# Uniform Decay of Solutions for a Nonlinear Viscoelastic Wave Equation with Boundary Dissipation

Authors: Shun-Tang Wu; Hsueh-Fang Chen
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We consider a nonlinear viscoelastic wave equationutt(t)-k0Δu(t)+∫0tg(t-s)div(a(x)∇u(s))ds+b(x)ut=f(u), with nonlinear boundary damping in a bounded domain Ω. Under appropriate assumptions imposed on g and with certain initial data, we establish the general decay rate of the solution energy which is not necessarily of exponential or polynomial type. This work generalizes and improves earlier results in the literature.


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## Body

## 1. Introduction

In this paper, we are concerned with the energy decay rate of the following viscoelastic problem with nonlinear boundary dissipation:(1.1)utt(t)-k0Δu(t)+∫0t‍g(t-s)div(a(x)∇u(s))ds+b(x)ut=f(u),inΩ×(0,∞),u=0,onΓ0×(0,∞),k0∂u∂ν-∫0t‍g(t-s)(a(x)∇u(s))·νds+h(ut)=0,onΓ1×(0,∞),u(0)=u0,ut(0)=u1,x∈Ω,
where k0>0 and Ω is a bounded domain in Rn(n≥1) with a smooth boundary Γ=Γ0∪Γ1. Here, Γ0 and Γ1 are closed and disjoint with meas(Γ0)>0, and ν is the unit outward normal to Γ. The relaxation function g is a positive and uniformly decaying function, h and a are functions satisfying some conditions given in (A2) and (A3), respectively, b:Ω→R+ is a function, and f(u)=|u|p-2u with
(1.2)1≤p≤2n-2,n>2,1≤p<∞,ifn=2.This type of equations usually arise in the theory of viscoelasticity. It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics are interesting and of great importance. From the mathematical point of view, their memory effects are modeled by integrodifferential equations. Hence, questions related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years.For example, Cavalcanti et al. [1] considered the following problem:
(1.3)utt-Δu+∫0t‍g(t-s)Δu(s)ds+a(x)ut+|u|γu=0,inΩ×(0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t≥0,
where Ω is a bounded domain in Rn(n≥1) with a smooth boundary, γ>0, and a:Ω→R+ is a function, which may be null on a part of Ω. The authors established an exponential decay estimate under the conditions that a(x)≥a0>0 on ω⊂Ω, with meas(ω)>0 and satisfying some geometry conditions and
(1.4)-ξ1g(t)≤g′(t)≤-ξ2g(t),t≥0.
Berrimi and Messaoudi [2] improved the result [1] by introducing a new function. They proved an exponential decay result under weaker conditions on both a and g. In fact, they allowed the function a to vanish on any part of Ω, and, consequently, the geometry condition imposed on a part of boundary is no longer needed. Later, the same authors [3] and Messaoudi [4] extended the result to a situation in which a source term is competing with the viscoelastic dissipation. In [5], Cavalcanti and Oquendo considered  the following:
(1.5)utt(t)-k0Δu(t)+∫0t‍g(t-s)div(a(x)∇u(s))ds+b(x)h(ut)+f(u)=0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t≥0.
Under some conditions on the relaxation function g, they improved the result of [1]. Indeed, they proved that the solution of (1.5) decays exponentially to zero when g is decaying exponentially and h is linear, and the solution decays polynomially to zero when g is decaying polynomially and h is nonlinear.On considering the boundary stabilization, Cavalcanti et al. [6] considered the following problem:
(1.6)utt(t)-Δu(t)+∫0t‍g(t-s)Δu(s)ds=0,inΩ×(0,∞),u=0,onΓ0×(0,∞),∂u∂ν-∫0t‍g(t-s)∂u∂ν(s)ds+h(ut)=0,onΓ1×(0,∞),u(0)=u0,ut(0)=u1,x∈Ω.
