρ2⁢b⁢∫01Ψt⁢q⁢Ψt⁢x⁢dx

+β⁢bδ⁢∫01[(δ⁢θx⁢x-γ⁢Ψt⁢x-μ1⁢θt-μ2⁢z⁢(x,1,t)-g*θx⁢x)+γ⁢Ψt⁢x]⁢q⁢(δ⁢θx-g*θx)⁢dx

+β⁢bδ⁢∫01(ρ3⁢θt+γ⁢Ψx)⁢q⁢(δ⁢θt⁢x-(g*θx)t)⁢dx.

By integration by parts, we obtain

J2′⁢(t)=b2⁢γ2⁢[q⁢Ψx2]x=0x=1-b2⁢γ2⁢∫01qx⁢Ψx2⁢dx-K⁢γ⁢b⁢∫01(Φx+Ψ)⁢q⁢Ψx⁢dx

-β⁢γ⁢b⁢∫01θt⁢x⁢q⁢Ψx⁢dx-γ⁢ρ2⁢b2⁢∫01qx⁢Ψt2⁢dx-β⁢b2⁢δ⁢∫01qx⁢(δ⁢θx-(g*θx))2⁢dx

-β⁢b⁢μ1δ⁢∫01θt⁢q⁢(δ⁢θx-(g*θx))⁢dx-β⁢b⁢μ2δ⁢∫01z⁢(x,1,t)⁢q⁢(δ⁢θx-(g*θx))⁢dx+β⁢b⁢ρ32⁢[q⁢θt2]x=0x=1

-β⁢b⁢ρ32⁢∫01qx⁢θt2⁢dx-β⁢b⁢ρ3δ⁢∫01θt⁢q⁢(g*θx)t⁢dx+γ⁢β⁢b⁢∫01Ψx⁢q⁢θt⁢x⁢dx-b⁢β⁢γδ⁢∫01Ψx⁢q⁢(g*θx)t⁢dx

=-b2⁢γ⁢[Ψx2⁢(1)+Ψx2⁢(0)]+2⁢b2⁢γ⁢∫01Ψx2⁢dx-K⁢γ⁢b⁢∫01(Φx+Ψ)⁢q⁢Ψx⁢dx+2⁢γ⁢ρ2⁢b⁢∫01Ψt2⁢dx

-β⁢b⁢γδ⁢∫01Ψx⁢q⁢(g*θx)t⁢dx-β⁢b⁢μ1⁢∫01θt⁢q⁢θx⁢dx+β⁢b⁢μ1δ⁢∫01θt⁢q⁢(g*θx)⁢dx+2⁢β⁢b⁢ρ3⁢∫01θt2⁢dx

-β⁢b⁢μ2⁢∫01z⁢(x,1,t)⁢q⁢θx⁢dx+β⁢b⁢μ2δ⁢∫01z⁢(x,1,t)⁢q⁢(g*θx)⁢dx-β⁢b⁢ρ3⁢[θt2⁢(1)+θt2⁢(0)]

-β⁢b⁢ρ3δ⁢∫01θt⁢q⁢(g*θx)t⁢dx+2⁢β⁢bδ⁢∫01(δ⁢θx-(g*θx))2⁢dx.

Note that

-β⁢b⁢γδ⁢∫01Ψx⁢q⁢(g*θx)t⁢dx=-β⁢b⁢γδ⁢∫01Ψx⁢q⁢(g⁢(t)⁢θx-g′⋄θx)⁢dx

≤2ε4⁢(b⁢β⁢γδ)2⁢g2⁢(0)⁢∫01θx2⁢dx+ε4⁢∫01Ψx2⁢dx-2ε4⁢(b⁢β⁢γδ)2⁢g⁢(0)⁢∫01(□⁡g′θx)⁢dx

and, similarly,

-β⁢b⁢ρ3δ⁢∫01θt⁢q⁢(g*θx)t⁢dx≤2ε4⁢(b⁢β⁢ρ3δ)2⁢g2⁢(0)⁢∫01θx2⁢dx+ε4⁢∫01θt2⁢dx-2ε4⁢(b⁢β⁢ρ3δ)2⁢g⁢(0)⁢∫01(□⁡g′θx)⁢dx.

