# Exponential Decay to the Degenerate Nonlinear Coupled Beams System with Weak Damping

Authors: R. F. C. Lobato; D. C. Pereira; M. L. Santos
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

We consider a nonlinear degenerate coupled beams system with weak damping. We show using the Nakao method that the solution of this system decays exponentially when the time tends to infinity.


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## Body

## 1. Introduction

For the last several decades, various types of equations have been employed as some mathematical models describing physical, chemical, biological, and engineering systems. Among them, the mathematical models of vibrating, flexible structures have been considerably stimulated in recent years by an increasing number of questions of practical concern. Research on stabilization of distributed parameter systems has largely focused on the stabilization of dynamic models of individual structural members such as strings, membranes, and beams.This paper is devoted to the study of the existence, uniqueness, and uniform decay rates of the energy of solution for the nonlinear degenerate coupled beams system with weak damping given by(1.1)K1(x,t)utt+Δ2u-M(‖u‖2+‖v‖2)Δu+ut=0inΩ×(0,T),(1.2)K2(x,t)vtt+Δ2v-M(‖u‖2+‖v‖2)Δv+vt=0inΩ×(0,T),(1.3)u=v=∂u∂η=∂v∂η=0on⁡Σ,(1.4)(u(x,0),v(x,0))=(u0,v0)inΩ,(1.5)(ut(x,0),vt(x,0))=(u1(x),v1(x))inΩ,
where Ω is a bounded domain of ℝn, n≥1, with smooth boundary Γ,   T>0 is a real arbitrary number, and η is the unit normal at Σ=Γ×(0,T) direct towards the exterior of Ω×(0,T). Here Ki∈C1([0,T];H01(Ω)∩L∞(Ω)),   i=1,2 and M∈C1([0,∞[), see Section 2 for more details.Problems related to the system (1.1)–(1.5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics. For instance, when we consider only one equation without the dissipative term, that is,
(1.6)K(x,t)utt+Δ2u-M(‖u‖2)Δu=0inΩ×(0,T)
and with K(x,t)=1, it is a generalization of one-dimensional model proposed by Woinowsky-Krieger [1] as a model for the transverse deflection u(x,t) of an extensible beam of natural length whose ends are held a fixed distance apart. The nonlinear term represents the change in the tension of the beam due to its extensibility. The model has also been discussed by Eisley [2], while related experimental results have been given by Burgreen [3]. Dickey [4] considered the initial-boundary value problem for one-dimensional case of (1.6) with K(x,t)=1 in the case when the ends of the beam are hinged. He showed how the model affords a description of the phenomenon of “dynamic buckling.” The one-dimensional case has also been studied by Ball [5]. He extended the work of Dickey [4] in several directions. In both cases he used the techniques of Lions [6] to prove that the initial boundary value problem is weakly well-posed. Menzala [7] studied the existence and uniqueness of solutions of (1.6) with K(x,t)=1, x∈ℝn, and M∈C1[0,∞[  and M(λ)≥m0>0, for all λ≥0. The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos [8] with K(x)≥k0>0,  x∈Ω. See also Pereira et al. [9]. The abstract model
(1.7)utt+A2u+M(|A1/2|2)Au=0
of (1.6), where A is a nonbounded self-adjoint operator in a conveniently Hilbert space has been studied by Medeiros [10]. He proved that the abstract model is well-posed  in the weak sense, since M∈C1[0,∞[  with M(λ)≥m0+m1λ, for all λ≥0,  where m0 and m1 are positive constants. Pereira [11] considered the abstract model (1.7) with dissipative term ut. He proved the existence, uniqueness, and exponential decay of the solutions with the following assumptions about M:
(1.8)M∈C0([0,∞[)withM(λ)≥-β,∀λ≥0,0<β<λ1,
where λ1 is the first eigenvalue of
(1.9)A2u-λAu=0.
Our main goal here is to extend the previous results for a nonlinear degenerate coupled beams system of type (1.1)–(1.5). We show the existence, uniqueness, and uniform exponential decay rates.Our paper is organized as follows. In Section2 we give some notations and state our main result. In Section 3 we obtain the existence and uniqueness for global weak solutions. To obtain the global weak solution we use the Faedo-Galerkin method. Finally, in Section 4 we use the Nakao method (see Nakao [12]) to derive the exponential decay of the energy.