The existence and uniform decay rate results were established under quite restrictive assumptions on damping term h and the kernel function g. Later, Cavalcanti et al. [7] generalized this result without imposing a growth condition on h and under a weaker assumption on g. Recently, Messaoudi and Mustafa [8] exploited some properties of convex functions [9] and the multiplier method to extend these results. They established an explicit and general decay rate result without imposing any restrictive growth assumption on the damping term h and greatly weakened the assumption on g. Very recently, problem (1.1) has been considered by Li et al. [10] with b(x)=0 and f(u)=-|u|γu,γ>0. They showed the global existence and uniqueness of global solution of problem (1.1) and established uniform decay rate of the energy under suitable conditions on the initial data and the relaxation function g. We refer the reader to related works [7, 11–16] dealing with boundary stabilization.Motivated by previous works, it is interesting to investigate the global existence and uniform decay result of solutions to problem (1.1) when a forcing source term is competing with the viscoelastic dissipation and nonlinear boundary damping under the weaker assumption on both b and g. In fact, we will allow the function b to be null on any part of Ω (including Ω itself) and the kernel function g is not necessarily decaying in an exponential or polynomial fashion. Therefore, our result allows a larger class of relaxation functions and improves the results in [10, 13] where only the exponential and polynomial rate was considered.The remainder of this paper is organized as follows. In Section2, we provide assumptions that will be used later and mention the local existence result Theorem 2.1. In Section 3, we prove our stability result that is given in Theorem 3.7.


## 2. Preliminary Results

In this section, we give assumptions and preliminaries that will be needed throughout the paper. First, we introduce the following set:(2.1)HΓ01={u∈H1(Ω):u|Γ0=0},
and endow HΓ01 with the Hilbert structure induced by H1(Ω). We have that HΓ01 is a Hilbert space. For simplicity, we denote ∥·∥q=∥·∥Lq(Ω) and ∥·∥q,Γ1=∥·∥Lq(Γ1),1≤q≤∞. According to (1.2), we have the imbedding: HΓ01↪L2(p+1)(Ω). Let B>0 be the optimal constant of Sobolev imbedding which satisfies the following inequality:
(2.2)∥u∥2(p+1)≤B∥∇u∥2,∀u∈HΓ01,
and we use the Trace-Sobolev imbedding: HΓ01↪Lk(Γ1),1≤k<2(n-1)/(n-2). In this case, the imbedding constant is denoted by B1, that is,
(2.3)∥u∥k,Γ1≤B1∥∇u∥2.
Next, we state the assumptions for problem (1.1) as follows.(A1)
g:[0,∞)→(0,∞) is a bounded C1 function satisfying
(2.4)g(0)>0,k0-∥a∥∞∫0∞‍g(s)ds=l>0,
and there exists a nonincreasing positive differentiable function ξ such that
(2.5)g′(t)≤-ξ(t)g(t),∀t≥0,∫0∞‍ξ(s)ds=∞.(A2)
h:R→R is a nondecreasing function with
(2.6)h(s)s≥α|s|2,∀s∈R,(2.7)|h(s)|≤β|s|,∀s∈R.(A3)
a:Ω→R is a nonnegative functions and a∈C1(Ω¯) such that
(2.8)a(x)≥a0>0,(2.9)|∇a(x)|2≤α12|a(x)|,
for some positive constant α1.By using the Galerkin method and procedure similar to that of [10, 16], we can have the following local existence result for problem (1.1).Theorem 2.1.
Let hypotheses (A1)–(A3) and (1.2) hold and assume that u0∈HΓ01∩H2(Ω),u1∈HΓ01. Then there exists a strong solution u of (1.1) satisfying
(2.10)u∈L∞([0,T);HΓ01∩H2(Ω)),ut∈L∞([0,T);HΓ01),utt∈L∞([0,T);L2(Ω)).
Furthermore, if u0∈HΓ01,u1∈L2(Ω), then there exists a weak solution u of (1.1) satisfying
(2.11)u∈C([0,T);HΓ01)∩C1([0,T);L2(Ω)),
for some T>0.


## 3. Global Existence and Energy Decay

In this section, we focus our attention on the uniform decay of weak solutions to problem (1.1). For this purpose, we define
(3.1)J(u(t))=12∫Ω‍(k0-a(x)∫0t‍g(s)ds)|∇u(t)|2dx+12(g∘∇u)(t)-1p∥u∥pp,
and the energy function
(3.2)E(t)=12∥ut∥22+J(u(t)),fort∈[0,T),
where
(3.3)(g∘∇u)(t)=∫Ω‍∫0t‍g(t-s)a(x)|∇u(t)-∇u(s)|2dsdx.Adopting the proof of [10], we still have the following results.Lemma 3.1.