Moreover,

-β⁢b⁢μ1⁢∫01θt⁢q⁢θx⁢dx≤ε42⁢∫01θt2⁢dx+2⁢β2⁢b2⁢μ12ε4⁢∫01θx2⁢dx,

-β⁢b⁢μ2⁢∫01z⁢(x,1,t)⁢q⁢θx⁢dx≤ε42⁢∫01z2⁢(x,1,t)⁢dx+2⁢β2⁢b2⁢μ22ε4⁢∫01θx2⁢dx,

-K⁢γ⁢b⁢∫01(Φx+Ψ)⁢q⁢Ψx⁢dx≤ε42⁢K2⁢∫01(Φx+Ψ)2⁢dx+γ2⁢b24⁢ε42⁢∫01Ψx2⁢dx,

2⁢β⁢bδ⁢∫01(δ⁢θx-(g*θx))2⁢dx≤8⁢β⁢bδ⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx+4⁢β⁢b⁢δ⁢∫01θx2⁢dx

+8⁢β⁢bδ⁢(∫0tg⁢(s)⁢ds)⁢∫01(□⁡gθx)⁢dx,

β⁢b⁢μ1δ⁢∫01θt⁢q⁢(g*θx)⁢dx≤ε42⁢∫01θt2⁢dx+2ε4⁢(β⁢b⁢μ1δ)2⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx

+ε42⁢∫01θt2⁢dx+2ε4⁢(β⁢b⁢μ1δ)2⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx

and

β⁢b⁢μ2δ⁢∫01z⁢(x,1,t)⁢q⁢(g*θx)⁢dx≤ε4⁢∫01z2⁢(x,1,t)⁢dx+2ε4⁢(β⁢b⁢μ2δ)2⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx

+2ε4⁢(β⁢b⁢μ2δ)2⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx.

This completes the proof of the lemma. ∎

Now, let us define the functional

I5⁢(t)=ρ2⁢ρ3⁢∫01∫0xθt⁢(t,y)⁢dy⁢Ψt⁢dx.

Lemma 4.7.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). For any ε2>0 and η1>0, we have

I5′⁢(t)≤-ρ2⁢γ4⁢∫01Ψt2⁢dx+ρ2γ⁢[δ2+2⁢(∫0tg⁢(s)⁢ds)2]⁢∫01θx2⁢dx+2⁢ρ2⁢μ22γ⁢∫01z2⁢(x,1,t)⁢dx

+ε2⁢(Cp+1)⁢∫01Ψx2⁢dx+2⁢ρ2γ⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx+b24⁢η1⁢Ψx2⁢(1)+ρ3⁢b4⁢η1⁢θt2⁢(1)

(4.11)+[β⁢ρ3+ρ2⁢μ12γ+ρ324⁢ε2⁢(2⁢K2+b2)+η1⁢(ρ32+ρ3⁢β2b)]⁢∫01θt2⁢dx+ε2⁢Cp⁢∫01Φx2⁢dx.

Proof.

Differentiating the functional I5 and using equations (2.1), we obtain

I5′⁢(t)=ρ2⁢∫01∫0x[δ⁢θy⁢y-γ⁢Ψt⁢y-μ1⁢θt-μ2⁢z⁢(y,1,t)-g*θy⁢y]⁢dy⁢Ψt⁢dx

+ρ3⁢∫01∫0xθt⁢dy⁢[b⁢Ψx⁢x-K⁢(Φx+Ψ)-β⁢θt⁢x]⁢dx

=-γ⁢ρ2⁢∫01Ψt2⁢dx+ρ2⁢δ⁢∫01θx⁢Ψt⁢dx-ρ2⁢∫01(g*θx)⁢Ψt⁢dx

-μ1⁢ρ2⁢∫01∫0xθt⁢dy⁢Ψt⁢dx-μ2⁢ρ2⁢∫01∫0xz⁢(y,1,t)⁢dy⁢Ψt⁢dx

-K⁢ρ3⁢∫01∫0xθt⁢dy⁢Ψ⁢dx-ρ3⁢b⁢∫01θt⁢Ψx⁢dx+K⁢ρ3⁢∫01θt⁢Φ⁢dx+β⁢ρ3⁢∫01θt2⁢dx

+ρ3⁢[∫0xθt⁢dy⁢(b⁢Ψx-β⁢θt)]x=0x=1.

Using Young’s and Poincaré’s inequalities, we find

I5′⁢(t)≤-γ⁢ρ22⁢∫01Ψt2⁢dx+ρ2⁢δ2γ⁢∫01θx2⁢dx+ε2⁢(1+Cp)⁢∫01Ψx2⁢dx+ε2⁢Cp⁢∫01Φx2⁢dx

+[β⁢ρ3+2⁢ρ2⁢μ12γ+ρ324⁢ε2⁢(2⁢K2+b2)]⁢∫01θt2⁢dx+2⁢ρ2⁢μ22γ⁢∫01z2⁢(x,1,t)⁢dx

-ρ2⁢∫01(g*θx)⁢Ψt⁢dx+ρ3⁢[∫0xθt⁢dy⁢(b⁢Ψx-β⁢θt)]x=0x=1.