## 2. Assumptions and Main Result

In what follows we are going to use the standard notations established in Lions [6].Let us consider the Hilbert spaceL2(Ω) endowed with the inner product
(2.1)(u,v)=∫Ω‍u(x)v(x)dx
and norm
(2.2)|u|=(u,v).
We also consider the Sobolev space H1(Ω) endowed with the scalar product
(2.3)(u,v)H1(Ω)=(u,v)+(∇u,∇v).
We define the subspace of H1(Ω), denoted by H01(Ω). This space endowed with the norm induced by the scalar product
(2.4)((u,v))H01(Ω)=(∇u,∇v)
is a Hilbert space.


### 2.1. Assumptions on the FunctionsKi, i=1,2, and M

To obtain the weak solution of the system (1.1)–(1.5) we consider the following hypothesis:
(2.5)Ki∈C1([0,T];H01(Ω)∩L∞(Ω)),i=1,2,withKi(x,t)≥0,∀(x,t)∈Ω×(0,T),andthereexistsγ>0suchthatKi(x,0)≥γ>0,(2.6)|∂Ki∂t|R≤δ+C(δ)Ki,i=1,2,∀δ>0,(2.7)M∈C1([0,∞[)withM(λ)≥-β,∀λ≥0,0<β<λ1,λ1isthefirsteingenvalueofthestationaryproblem,Δ2u-λ(-Δu)=0.Remark 2.1.
Letλ1 be the first eingevalue of Δ2u-λ(-Δu)=0; then (see Miklin [13])
(2.8)λ1=inf⁡w∈H02(Ω)|Δw|2|∇w|2>0.


## 2.1. Assumptions on the FunctionsKi, i=1,2, and M

To obtain the weak solution of the system (1.1)–(1.5) we consider the following hypothesis:
(2.5)Ki∈C1([0,T];H01(Ω)∩L∞(Ω)),i=1,2,withKi(x,t)≥0,∀(x,t)∈Ω×(0,T),andthereexistsγ>0suchthatKi(x,0)≥γ>0,(2.6)|∂Ki∂t|R≤δ+C(δ)Ki,i=1,2,∀δ>0,(2.7)M∈C1([0,∞[)withM(λ)≥-β,∀λ≥0,0<β<λ1,λ1isthefirsteingenvalueofthestationaryproblem,Δ2u-λ(-Δu)=0.Remark 2.1.
Letλ1 be the first eingevalue of Δ2u-λ(-Δu)=0; then (see Miklin [13])
(2.8)λ1=inf⁡w∈H02(Ω)|Δw|2|∇w|2>0.