For anyu∈C1(0,T;H1(Ω)), we have
(3.4)∫Ω‍a(x)∫0t‍g(t-s)∇u(s)∇ut(t)dsdx=-12∫Ω‍a(x)g(t)|∇u(t)|2dx+12(g′∘∇u)(t)-12ddt[(g∘∇u)(t)-∫Ω‍a(x)∫0t‍g(s)ds|∇u(t)|2dx].Lemma 3.2.
Letu be the solution of (1.1), then, under assumptions (A1)-(A2), E(t) is a nonincreasing function on [0,T) and
(3.5)E′(t)=-∫Γ1‍uth(ut)dΓ+12(g′∘∇u)(t)-12∫Ω‍a(x)g(t)|∇u(t)|2dx-∫Ω‍b(x)|ut(t)|2dx≤0.Next, we define a functionalF, which helps in establishing the desired results. Setting
(3.6)F(x)=12x2-Bpplp/2xp,x>0.Remark 3.3.
As in [17], we can verify that the functional F is increasing in (0,λ0), decreasing in (λ0,∞), and F has a maximum at λ0=(lp/2/Bp)1/(p-2) with the following maximum value:
(3.7)E1=p-22plp/(p-2)B-2p/(p-2).
Further, from (3.1), (3.2), (2.4), and the definition of F by (3.6), we have
(3.8)E(t)≥J(u(t))≥12l∥∇u(t)∥2+12(g∘∇u)(t)-Bpplp/2(l∥∇u∥22+(g∘∇u)(t))p=F(l∥∇u∥22+(g∘∇u)(t)),t≥0.Lemma 3.4.
Suppose that (A1)-(A2) and (1.2) hold. Assume further that u0∈HΓ01,u1∈L2(Ω), and satisfy l∥∇u0∥22<λ02  and  E(0)<E1. Then, it holds that
(3.9)l∥∇u(t)∥22+(g∘∇u)(t)<λ02,
for all t∈[0,T). Moreover, one has  J(u(t))<E1 and
(3.10)l∥∇u(t)∥22+(g∘∇u)(t)≤2pE(t)p-2<2pE(0)p-2<2pE1p-2,
for all t∈[0,T).Proof.
Using (3.8) and considering E(t) is a nonincreasing function, we obtain
(3.11)F(l∥∇u∥22+(g∘∇u)(t))≤E(t)≤E(0)<E1,t∈[0,T).
Further, from Remark 3.3, we observe that F is increasing in (0,λ0), decreasing in (λ0,∞), and F(λ)→-∞ as λ→∞. Thus, as E(0)<E1, there exist λ1′<λ0<λ1 such that F(λ1′)=F(λ1)=E(0), which together with l∥∇u0∥22<λ02 infer that
(3.12)F(l∥∇u0∥22)≤E(0)=F(λ1′).
This impliesthat l1/2∥∇u0∥2≤λ1′.
Next, we will prove that(3.13)l∥∇u(t)∥22+(g∘∇u)(t)≤λ1′.
To establish (3.13), we argue by contradiction. Suppose that (3.13) does not hold, then there exists t*∈(0,T) such that
(3.14)l∥∇u(t*)∥22+(g∘∇u)(t*)>λ1′.Case 1. If λ1′<l∥∇u(t*)∥22+(g∘∇u)(t*)<λ0, then
(3.15)F(l∥∇u(t*)∥22+(g∘∇u)(t*))>F(λ1′)=E(0)≥E(t*).
This contradicts (3.11).
Case 2. If l∥∇u(t*)∥22+(g⋄∇u)(t*)≥λ0, then by continuity of l∥∇u(t)∥22+(g∘∇u)(t), there exists 0<t1<t* such that
(3.16)λ1′<l∥∇u(t1)∥22+(g∘∇u)(t1)<λ0,
then
(3.17)F(l∥∇u(t1)∥22+(g∘∇u)(t1))>F(λ1′)=E(0)≥E(t1).
This is also a contradiction of (3.11). Thus, we have proved the inequality (3.13).