We can estimate the terms in the right-hand side as follows:

ρ2⁢∫01(g*θx)⁢Ψt⁢dx=-ρ2⁢∫01(g⋄θx)⁢Ψt⁢dx+ρ2⁢∫0tg⁢(s)⁢ds⁢∫01θx⁢Ψt⁢dx

≤ρ2⁢γ4⁢∫01Ψt2⁢dx+2⁢ρ2γ⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx+2⁢ρ2γ⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx

and

ρ3⁢[∫0xθt2⁢dy⁢(b⁢Ψx-β⁢θt)]x=0x=1≤b24⁢η1⁢Ψx2⁢(1)+ρ3⁢b4⁢η1⁢θt2⁢(1)+η1⁢(ρ32+ρ3⁢β2b)⁢∫01θt2⁢dx.

The proof is completed. ∎

Finally, as in [13], we introduce the functional

I6⁢(t)=∫01∫01e-2⁢τ⁢ρ⁢z2⁢(x,ρ,t)⁢dρ⁢dx.

Lemma 4.8.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). Then we have

(4.12)I6′⁢(t)≤-2⁢I6⁢(t)-cτ⁢∫01z2⁢(x,1,t)⁢dx+1τ⁢∫01θt2⁢dx,

where c is a positive constant.

Proof.

Differentiating the functional I6, we have

I6′⁢(t)=-2τ⁢∫01∫01e-2⁢τ⁢ρ⁢z⁢zρ⁢(x,ρ,t)⁢dρ⁢dx

=-2⁢∫01∫01e-2⁢τ⁢ρ⁢z2⁢(x,ρ,t)⁢dρ⁢dx-1τ⁢∫01∫01∂∂⁡ρ⁢(e-2⁢τ⁢ρ⁢z2⁢(x,ρ,t))⁢dρ⁢dx

=-2⁢I6⁢(t)+1τ⁢∫01θt2⁢dx-1τ⁢∫01e-2⁢τ⁢z2⁢(x,1,t)⁢dx.

The above equality implies that there exists a positive constant c such that (4.12) holds. ∎

Now, we define the Lyapunov functional ℒ as follows:

ℒ⁢(t)=N⁢E⁢(t)+N1⁢I1⁢(t)+N2⁢I2⁢(t)+υ4⁢I3⁢(t)+υ⁢I4⁢(t)+N5⁢I5⁢(t)+I6⁢(t)

+υ⁢ε4⁢(1K+ρ3⁢Kρ12⁢b)⁢J1⁢(t)+12⁢ε4⁢J2⁢(t),t≥0,

where N,N1,N2,N5 are positive constants to be chosen properly later and υ=min⁡{γ,β}. For large N and for some m,M>0, we can verify that

(4.13)m⁢E⁢(t)≤ℒ⁢(t)≤M⁢E⁢(t),t≥0.

Taking into account (4.1), (4.4), (4.5), (4.6), (4.7), (4.9), (4.10), (4.11), (4.12) and the relation

(4.14)∫01Φx2⁢dx≤2⁢∫01(Φx+Ψ)2⁢dx+2⁢Cp⁢∫01Ψx2⁢dx,

we arrive at

ℒ′⁢(t)≤-{ρ1⁢υ4-N1⁢ε1-2⁢ε4⁢υ⁢[1+ρ1⁢(1K+ρ3⁢Kρ12⁢b)]}⁢∫01Φt2⁢dx

-{ρ2⁢γ⁢N54-N1⁢(3⁢ρ22+ρ12⁢Cp4⁢ε1)-3⁢ρ2⁢υ4-γ⁢ρ2⁢bε4}⁢∫01Ψt2⁢dx

-{3⁢υ⁢K4-2⁢ε2⁢Cp⁢N5-ε4⁢[6⁢υ⁢K⁢(1K+ρ3⁢Kρ12⁢b)+2⁢υ+K22]}⁢∫01(Φx+Ψ)2⁢dx

-{N1b-ε2(N2+N5(1+Cp+2Cp2))-12⁢ε4[2b2γ+γ2⁢b24⁢ε42+ε4]