## 3. Existence and Uniqueness Results

Now, we are in a position to state our result about the existence of weak solution to the system (1.1)–(1.5).Theorem 3.1.
Let one take(u0,v0)∈(H01(Ω)∩H4(Ω))2  and   (u1,v1)∈(H02(Ω))2, and let one suppose that assumptions (2.5), (2.6) and (2.7) hold. Then, there exist unique functions u,v:[0,T]→L2(Ω) in the class
(3.1)(u,v)∈((Lloc⁡∞(0,∞):H02(Ω)∩H4(Ω)))2,(ut,vt)∈((Lloc⁡∞(0,∞):H02(Ω)))2,(utt,vtt)∈((Lloc⁡∞(0,∞):L2(Ω)))2
satisfying
(3.2)K1(x,t)utt+Δ2u-M(‖u‖2+‖v‖2)Δu+ut=0inLloc⁡2(0,∞;L2(Ω)),K2(x,t)vtt+Δ2v-M(‖u‖2+‖v‖2)Δv+vt=0inLloc⁡2(0,∞;L2(Ω)),(3.3)(u(x,0),v(x,0))=(u0(x),v0(x))inΩ,(ut(x,0),vt(x,0))=(u1(x),v1(x))inΩ.Proof.
SinceKi≥0, i=1,2, we first perturb the system (1.1)–(1.5) with the terms ɛutt,ɛvtt, with 0<ɛ<1, and we apply the Faedo-Galerkin method to the perturbed system. After we pass to the limit with ɛ→0 in the perturbed system and we obtain the solution for the system (1.1)–(1.5).
(1) Perturbed System
Consider the perturbed system(3.4)(K1+ɛ)uttɛ+Δuɛ+M(‖uɛ‖2+‖vɛ‖2)(-Δuɛ)+utɛ=0inΩ×(0,T),(K2+ɛ)vttɛ+Δvɛ+M(‖uɛ‖2+‖vɛ‖2)(-Δvɛ)+vtɛ=0inΩ×(0,T),uɛ=vɛ=∂uɛ∂η=∂vɛ∂η=0on⁡Σ,(uɛ(x,0),vɛ(x,0))=(u0(x),v0(x))inΩ,(utɛ(x,0),vtɛ(x,0))=(u1(x),v1(x))inΩ.
Let (wν)ν∈ℕ be a basis of H02(Ω) formed by the eigenvectors of the operator -Δ, that is, -Δwν=λνwν, with λν→∞ when ν→∞. Let Vm=[w1,w2,…,wm] be the subspace generated by the first m vectors of (wν)ν∈ℕ.
For each fixedɛ, we consider
(3.5)umɛ(t)=∑j=1m‍gjɛm(t)wj∈Vm,vmɛ(t)=∑j=1m‍hjɛm(t)wj∈Vm
as solutions of the approximated perturbed system
(3.6)((K1+ɛ)uttmɛ(t),w)+(-Δumɛ(t),-Δw)+M(‖umɛ(t)‖2+‖vmɛ(t)‖2)(-Δumɛ(t),w)+(utmɛ(t),w)=0,∀w∈Vm,(3.7)((K2+ɛ)vttmɛ(t),z)+(-Δvmɛ(t),-Δz)+M(‖umɛ(t)‖2+‖vmɛ(t)‖2)(-Δvmɛ(t),z)+(vtmɛ(t),z)=0,∀z∈Vm,(3.8)(umɛ(0),vmɛ(0))=(u0m,v0m)⟶(u0,v0)in(H02(Ω)∩H4(Ω))2,(3.9)(utmɛ(0),vtmɛ(0))=(u1m,v1m)⟶(u1,v1)in(H02(Ω))2.
The local existence of the approximated solutions (umɛ,vmɛ) is guaranteed by the standard results of ordinary differential equations. The extension of the solutions (umɛ,vmɛ) to the whole interval [0,T] is a consequence of the first estimate below.
The First Estimate
Settingw=utmɛ and z=vtmɛ in (3.6) and (3.7), respectively, integrating over (0,t), and taking the convergences (3.8) and (3.9) in consideration, we arrive at