To prove (3.10), we note for λ<λ0,  such that
(3.18)F(λ)=λ2(12-Bpplp/2λp-2)≥λ2(12-Bpplp/2λ0p-2)≥p-22pλ2,
because of λ0=(lp/2/Bp)1/(p-2). Thanks to (l∥∇u(t)∥22+(g∘∇u)(t))1/2<λ0 by (3.9), we obtain
(3.19)p-22p(l∥∇u∥22+(g∘∇u)(t))≤F(l∥∇u∥22+(g∘∇u)(t))≤J(u(t))≤E(t)<E(0)<E1.
Therefore, we complete the proof of Lemma 3.4.Theorem 3.5.
Letu0∈HΓ01,u1∈L2(Ω), and (A1)-(A2) and (1.2) hold. Assume further that l∥∇u0∥22<λ02 and E(0)<E1, then the problem (1.1) admits a global solution. Furthermore, for all t∈[0,∞), one has
(3.20)l∥∇u(t)∥22+(g∘∇u)(t)≤2pE(t)p-2<2pE(0)p-2<2pE1p-2,(3.21)∥u∥pp≤L(l∥∇u∥22+(g∘∇u)(t)),
with L=(Bp/l)(2pE(0)/(p-2)l)(p-2)/2.Proof.
It follows from (3.19) and (3.8) that
(3.22)12∥ut∥22+p-22p(l∥∇u∥22+(g∘∇u)(t))≤12∥ut∥22+F(l∥∇u∥22+(g∘∇u)(t))≤12∥ut∥22+J(u(t))=E(t)<E(0)<E1.
Thus, we have the inequality (3.20) and we also establish the boundedness of ut in L2(Ω) and the boundedness of u in HΓ01. Moreover, from (2.2) and (3.22), we also obtain the boundedness of u in Lp(Ω). Hence, it must have T=∞.
Additionally, using (2.2) and (3.20), we obtain
(3.23)∥u∥pp≤Bp∥∇u∥2p≤1l(Bp(2pE(0)(p-2)l)(p-2)/2)l∥∇u∥22≤L(l∥∇u∥22+(g∘∇u)(t)),
for all t≥0.Now, we will investigate the asymptotic behavior of the energy functionE(t). First, we define some functionals and establish Lemma 3.6. Let
(3.24)G(t)=E(t)+ε1Φ(t)+ε2Ψ(t),
where
(3.25)Φ(t)=∫Ω‍utudx,(3.26)Ψ(t)=∫Ω‍a(x)ut∫0t‍g(t-s)(u(s)-u(t))dsdx,
and ε1, ε2 are some positive constants to be be specified later.Lemma 3.6.
There exist two positive constantsβ1 and β2 such that the relation
(3.27)β1E(t)≤G(t)≤β2E(t)
holds, for ε1,ε2>0 small enough.Proof.
By Hölder’s inequality, Young’s inequality, (2.2), and (2.8), we deduce that
(3.28)|Φ(t)|≤12∥ut∥22+B22∥∇u∥22,|Ψ(t)|≤12∥ut∥22+12∫Ω‍(a(x)∫0t‍g(t-s)(u(t)-u(s))ds)2dx≤12∥ut∥22+∥a∥∞22∫0t‍g(s)ds∫Ω‍∫0t‍g(t-s)|u(t)-u(s)|2dsdx≤12∥ut∥22+(k0-l)∥a∥∞B22a0(g∘∇u)(t).
Hence, taking (3.24) and (3.28) into account, we have
(3.29)G(t)=E(t)+ε1Φ(t)+ε2Ψ(t)≤E(t)+c1∥ut∥22+c2∥∇u∥22+c3(g∘∇u)(t),G(t)≥E(t)-c4(∥ut∥22+∥∇u∥22+(g∘∇u)(t)),
where c1=(ε1+ε2)/2,c2=ε1B2/2,c3=(k0-l)∥a∥∞B2ε2/2a0, and c4=max(c1,c2,c3). Thus, using (3.22) and selecting ε1,ε3>0 small enough, there exist two positive constants β1 and β2 such that
(3.30)β1E(t)≤G(t)≤β2E(t).Theorem 3.7.
Let (A1)–(A3) and (1.2) hold. Assume that u0∈HΓ01,u1∈L2(Ω),l∥∇u0∥22<λ02, and E(0)<E1. Then, for any t0>0, there exist two positive constants K and k such that the solution of (1.1) satisfies
(3.31)E(t)≤Ke-k∫t0t‍ξ(s)ds,fort≥t0.Proof.