-ε4(Kυ+6CpKυ)(1K+ρ3⁢Kρ12⁢b)-3⁢b⁢υ8-ε4(υ+2υCp)}∫01Ψx2dx

-{Nm0-N2(ρ3+γ24⁢ε2)-[υ4⁢ε4(δ2+μ12)+β⁢b⁢ρ3ε4+54]-β2⁢υ8⁢b

-N5[βρ3+ρ2⁢μ12γ+ρ324⁢ε2(2K2+b2)+η1(ρ32+ρ3⁢β2b)]-1τ}∫01θt2dx

-{λ⁢N22-N1⁢β22⁢ρ2-12⁢ε4⁢(υ⁢g2⁢(0)+C1⁢(ε4))-N5⁢ρ2γ⁢(δ2+2⁢g¯2)}⁢∫01θx2⁢dx

+{N2⁢g¯λ+β⁢b⁢g¯ε42⁢δ2⁢(4⁢δ⁢ε4+β⁢b⁢μ12+β⁢b⁢μ22)+2⁢ρ2⁢g¯⁢N5γ}⁢∫01(□⁡gθx)⁢dx

+{β⁢N2-υ⁢g⁢(0)2⁢ε4-g⁢(0)⁢b2⁢β2⁢(γ2+ρ32)ε42⁢δ2}⁢∫01(□⁡g′θx)⁢dx-2⁢I6⁢(t)

(4.15)-{N⁢m0+cτ-μ22⁢N2⁢Cpλ-μ22⁢υ4⁢ε4-34-2⁢ρ2⁢μ22⁢N5γ}⁢∫01z2⁢(x,1,t)⁢dx,

where m0=min⁡{β⁢(μ1-ξ2⁢τ-μ22),β⁢(ξ2⁢τ-μ22)}.

At this point, we need to choose our constants very carefully. First, let us pick η1=N5⁢ε4υ and choose

ε4≤min⁡{ρ116⁢[1+ρ1⁢(1K+ρ3⁢Kρ12⁢b)]-1,3⁢υ⁢K8⁢[6⁢υ⁢K⁢(1K+ρ3⁢Kρ12⁢b)+2⁢υ+K22]-1}.

Second, we select N1 sufficiently large such that

N1⁢b2≥12⁢ε4⁢[2⁢b2⁢γ+γ2⁢b24⁢ε42+ε4]+ε4⁢K⁢υ⁢(1+6⁢Cp)⁢(1K+ρ3⁢Kρ12⁢b)+3⁢b⁢υ8+ε4⁢υ⁢(1+2⁢Cp),

and then we choose ε1 small enough so that

ε1≤ρ1⁢υ16⁢N1.

Next, we choose N5 sufficiently large so that

N5≥8ρ2⁢γ⁢[N1⁢(3⁢ρ22+ρ12⁢Cp4⁢ε1)+γ⁢ρ2⁢bε4+3⁢ρ2⁢υ4],

and we select also N2 sufficiently large so that

λ⁢N24>N1⁢β22⁢ρ2+12⁢ε4⁢(υ⁢g2⁢(0)+C1⁢(ε4))+N5⁢ρ2γ⁢(δ2+2⁢g¯2).

Furthermore, we select ε2 so that

ε2<min⁡{N1⁢b2⁢(N2+N5⁢(1+Cp+2⁢Cp2)),3⁢K⁢υ16⁢N5⁢Cp}.

Finally, we choose N large enough so that (4.13) remains valid and (4.15) takes the form

ℒ′⁢(t)≤-C1⁢∫01(Φt2+Ψt2+(Φx+Ψ)2+Ψx2+θt2+θx2+∫01z2⁢(x,ρ,t)⁢dρ)⁢dx+C2⁢∫01(□⁡gθx)⁢dx

(4.16)≤-C⁢E⁢(t)+C3⁢∫01(□⁡gθx)⁢dx,

where C1,C2,C3 and C are positive constants.

4.2 The case μ2=μ1

If μ1=μ2=μ, then we can choose ξ=τ⁢μ in (2.5) and Lemma 4.1 takes the following form.

Lemma 4.9.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). Assume that μ1=μ2=μ, g satisfies Hypotheses 1 and 2, and ξ=τ⁢μ. Then the energy functional defined by (2.3) is a non-increasing function and satisfies

(4.17)E′⁢(t)≤-β2⁢g⁢(t)⁢∫01θx2⁢dx+β2⁢∫01(□⁡g′θx)⁢dx≤0,t≥0.

The proof of Lemma 4.9 is an immediate consequence of Lemma 4.1, by choosing ξ=τ⁢μ.

If μ1=μ2=μ, we need some additional negative term of ∫01θt2⁢dx. For this purpose, let us introduce the functional

I7⁢(t)=-ρ3⁢∫01θt⁢(g⋄θ)⁢dx.

Then, we have the following estimate.

Lemma 4.10.

Let (Φ,Ψ,θ,z) be a solution of (2.1)–(2.2). Then for any ε7>0 and η2>0, we have

I7′⁢(t)≤-(ρ3⁢∫0tg⁢(s)⁢ds-η2)⁢∫01θt2⁢dx+ε7⁢∫01Ψt2⁢dx+ε7⁢[1+(∫0tg⁢(s)⁢ds)2]⁢∫01θx2⁢dx

(4.18)+ε7⁢∫01z2⁢(x,1,t)⁢dx+C2⁢(ε7)⁢∫01(□⁡gθx)⁢dx-ρ322⁢η2⁢g⁢(0)⁢Cp⁢∫01(□⁡g′θx)⁢dx,

where

C2⁢(ε7)=∫0tg⁢(s)⁢ds4⁢ε7⁢(δ2+γ2+4⁢ε72+2+μ22⁢Cp)+μ12⁢Cp2⁢η2⁢∫0tg⁢(s)⁢ds.