(3.10)(K1,(utmɛ)2(t))+ɛ|utmɛ(t)|2+|Δumɛ(t)|2+(K2,(vtmɛ)2(t))+ɛ|vtmɛ|2+|Δvmɛ(t)|2+M^(‖umɛ(t)‖2+‖vmɛ(t)‖2)+2∫0t‍[|utmɛ(s)|2+|vtmɛ(s)|2]ds≤∫0t‍[|(∂K1∂t,(utmɛ)2(s))|R+|(∂K2∂t,(vtmɛ)2(s))|R]ds+(K1(0),u1m2)+ɛ|u1m|2+|Δu0m|2+(K2(0),v1m2)+ɛ|v1m|2+|Δv0m|2+M^(‖u0m‖2+‖v0m‖2),
where
(3.11)M^(s)=∫0s‍M(τ)dτ.
From (2.7) and (2.8), we have
(3.12)M^(‖umɛ(t)‖2+‖vmɛ(t)‖2)≥-βλ1(|Δumɛ(t)|2+|Δvmɛ(t)|2).
Since β<λ1 and so by (2.5)–(2.7) and convergences (3.8), (3.9), and (3.12), we obtain
(3.13)(K1,(utmɛ)2(t))+(K2,(utmɛ)2(t))+ɛ(|utmɛ(t)|2+|vtmɛ|2(t))+(1-βλ1)(|Δumɛ(t)|2+|Δvmɛ(t)|2)+(2-δ)∫0t‍[|utmɛ(s)|2+|utmɛ(s)|2]ds≤C0+C(δ)∫0t‍[(K1,(utmɛ)2(s))+(K2,(vtmɛ)2(s))]ds
with 0<δ<1 and C0 being a positive constant independent of ɛ, m, and t.
Employing Gronwall’s lemma in (3.13), we obtain the first estimate
(3.14)(K1,(utmɛ)2(t))+(K2,(utmɛ)2(t))+ɛ(|utmɛ(t)|2+|vtmɛ|2(t))+(1-βλ1)(|Δumɛ(t)|2+|Δvmɛ(t)|2)+(2-δ)∫0t‍[|utmɛ(s)|2+|utmɛ(s)|2]ds≤C1,
where C1 is a positive constant independent of ɛ, m, and t. Then, we can conclude that
(3.15)(K11/2utmɛ),(K21/2vtmɛ)areboundedinL∞(0,T;L2(Ω)),(ɛutmɛ),(ɛvtmɛ)areboundedinL∞(0,T;L2(Ω)),(umɛ),(vmɛ)areboundedinL∞(0,T;H02(Ω)),(utmɛ),(vtmɛ)areboundedinL2(0,T;L2(Ω)).
The Second Estimate
Substitutingw=-Δutmɛ(t) and z=-Δvtmɛ(t) in (3.6) and (3.7), respectively, it holds that
(3.16)ddt[((K1,(utmɛ)2(t)))+((K2,(vtmɛ)2(t)))+ɛ(‖utmɛ(t)‖2+‖vtmɛ(t)‖2)+‖Δumɛ(t)‖2+‖Δvmɛ(t)‖2]+2(‖utmɛ(t)‖2+‖vtmɛ(t)‖2)=((∂K1∂t,(utmɛ)2(t)))+((∂K2∂t,(vtmɛ)2(t)))-2M(‖umɛ(t)‖2+‖vmɛ(t)‖2)⋅[((-Δumɛ(t),utmɛ(t)))+((-Δvmɛ(t),vtmɛ(t)))].
Integrating (3.16) over (0,t), 0<t<T, and taking (2.5)–(2.7) and (3.8), (3.9), and first estimate into account, we infer
(3.17)((K1,(utmɛ)2(t)))+((K2,(vtmɛ)2(t)))+ɛ(‖utmɛ(t)‖2+‖vtmɛ(t)‖2)+‖Δumɛ(t)‖2+‖Δvmɛ(t)‖2+2(‖utmɛ(t)‖2+‖vtmɛ(t)‖2)+(2-2δ)∫0t‍(‖utmɛ(t)‖2+‖vtmɛ(t)‖2)ds≤C2,
where C2 is a positive constant independent of ɛ, m, and t. From the above estimate we conclude that
(3.18)(K11/2utmɛ),(K21/2vtmɛ)areboundedinL∞(0,T;H01(Ω)),(ɛutmɛ),(ɛvtmɛ)areboundedinL∞(0,T;H01(Ω)),(umɛ),(vmɛ)areboundedinL∞(0,T;H02(Ω)∩H4(Ω)),(utmɛ),(vtmɛ)areboundedinL2(0,T;H01(Ω)).The Third Estimate
Differentiating (3.6) and (3.7) with respect to t and setting w=uttmɛ and vttmɛ, respectively, we arrive at
(3.19)ddt[K1,(uttmɛ(t))2+(K2,(vttmɛ(t))2)+ɛ(|uttmɛ(t)|2+|vttmɛ|2)+|Δutmɛ(t)|2+|Δvtmɛ(t)|2]+2(|utmɛ(s)|2+|vtmɛ(s)|2)=-2M(‖umɛ(t)‖2+‖vmɛ(t)‖2)⋅[(-Δutmɛ(t),uttmɛ(t))+(-Δvtmɛ(t),vttmɛ(t))]-4[(umɛ(t),-Δutmɛ(t))+(vmɛ(t),-Δvtmɛ(t))]⋅M′(‖umɛ(t)‖2+‖vmɛ(t)‖2)⋅[(-Δutmɛ(t),uttmɛ(t))+(-Δvtmɛ(t),vttmɛ(t))]+(∂K1∂t,(uttmɛ)2(t))+(∂K2∂t,(vttmɛ)2(t)).