First, we estimate the derivative ofG(t). From (3.25) and using (1.1), we have
(3.32)Φ′(t)=∥ut∥22-k0∥∇u∥22+∫Ω‍∇u(t)a(x)∫0t‍g(t-s)∇u(s)dsdx-∫Γ1‍h(ut)udΓ-∫Ω‍b(x)utudx+∥u∥pp.
The third, the fourth, and the fifth terms on the right-hand side of (3.32) can be estimated as follows. From Hölder’s inequality, Young’s inequality, and (2.4), for η>0, we have
(3.33)∫Ω‍∇u(t)a(x)∫0t‍g(t-s)∇u(s)dsdx≤k02∥∇u∥22+12k0∫Ω‍(a(x)∫0t‍g(t-s)(∇u(s)-∇u(t)+∇u(t))ds)2dx≤[k02+12k0(1+η)(k0-l)2]∥∇u∥22+12k0(1+1η)(k0-l)(g∘∇u)(t).
Employing Hölder’s inequality, Young’s inequality, (2.2), (2.3), and (2.7), for δ1,δ2>0, we see that
(3.34)|∫Γ1‍h(ut)udΓ|≤δ1B*2∥∇u∥22+β24δ1∫Γ1‍ut2dΓ,∫Ω‍b(x)utudx≤B2∥b∥∞δ2∥∇u∥22+14δ2∫Ω‍b(x)ut2dx.
A substitution of (3.33)-(3.34) into (3.32) yields
(3.35)Φ′(t)≤∥ut∥22-(k02-12k0(1+η)(k0-l)2-δ1B*2-B2∥b∥∞δ2)∥∇u∥22+12k0(1+1η)(k0-l)(g∘∇u)(t)+β24δ1∫Γ1‍ut2dΓ+14δ2∫Ω‍b(x)ut2dx+∥u∥pp.
Letting η=l/(k0-l)>0 and δ1=l/(8B*2),δ2=l/8B2∥b∥∞ in the above inequality, we obtain
(3.36)Φ′(t)≤-l4∥∇u∥22+∥ut∥22+(k0-l)2l(g∘∇u)(t)+2β2B*2l∫Γ1‍ut2dΓ+2B2∥b∥∞l∫Ω‍b(x)ut2dx+∥u∥pp.
Next, we estimate Ψ′(t). Taking the derivative of Ψ(t) in (3.26) and using (1.1) to obtain
(3.37)Ψ′(t)=∫Ω‍k0a(x)∇u(t)∫0t‍g(t-s)(∇u(t)-∇u(s))dsdx+∫Ω‍k0∇u(t)·∇a(x)∫0t‍g(t-s)(u(t)-u(s))dsdx-∫Ω‍(∫0t‍g(t-s)a(x)∇u(s)·∇a(x)ds)(∫0t‍g(t-s)(u(t)-u(s))ds)dx-∫Ω‍a(x)(∫0t‍g(t-s)a(x)∇u(s)ds)(∫0t‍g(t-s)(∇u(t)-∇u(s))ds)dx+∫Ω‍a(x)b(x)ut∫0t‍g(t-s)(u(t)-u(s))dsdx+∫Γ1‍a(x)h(ut)∫0t‍g(t-s)(u(t)-u(s))dsdΓ-∫Ω‍a(x)|u|p-2u∫0t‍g(t-s)(u(t)-u(s))dsdx-∫Ω‍a(x)ut∫0t‍g′(t-s)(u(t)-u(s))dsdx-(∫0t‍g(s)ds)∫Ω‍a(x)|ut2|dx.
As in deriving (3.36), in what follows we will estimate the right-hand side of (3.37). Using Young’s inequality, Hölder’s inequality, (2.4), and (2.9), for δ>0, we have
(3.38)|∫Ω‍k0a(x)∇u(t)∫0t‍g(t-s)(∇u(t)-∇u(s))dsdx|≤k02δ∥∇u∥22+14δ∫Ω‍(a(x)∫0t‍g(t-s)(∇u(t)-∇u(s))ds)2dx≤k02δ∥∇u∥22+∥a∥∞4δ∫0t‍g(s)ds∫Ω‍a(x)∫0t‍g(t-s)|∇u(t)-∇u(s)|2dsdx≤k02δ∥∇u∥22+k0-l4δ(g∘∇u)(t),∫Ω‍k0∇u(t)·∇a(x)∫0t‍g(t-s)(u(t)-u(s))dsdx≤k0α1∫Ω‍|∇u|a(x)(∫0t‍g(s)ds)1/2(∫0t‍g(t-s)(u(t)-u(s))2ds)1/2dx≤k02α12δ∥∇u∥22+(k0-l)B24δa0(g∘∇u)(t).