Proof.

A simple differentiation leads to

I7′⁢(t)=-ρ3⁢∫01θt⁢(g⋄θ)t⁢dx-ρ3⁢∫01θt⁢t⁢(g⋄θ)⁢dx

=-(ρ3∫0tg(s)ds)∫01θt2dx-ρ3∫01θt(g′⋄θ)dx-∫01(∫0tg(t-s)θx(s)ds)(g⋄θx)dx

(4.19)+δ⁢∫01θx⁢(g⋄θx)⁢dx-γ⁢∫01Ψt⁢(g⋄θx)⁢dx+μ1⁢∫01θt⁢(g⋄θ)⁢dx+μ2⁢∫01z⁢(x,1,t)⁢(g⋄θ)⁢dx.

The terms in the right-hand side of (4.19) are estimated as follows. Using Young’s inequality and Lemma 2.2, for all η2>0, we obtain

-ρ3⁢∫01θt⁢(g′⋄θ)⁢dx≤η22⁢∫01θt2⁢dx+ρ322⁢η2⁢∫0t(-g′⁢(s)⁢d⁢s)⁢∫01(-□⁡g′θ)⁢dx

≤η22⁢∫01θt2⁢dx-ρ322⁢η2⁢g⁢(0)⁢Cp⁢∫01(□⁡g′θx)⁢dx.

Similarly, for any ε7>0, we have

δ⁢∫01θx⁢(g⋄θx)⁢dx≤ε7⁢∫01θx2⁢dx+δ24⁢ε7⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx,

-γ⁢∫01Ψt⁢(g⋄θx)⁢dx≤ε7⁢∫01Ψt2⁢dx+γ24⁢ε7⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx,

μ1⁢∫01θt⁢(g⋄θ)⁢dx≤η22⁢∫01θt2⁢dx+μ12⁢Cp2⁢η2⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx,

μ2⁢∫01z⁢(x,1,t)⁢(g⋄θ)⁢dx≤ε7⁢∫01z2⁢(x,1,t)⁢dx+μ22⁢Cp4⁢ε7⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx.

Finally,

-∫01(∫0tg⁢(t-s)⁢θx⁢(s)⁢ds)⁢(g⋄θx)⁢dx≤ε72⁢∫01(∫0tg⁢(t-s)⁢(θx⁢(t)-θx⁢(x)-θx⁢(t))⁢ds)2⁢dx+12⁢ε7⁢∫01(g⋄θx)2⁢dx

≤ε7⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx+(ε7+12⁢ε7)⁢∫01(g⋄θx)2⁢dx

≤ε7⁢(∫0tg⁢(s)⁢ds)2⁢∫01θx2⁢dx+(ε7+12⁢ε7)⁢∫0tg⁢(s)⁢ds⁢∫01(□⁡gθx)⁢dx.

Therefore, the assertion of the lemma follows by combining all the above estimates. ∎

Now, we define the Lyapunov functional ℒ as

ℒ⁢(t)=N⁢E⁢(t)+N1⁢I1⁢(t)+N2⁢I2⁢(t)+υ4⁢I3⁢(t)+υ⁢I4⁢(t)+N5⁢I5⁢(t)+N6⁢I6⁢(t)+N7⁢I7⁢(t)

+υ⁢ε4⁢(1K+ρ3⁢Kρ12⁢b)⁢J1⁢(t)+12⁢ε4⁢J2⁢(t),t≥0,

where N,N1,N2,N5,N6,N7 are positive real numbers which will be chosen later.

Since g is continuous and g⁢(0)>0, for any t≥t0>0, we have

∫0tg⁢(s)⁢ds≥∫0t0g⁢(s)⁢ds=g0.