Integrating (3.19) over (0,t), and using (2.5), (3.8), (3.9), and the norms |uttmɛ(0)|2≤C3 and |vttmɛ(0)|2≤C4 after employing Gronwall’s lemma, we obtain the third estimate
(3.20)(K1,(uttmɛ)2(t))+(K2,(vttmɛ)2(t))+ɛ(|uttmɛ(t)|2+|vttmɛ(t)|2)+|Δutmɛ(t)|2+|Δvtmɛ(t)|2]+(2-2δ)∫0t‍(|uttmɛ(t)|2+|vttmɛ(t)|2)ds≤C5,
where C5 is a positive constant independent of ɛ, m, and t. From the above estimate we conclude that
(3.21)(K11/2uttmɛ),(K21/2vttmɛ)areboundedinL∞(0,T;L2(Ω)),(ɛuttmɛ),(ɛvttmɛ)areboundedinL∞(0,T;L2(Ω)),(utmɛ),(vtmɛ)areboundedinL∞(0,T;H02(Ω)),(uttmɛ),(vttmɛ)areboundedinL2(0,T;L2(Ω)).(2) Limits of Approximated Solutions
From the Aubin-Lions theorem (see [6]) we deduce that there exist subsequences of (umɛ)m∈ℕ and (vmɛ)m∈ℕ such that
(3.22)umɛ⟶uɛstronglyinL2(0,T;H01(Ω)),vmɛ⟶vɛstronglyinL2(0,T;H01(Ω)),
and since M is continuous, it follows that
(3.23)M(‖umɛ(t)‖2+‖vmɛ(t)‖2)⟶M(‖uɛ(t)‖2+‖vɛ(t)‖2).
From the above estimate we can conclude that there exist subsequences of (umɛ)m∈ℕ and (vmɛ)m∈ℕ, that we denote also by (umɛ)m∈ℕ and (vmɛ)m∈ℕ such that as m→∞ and ɛ→0 we have
(3.24)umɛ⟶u,vmɛ⟶vweakstarinL∞(0,T;H02(Ω)∩H4(Ω)),utmɛ⟶ut,vtmɛ⟶vtweakstarL∞(0,T;H02(Ω)),uttmɛ⟶utt,vttmɛ⟶vttweakstarL2(0,T;L2(Ω)),Δumɛ⟶Δu,Δvmɛ⟶ΔvweakstarinL∞(0,T;L2(Ω)),K1uttmɛ⟶K1utt,K2vttmɛ⟶K2vttweakstarinL∞(0,T;L2(Ω)),Δ2umɛ⟶Δ2u,Δ2vmɛ⟶Δ2vweakstarinL2(0,T;L2(Ω)),ɛuttmɛ⟶ɛutt,ɛvttmɛ⟶ɛvttweakstarinL∞(0,T;L2(Ω)),M(‖umɛ(t)‖2+‖vmɛ(t)‖2)(-Δumɛ-Δvmɛ)⟶M(‖u(t)‖2+‖v(t)‖2)(-Δu-Δv)weakstarinL∞(0,T;L2(Ω)).
Now, multiplying (3.6), (3.7) by θ∈𝒟(0,T) and integrating over (0,T), we arrive at
(3.25)∫0T‍((K1+ɛ)uttmɛ(t),w)θ(t)dt+∫0T‍(-Δumɛ(t),-Δw)θ(t)dt+∫0T‍M(‖umɛ(t)‖2+‖vmɛ(t)‖2)(-Δumɛ(t),w)θdt+∫0T‍(utmɛ,w)θdt=0,∀w∈Vm,∀θ∈D(0,T),∫0T‍((K2+ɛ)vttmɛ(t),z)θ(t)dt+∫0T‍(-Δvmɛ(t),-Δz)θ(t)dt+∫0T‍M(‖umɛ(t)‖2+‖vmɛ(t)‖2)(-Δvmɛ(t),z)θdt+∫0T‍(vtmɛ,z)θdt=0,∀z∈Vm,∀θ∈D(0,T).
The convergences (3.24) are sufficient to pass to the limit in (3.25) in order to obtain
(3.26)K1utt+Δ2u+M(‖u(t)‖2+‖v(t)‖2)(-Δu)+ut=0inLloc⁡∞(0,∞;L2(Ω)),K2vtt+Δ2v+M(‖u(t)‖2+‖v(t)‖2)(-Δv)+vt=0inLloc⁡∞(0,∞;L2(Ω)),
and (u,v) satisfies (3.1).
The uniqueness and initial conditions follow by using the standard arguments as in Lions [6]. The proof is now complete.