Again, exploiting (2.9), Young’s inequality, Hölder’s inequality, and (2.4), we obtain
(3.39)|∫Ω‍(∫0t‍g(t-s)a(x)∇u(s)·∇a(x)ds)(∫0t‍g(t-s)(u(t)-u(s))ds)dx|≤α12δ∫Ω‍a2(x)(∫0t‍g(t-s)|∇u(s)|ds)2dx+14δ∫Ω‍a(x)(∫0t‍g(t-s)(u(t)-u(s))ds)2dx≤2α12δ(k0-l)∫Ω‍a(x)∫0t‍g(t-s)(|∇u(s)-∇u(t)|2+|∇u(t)|2)dsdx+14δ∫Ω‍a(x)(∫0t‍g(t-s)(u(t)-u(s))ds)2dx≤2α12δ(k0-l)2∥∇u∥22+(2α12δ(k0-l)+(k0-l)B24δa0)(g∘∇u)(t),∫Ω‍a(x)(∫0t‍g(t-s)a(x)∇u(s)ds)(∫0t‍g(t-s)(∇u(t)-∇u(s))ds)dx≤δ∫Ω‍a2(x)(∫0t‍g(t-s)|∇u(s)|ds)2dx+14δ∫Ω‍a2(x)(∫0t‍g(t-s)(∇u(t)-∇u(s))ds)2dx≤2δ(k0-l)2∥∇u∥22+(2δ+14δ)(k0-l)(g∘∇u)(t),|∫Ω‍a(x)b(x)ut∫0t‍g(t-s)(u(t)-u(s))dsdx|≤δ∥b∥∞∫Ω‍b(x)ut2dx+(k0-l)∥a∥∞B24a0δ(g∘∇u)(t).
Utilizing Hölder’s inequality, Young’s inequality, (2.3), and (2.7), the sixth term on the right-hand side of (3.37) can be estimated as
(3.40)|∫Γ1‍a(x)h(ut)∫0t‍g(t-s)(u(t)-u(s))dsdΓ|≤δβ2∫Γ1‍ut2dΓ+(k0-l)∥a∥∞B*24a0δ(g∘∇u)(t).
As for the seventh and the eighth terms on the right-hand side of (3.37), using Hölder’s inequality, Young’s inequality, (2.2), (3.20), and (2.4), we obtain
(3.41)|∫Ω‍a(x)|u|p-2u∫0t‍g(t-s)(u(t)-u(s))dsdx|≤δ∥u∥2(p-1)2(p-1)+(k0-l)∥a∥∞B24a0δ(g∘∇u)(t)≤δB2(p-1)(2pE(0)l(p-2))p-2∥∇u∥22+(k0-l)∥a∥∞B24a0δ(g∘∇u)(t),|∫Ω‍a(x)ut∫0t‍g′(t-s)(u(t)-u(s))dsdx|≤δ∥ut∥22-g(0)∥a∥∞2B24a0δ(g′∘∇u)(t).