Then, using estimates (4.4), (4.5), (4.6), (4.7), (4.9), (4.10), (4.11), (4.12), (4.17), (4.18) and the algebraic inequality (4.14), we get

ℒ′⁢(t)≤-{ρ1⁢υ4-N1⁢ε1-2⁢ε4⁢υ⁢[1+ρ1⁢(1K+ρ3⁢Kρ12⁢b)]}⁢∫01Φt2⁢dx-2⁢N6⁢I6⁢(t)

-{λ⁢N22-N1⁢β22⁢ρ2-N7⁢ε7⁢(1+g¯2)-12⁢ε4⁢(υ⁢g2⁢(0)+C1⁢(ε4))-N5⁢ρ2γ⁢(δ2+2⁢g¯2)}⁢∫01θx2⁢dx

-{N7(ρ3g0-η2)-N2(ρ3+γ24⁢ε2)-[υ4⁢ε4(δ2+μ2)+β⁢b⁢ρ3ε4+54]-β2⁢υ8⁢b

-N5[βρ3+2⁢ρ2⁢μ2γ+ρ324⁢ε2(2K2+b2)+η1(ρ32+ρ3⁢β2b)]-N6τ}∫01θt2dx

-{3⁢υ⁢K4-2⁢ε2⁢Cp⁢N5-ε4⁢[6⁢υ⁢K⁢(1K+ρ3⁢Kρ12⁢b)+2⁢υ+K22]}⁢∫01(Φx+Ψ)2⁢dx

-{ρ2⁢γ⁢N54-N1⁢(3⁢ρ22+ρ12⁢Cp4⁢ε1)-N7⁢ε7-3⁢ρ2⁢υ4-γ⁢ρ2⁢bε4}⁢∫01Ψt2⁢dx

-{N1b-ε2(N2+N5(1+Cp+2Cp2))-12⁢ε4[2b2γ+γ2⁢b24⁢ε42+ε4]

-ε4(Kυ+6CpKυ)(1K+ρ3⁢Kρ12⁢b)-3⁢b⁢υ8-ε4(υ+2υCp)}∫01Ψx2dx

+{N2⁢g¯2⁢λ+C2⁢(ε7)+β⁢b⁢g¯ε42⁢δ2⁢(4⁢δ⁢ε4+2⁢β⁢b⁢μ2)+2⁢ρ2⁢g¯⁢N5γ}⁢∫01(□⁡gθx)⁢dx

+{β⁢N2-ρ32⁢g⁢(0)⁢Cp2⁢η2⁢N7-υ⁢g⁢(0)2⁢ε4-g⁢(0)⁢b2⁢β2⁢(γ2+ρ32)ε42⁢δ2}⁢∫01(□⁡g′θx)⁢dx

(4.20)-{N6⁢cτ-μ2⁢N2⁢Cpλ-N7⁢ε7-μ2⁢υ4⁢ε4-34-2⁢ρ2⁢μ2⁢N5γ}⁢∫01z2⁢(x,1,t)⁢dx.

Now, our goal is to choose our constants in (4.20) in order to get the negative coefficients on the right-hand side of (4.20). To this end, let us pick η1=N5⁢ε4υ and η2=14⁢N7, and we pick ε4,N1,ε1,N5,N2,ε2 in the same order with the same values as the case μ2<μ1, respectively. Then we pick N6 large enough such that

N6⁢c2⁢τ>N2⁢μ2⁢Cpλ+μ2⁢υ4⁢ε4+34+2⁢ρ2⁢μ2⁢N5γ.

After that, we choose N7 sufficiently large so that

N7⁢ρ3⁢g02-18>N2⁢(ρ3+γ24⁢ε2)+υ⁢(δ2+μ2)4⁢ε4+β⁢b⁢ρ3ε4+54+N6τ+β2⁢υ8⁢b

+N5⁢[β⁢ρ3+2⁢ρ2⁢μ2γ+ρ324⁢ε2⁢(2⁢K2+b2)+η1⁢(ρ32+ρ3⁢β2b)].

Furthermore, choosing ε7 sufficiently small so that

ε7<min⁡{N6⁢c2⁢τ⁢N7,ρ2⁢γ⁢N58⁢N7,λ⁢N24⁢N7⁢(1+g¯2)}.

Once all the above constants are fixed, we pick N large enough so that there exists two positive constants C^ and C3^ such that

(4.21)ℒ′⁢(t)≤-C^⁢E⁢(t)+C3^⁢∫01(□⁡gθx)⁢dx,t≥t0.

From (4.16) and (4.21), we know that the Lyapunov functionals ℒ are of the same form for both cases μ2<μ1 and μ2=μ1, that is,

(4.22)ℒ′⁢(t)≤-C⁢E⁢(t)+C3⁢∫01(□⁡gθx)⁢dx,t≥t0.

Continuity of the proof of Theorem 2.4.

Multiplying (4.22) by ζ⁢(t) gives

(4.23)ζ⁢(t)⁢ℒ′⁢(t)≤-C⁢ζ⁢(t)⁢E⁢(t)+C3⁢ζ⁢(t)⁢∫01(□⁡gθx)⁢dx.

The last term can be estimated, using Hypothesis 2, to obtain

ζ⁢(t)⁢∫01(□⁡gθx)⁢dx≤-∫01(□⁡g′θx)⁢dx≤-2β⁢E′⁢(t).