## 4. Asymptotic Behavior

In this section we study the asymptotic behavior of solutions to the system (1.1)–(1.5). We show using the Nakao method that the system (1.1)–(1.5) is exponentially stable. The main result of this paper is given by the following theorem.Theorem 4.1.
Let one take(u0,v0)∈(H01(Ω)∩H4(Ω))2,  and   (u1,v1)∈(H02(Ω))2 and let one suppose that assumptions (2.5), (2.6), and (2.7) hold. Then, the solution (u,v) of system (1.1)–(1.5) satisfies
(4.1)|K11/2ut(t)|2+|K21/2vt(t)|2+|Δu(t)|2+|Δv(t)|2+∫tt+1(|ut(s)|2+|vt(s)|2)ds≤α1e-α2t,
for all t≥1, where α1 and α2 are positive constants.Proof.
Multiplying (3.2) by ut(t) and vt(t), respectively, and integrating over Ω, we obtain
(4.2)12ddt[|K11/2ut(t)|2+|K21/2vt(t)|2+|Δu(t)|2+|Δv(t)|2+M^(‖u(t)‖2+‖v(t)‖2)]+|ut(t)|2+|vt(t)|2=(∂K1∂t,ut2(t))+(∂K2∂t,vt2(t)),
where
(4.3)M^(s)=∫0s‍M(τ)dτ.
Using (2.6) and considering δ>0 sufficiently small, we get
(4.4)12ddt[|K11/2ut(t)|2+|K21/2vt(t)|2+|Δu(t)|2+|Δv(t)|2+M^(‖u(t)‖2+‖v(t)‖2)]+(1-δ-K0C(δ))(|ut(t)|2+|vt(t)|2)≤0,
where K0=max{K1¯,K2¯} with
(4.5)Ki¯=max⁡s∈[t,t+1]{esssup⁡Ki(x,s)},i=1,2.
Integrating (4.4) from 0tot, we have
(4.6)E(t)+(1-δ-K0C(δ))∫0t‍(|ut(s)|2+|vt(s)|2)ds≤E(0),
where
(4.7)E(t)=12[|K11/2ut(t)|2+|K21/2vt(t)|2+|Δv(t)|2+|Δv(t)|2+M^(‖u(t)‖2+‖v(t)‖2)]
is the energy associated with the system (1.1)–(1.5). From (4.4) we conclude that
(4.8)ddtE(t)≤0∀t∈(0,∞),
that is, E(t) is bounded and increasing in (0,∞).
Integrating (4.4) from τ1toτ2, 0<τ1<τ2<∞, we arrive at
(4.9)E(τ2)+(1-δ-K0C(δ))∫τ1τ2‍(|ut(s)|2+|vt(s)|2)ds≤E(τ1).
Taking τ1=t and τ2=t+1 in (4.9), we get
(4.10)∫tt+1‍(|ut(s)|2+|vt(s)|2)ds≤11-δ-K0C(δ)[E(t)-E(t+1)]=F2(t).
Therefore, there exist two points t1∈[t,t+1/4] and t2∈[t+3/4,t+1], such that
(4.11)|ut(ti)|+|vt(ti)|≤4F(t),i=1,2.
Making the inner product in L2(Ω) of (1.1) and (1.2) by u(t) and v(t), respectively, and summing up the result we obtain
(4.12)ddt(K1ut(t),u(t))+ddt(K2vt(t),v(t))-|K1ut(t)|2-|K2vt(t)|2+|Δu(t)|2+|Δv(t)|2+M(‖u(t)‖2+‖v(t)‖2)(‖u(t)‖2+‖v(t)‖2)+(ut(t),u(t))+(vt(t),v(t))=(∂K1∂tut(t),u(t))+(∂K2∂tvt(t),v(t)).
Integrating from t1 to t2 and using (2.6), and (2.7) we have
(4.13)(1-βλ1)∫t1t2‍(|Δu(s)|2+|Δv(s)|2)ds≤(K1ut(t1),u(t1))-(K1ut(t2),u(t2))+(K2vt(t1),v(t1))-(K2vt(t2),v(t2))+(1+δ+K0C(δ))∫t1t2‍(|ut(s)||u(s)|+|vt(s)||v(s)|)ds+K0∫t1t2‍(|ut(s)|2+|vt(s)|2)ds.