Combining these estimates (3.38)–(3.41), (3.37) becomes
(3.42)Ψ′(t)≤-(a0∫0t‍g(s)ds-δ)∥ut∥22+δc5∥∇u∥22+c6(g∘∇u)(t)-g(0)∥a∥∞2B24a0δ(g′∘∇u)(t)+δ∥b∥∞∫Ω‍b(x)ut2dx+δβ2∫Γ1‍ut2dΓ,
where c5=k02(α12+1)+2(α12+1)(k0-l)2+B2(p-1)(2pE(0)/l(p-2))p-2 and c6=(k0-l)(B2/2δa0+2α12δ+(2δ+1/2δ)+∥a∥∞(B*2+3B2)/4a0δ). Hence, we conclude from (3.24), (3.5), (3.36), (3.42), and (2.6) that
(3.43)G′(t)=E′(t)+ε1Φ′(t)+ε2Ψ′(t)≤-(ε2(a0g0-δ)-ε1)∥ut∥22-(ε1l4-ε2δc5)∥∇u∥22+(ε2c6+(k0-l)ε12l)(g∘∇u)(t)-(1-2ε1B2∥b∥∞l-ε2δ∥b∥∞)∫Ω‍b(x)ut2dx-(α-2B*2ε1β2l-ε2δβ2)∫Γ1‍|ut|2dΓ-(12-ε2g(0)∥a∥∞2B24a0δ)(-g′∘∇u)(t)+ε1∥u∥pp,∀t≥t0,
where we have used the fact that for any t0>0,
(3.44)∫0t‍g(s)ds≥∫0t0‍g(s)ds=g0,∀t≥t0,
because gis positive and continuous with g(0)>0. At this point, we choose δ>0 small enough so that
(3.45)4δc5l<a0g02<a0g0-δ.
Whence δ is fixed, the choice of any two positive constants ε1 and ε2 satisfying
(3.46)4ε2c5δl<ε1<a0g02ε2
will make
(3.47)k1=ε1l4-ε2δc5>0,k2=ε2(a0g0-δ)-ε1>0.
Then, we choose ε1 and ε2 so small that (3.27) and (3.45) remain valid, further
(3.48)k3=1-2ε1B2∥b∥∞l-ε2δ∥b∥∞>0k4=α-2B*2ε1β2l-ε2δβ2>0k5=12-ε2g(0)∥a∥∞2B24a0δ>0.
Hence, for all t≥t0, we arrive at
(3.49)G′(t)≤-k1∥∇u∥22-k2∥ut∥22+c7(g∘∇u)(t)+c8(g′∘∇u)(t)-k3∫Ω‍b(x)ut2dx-k4∫Γ1‍|ut|2dΓ+ε1∥u∥p+1p+1,
which yields (if needed, one can choose ε1 sufficiently small)
(3.50)G′(t)≤-c9E(t)+c10(g∘∇u)(t),
where ci,i=7,8,9,10 are some positive constants. It follows from (3.50), (2.5), and (3.5) that
(3.51)ξ(t)G′(t)≤-c9ξ(t)E(t)+c10ξ(t)(g∘∇u)(t)≤-c9ξ(t)E(t)-c10(g′∘∇u)(t)≤-c9ξ(t)E(t)-2c10E′(t),fort≥t0.
That is,
(3.52)L′(t)≤-c9ξ(t)E(t)≤-kξ(t)L(t),fort≥t0,
where L(t)=ξ(t)G(t)+2c10E(t) is equivalent to E(t) by Lemma 3.6 and k is a positive constant. A integration of (3.52) leads to
(3.53)L(t)≤L(t0)e-k∫t0t‍ξ(s)ds,fort≥t0.
Again, employing L(t) is equivalent to E(t) leads to
(3.54)E(t)≤Ke-k∫t0t‍ξ(s)ds,fort≥t0,
where K is a positive constant. This completes the proof.Remark 3.8.
We illustrate the energy decay rate given by Theorem3.7 through the following examples which are introduced in [18].(i)
If(3.55)ξ(t)=α,α>0,
then (3.54) gives the exponential decay estimate
(3.56)E(t)≤Ke-kαt.
Similarly, if
(3.57)ξ(t)=α(1+t)-1,α>0,
then we obtain the polynomial decay estimate
(3.58)E(t)≤K(1+t)-αk.(ii)
If(3.59)g(t)=αe-α1(ln(1+t))ν,
with α,α1,ν>1, then (2.5) holds for
(3.60)ξ(t)=α1ν(ln(1+t))ν-11+t.
Thus (3.54) gives the estimate
(3.61)E(t)≤Ke-kα1(ln(1+t))ν.(iii)
If(3.62)g(t)=α(2+t)ν(ln(2+t))α1,
where α>0 and ν>1 and α1∈R(orν=1andα1>1), then for
(3.63)ξ(t)=ν(ln(2+t))+α1(2+t)(ln(2+t))α1,
we obtain from (3.54) that
(3.64)E(t)≤K[(2+t)ν(ln(2+t))α1]k.



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*Source: 421847-2012-10-04.xml*