Thus, for some positive constant C4, (4.23) becomes

(4.24)ζ⁢(t)⁢ℒ′⁢(t)≤-C⁢ζ⁢(t)⁢E⁢(t)-C4⁢E′⁢(t).

It is clear that

L⁢(t)=ζ⁢(t)⁢ℒ⁢(t)+C4⁢E⁢(t)∼E⁢(t).

Therefore, using (4.24) and the fact that ζ′⁢(t)≤0, we arrive at

(4.25)L′⁢(t)=ζ′⁢(t)⁢ℒ⁢(t)+ζ⁢(t)⁢ℒ′⁢(t)+C4⁢E′⁢(t)≤-C⁢ζ⁢(t)⁢E⁢(t).

A simple integration of (4.25) over (t0,t) leads to

(4.26)L⁢(t)≤L⁢(t0)⁢e-C⁢∫t0tζ⁢(s)⁢ds,t≥t0.

Recalling (4.13), estimate (4.26) yields the desired result (2.7). ∎

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11301277

Award Identifier / Grant number: 41475091

Funding source: Natural Science Foundation of Jiangsu Province

Award Identifier / Grant number: BK20151523

Funding statement: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11301277, 41475091), the Natural Science Foundation of Jiangsu Province (Grant No. BK20151523), the Six Talent Peaks Project in Jiangsu Province (Grant No. 2015-XCL-020), and the Qing Lan Project of Jiangsu Province.

Acknowledgements

The authors express sincere thanks to the editors and anonymous reviewers for their constructive comments and suggestions that helped to improve this paper.

References

[1] F. Boulanouar and S. Drabla, General boundary stabilization result of memory-type thermoelasticity with second sound, Electron. J. Differential Equations 2014 (2014), Paper No. 202. Search in Google Scholar

[2] M. M. Cavalcanti, Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differential Integral Equations 14 (2001), no. 1, 85–116. 10.57262/die/1356123377Search in Google Scholar

[3] M. M. Cavalcanti, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys. 65 (2014), no. 6, 1189–1206. 10.1007/s00033-013-0380-7Search in Google Scholar

[4] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations 2002 (2002), Paper No. 44. Search in Google Scholar

[5] M. M. Cavalcanti and A. Guesmia, General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type, Differential Integral Equations 18 (2005), no. 5, 583–600. 10.57262/die/1356060186Search in Google Scholar

[6] M. M. Cavalcanti and H. P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (2003), no. 4, 1310–1324. 10.1137/S0363012902408010Search in Google Scholar

[7] M. de Lima Santos, Decay rates for solutions of a Timoshenko system with a memory condition at the boundary, Abstr. Appl. Anal. 7 (2002), no. 10, 531–546. 10.1155/S1085337502204133Search in Google Scholar

[8] A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping, J. Dyn. Control Syst. 16 (2010), no. 2, 189–210. 10.1007/s10883-010-9089-5Search in Google Scholar

[9] A. Djebabla and N. Tatar, Stabilization of the Timoshenko beam by thermal effect, Mediterr. J. Math. 7 (2010), no. 3, 373–385. 10.1007/s00009-010-0058-8Search in Google Scholar

[10] A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel, Math. Comput. Modelling 54 (2011), no. 1–2, 301–314. 10.1016/j.mcm.2011.02.013Search in Google Scholar

[11] D. Feng, D. Shi and W. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation, Sci. China Ser. A 41 (1998), no. 5, 483–490. 10.1007/BF02879936Search in Google Scholar

[12] J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim. 25 (1987), no. 6, 1417–1429. 10.1137/0325078Search in Google Scholar

[13] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys. 62 (2011), no. 6, 1065–1082. 10.1007/s00033-011-0145-0Search in Google Scholar

[14] M. Kirane, B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal. 10 (2011), no. 2, 667–686. 10.3934/cpaa.2011.10.667Search in Google Scholar

[15] G. Li, D. Wang and B. Zhu, Well-posedness and decay of solutions for a transmission problem with history and delay, Electron. J. Differential Equations 2016 (2016), Paper No. 23. Search in Google Scholar

[16] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969. Search in Google Scholar

[17] W. Liu, General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback, J. Math. Phys. 54 (2013), no. 4, Article ID 043504. 10.1063/1.4799929Search in Google Scholar

[18] W. Liu and K. Chen, Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions, Math. Nachr. 289 (2016), no. 2–3, 300–320. 10.1002/mana.201400343Search in Google Scholar

[19] W. Liu, K. Chen and J. Yu, Existence and general decay for the full von Kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, IMA J. Math. Control Inform. (2015), 10.1093/imamci/dnv056. 10.1093/imamci/dnv056Search in Google Scholar