Let us consider C>0 such that
(4.14)|u(s)|≤C|Δu(s)|,|v(s)|≤C|Δv(s)|
and we take d>0 sufficiently small Then we have. 							(4.15)(1+δ+K0C(δ))(|ut(s)||u(s)|+|vt(s)||v(s)|)≤(1+δ+K0C(δ))2d(|ut(s)|2+|vt(s)|2)+d(|Δu(s)|2+|Δv(s)|2),|(K1ut(t1),u(t1))+(K2vt(t1),v(t1))-(K1ut(t2),u(t2))-(K2vt(t2),v(t2))|≤CK0esssup⁡s∈[t,t+1]|Δu(s)|(|ut(t1)|+|ut(t2)|)+CK0esssup⁡s∈[t,t+1]|Δv(s)|(|vt(t1)|+|vt(t2)|).
Thus, substituting (4.15) into (4.13), we arrive at
(4.16)(1-βλ1)∫t1t2(|Δu(s)|2+|Δv(s)|2)ds≤K0∫t1t2(|ut(s)|2+|vt(s)|2)ds+d∫t1t2(|Δu(s)|2+|Δv(s)|2)ds+CK0esssup⁡s∈[t,t+1]|Δu(s)|(|ut(t1)|+|ut(t2)|)+CK0esssup⁡s∈[t,t+1]|Δv(s)|(|vt(t1)|+|vt(t2)|).
Applying (4.10) and (4.11) in (4.16), we have
(4.17)∫t1t2‍(|Δu(s)|2+|Δv(s)|2)ds≤C1[F2(t)+esssup⁡s∈[t,t+1](|Δu(s)|+|Δv(s)|)F(t)]=G2(t),
where C1 is a positive constant independent of t. Therefore, from (4.10) and (4.17) we obtain
(4.18)∫t1t2‍(|ut(s)|2+|vt(s)|2+|Δu(s)|2+|Δv(s)|2)ds≤F2(t)+G2(t).
Hence, there exists t*∈[t1,t2] such that
(4.19)|ut(t*)|2+|vt(t*)|2+|Δu(t*)|2+|Δv(t*)|2≤2[F2(t)+G2(t)].
Consequently,
(4.20)M^(‖u(t)‖2+‖v(t)‖2)≤C2[F2(t)+G2(t)],
where
(4.21)C2=2m0C~,m0=max⁡0≤s≤(‖u(t*)‖2+‖v(t*)‖2)<∞M(s)
and C~ is a positive constant such that ∥u(t*)∥2≤C~|Δu(t*)|2.
From (4.19) and (4.20), we have
(4.22)E(t*)≤C3[F2(t)+G2(t)].
Since E(t) is increasing, we have
(4.23)esssup⁡s∈[t,t+1]E(s)≤E(t*)+(1+δ+K0C(δ))∫t1t2‍(|ut(s)|2+|vt(s)|2)ds.
Now, by (4.10), (4.22), and (4.23) we get
(4.24)E(t)≤C4[E(t)-E(t+1)],
where C4 is a positive constant. Then, by the Nakao lemma (see [12]) we conclude that
(4.25)E(t)≤b1e-α2t,∀t≥1,
where b1 and α2 are positive constants, that is,
(4.26)|K1ut(t)|2+|K2vt(t)|2+|Δu(t)|2+|Δv(t)|2M^(‖u(t)‖2+‖v(t)‖2)≤2b1e-α2t.
Using (2.7) we obtain
(4.27)|K1ut(t)|2+|K2vt(t)|2+|Δu(t)|2+|Δv(t)|2≤2b1m1e-α2t,
where
(4.28)m1=1-βλ1>0.
From (4.10) we have
(4.29)∫tt+1‍(|ut(s)|2+|vt(s)|2)ds≤11-δ-K0C(δ)[E(t)-E(t+1)]≤E(t)≤b1e-α2t.
Therefore, from (4.27) and (4.29) we conclude that
(4.30)|K1ut(t)|2+|K2vt(t)|2+|Δu(t)|2+|Δv(t)|2+∫tt+1‍(|ut(s)|2+|vt(s)|2)ds≤α1e-α2t,∀t≥1,
where α1 and α2 are positive constants. Now, the proof is complete.



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*Source: 659289-2012-08-15.xml*