[20] W. Liu, Y. Sun and G. Li, On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term, preprint (2013), http://arxiv.org/abs/1303.4246; to appear in Topol. Methods Nonlinear Anal. 10.12775/TMNA.2016.077Search in Google Scholar

[21] Z. Liu and C. Peng, Exponential stability of a viscoelastic Timoshenko beam, Adv. Math. Sci. Appl. 8 (1998), no. 1, 343–351. Search in Google Scholar

[22] S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arab. J. Math. (Springer) 2 (2013), no. 2, 199–207. 10.1007/s40065-012-0061-ySearch in Google Scholar

[23] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound—Global existence and exponential stability, Math. Methods Appl. Sci. 32 (2009), no. 5, 505–534. 10.1002/mma.1049Search in Google Scholar

[24] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl. 348 (2008), no. 1, 298–307. 10.1016/j.jmaa.2008.07.036Search in Google Scholar

[25] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl. 276 (2002), no. 1, 248–278. 10.1016/S0022-247X(02)00436-5Search in Google Scholar

[26] J. E. Muñoz Rivera and R. Racke, Global stability for damped Timoshenko systems, Discrete Contin. Dyn. Syst. 9 (2003), no. 6, 1625–1639. 10.3934/dcds.2003.9.1625Search in Google Scholar

[27] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim. 45 (2006), no. 5, 1561–1585. 10.1137/060648891Search in Google Scholar

[28] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations 21 (2008), no. 9–10, 935–958. 10.57262/die/1356038593Search in Google Scholar

[29] C. A. Raposo, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett. 18 (2005), no. 5, 535–541. 10.1016/j.aml.2004.03.017Search in Google Scholar

[30] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comput. 217 (2010), no. 6, 2857–2869. 10.1016/j.amc.2010.08.021Search in Google Scholar

[31] B. Said-Houari and R. Rahali, A stability result for a Timoshenko system with past history and a delay term in the internal feedback, Dynam. Systems Appl. 20 (2011), no. 2–3, 327–353. Search in Google Scholar

[32] D.-H. Shi, S. H. Hou and D.-X. Feng, Feedback stabilization of a Timoshenko beam with an end mass, Internat. J. Control 69 (1998), no. 2, 285–300. 10.1080/002071798222848Search in Google Scholar

[33] M. A. Shubov, Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model, Math. Nachr. 241 (2002), 125–162. 10.1002/1522-2616(200207)241:1<125::AID-MANA125>3.0.CO;2-3Search in Google Scholar

[34] A. Soufyane and A. Wehbe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electron. J. Differential Equations 2003 (2003), Paper No. 29. Search in Google Scholar

[35] F. Tahamtani and A. Peyravi, Asymptotic behavior and blow-up of solutions for a nonlinear viscoelastic wave equation with boundary dissipation, Taiwanese J. Math. 17 (2013), no. 6, 1921–1943. 10.11650/tjm.17.2013.3034Search in Google Scholar

[36] S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismaticbars, Philos. Mag. 41 (1921), 744–746. 10.1080/14786442108636264Search in Google Scholar

[37] D. Wang, G. Li and B. Zhu, Well-posedness and general decay of solution for a transmission problem with viscoelastic term and delay, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 1202–1215. 10.22436/jnsa.009.03.46Search in Google Scholar

[38] S.-T. Wu, Asymptotic behavior for a viscoelastic wave equation with a delay term, Taiwanese J. Math. 17 (2013), no. 3, 765–784. 10.11650/tjm.17.2013.2517Search in Google Scholar

[39] S.-T. Wu, General decay of solutions for a nonlinear system of viscoelastic wave equations with degenerate damping and source terms, J. Math. Anal. Appl. 406 (2013), no. 1, 34–48. 10.1016/j.jmaa.2013.04.029Search in Google Scholar

Received: 2016-04-09

Revised: 2016-06-27

Accepted: 2016-07-10

Published Online: 2016-10-11

Published in Print: 2018-11-01

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms

Abstract

1 Introduction

2 Preliminaries and main results

Lemma 2.1 ([5, 6]).

Lemma 2.2 ([8]).

Hypothesis 1.

Hypothesis 2.

Theorem 2.3.

Theorem 2.4.

Remark.

3 Proof of Theorem 2.3

Step 1: Faedo–Galerkin approximations.

Step 2: Energy estimates.

4 Proof of Theorem 2.4

4.1 The case μ2<μ1

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Lemma 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Lemma 4.8.

Proof.

4.2 The case μ2=μ1

Lemma 4.9.

Lemma 4.10.

Proof.

Continuity of the proof of Theorem 2.4.

Acknowledgements

References

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