# Asymptotic Energy Estimates for Nonlinear Petrovsky Plate Model Subject to Viscoelastic Damping

Authors: Xiuli Lin; Fushan Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the nonlinear Petrovsky plate model under the presence of long-time memory. Under suitable conditions, we show that the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity.


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## Body

## 1. Introduction

LetΩ be a bounded domain of ℝ2 with a sufficiently smooth boundary Γ=Γ0∪Γ1. We will assume that Γ0∩Γ1=∅ and Γ0,Γ1 have positive measures. The vector ν=(ν1,ν2) is the unit exterior normal and τ=(-ν2,ν1) represents the tangential direction to Γ. Here, the variable w represents displacement of a plate occupying the domain Ω. The governing equation is given by
(1.1)w′′(t)-γΔw′′(t)+Δ2w(t)+Δ2∫0∞‍h′(s)wt(s)ds=div[𝒞(σ(t))∇w(t)]inΩ×(0,∞),
where the notations wt(s)=w(t-s),′=∂t. The positive constant γ is the proportional to the thickness of the plate, h(t) is relaxation function, div stands for scalar divergence of a vector field, the stress resultant σ(t) is given by
(1.2)σ(t):=f(∇w(t)),
and the nonlinear function f:ℝ2→M is defined as f(s)=(1/2)s⊗s for all s∈ℝ2.𝒞 is a linear operator defined on M with value on M, the space of 2×2 symmetric matrices, and defined as
(1.3)𝒞(σ)=Ed(1-μ2)[(μtrσ)I+(1-μ)σ]
for any σ∈M, where Iis the identity matrix and trσ denotes the trace of σ. Moreover, d>0 is the density of the shell, E>0 denotes the Young modulus, and μ(0<μ<1/2) is the Poisson’s ratio.With (1.1), we associate the boundary conditions on the portion of the boundary Γ0,
(1.4)w=∂νw=0onΓ0×(0,∞),
and the boundary conditions on the remaining portion of the boundary Γ1,
(1.5)ℬ1w(t)+ℬ1∫0∞‍h′(s)wt(s)ds=0onΓ1×(0,∞),ℬ2w(t)-γ∂νw′′(t)+ℬ2∫0∞‍h′(s)wt(s)ds-𝒞(σ(t))ν·∇w(t)=0onΓ1×(0,∞),
where
(1.6)ℬ1w=Δw+(1-μ)B1w,ℬ2w=∂νΔw+(1-μ)∂τB2w
and the boundary operators B1 and B2 are defined by
(1.7)B1w=2ν1ν2wxy-ν12wyy-ν22wxx,B2w=(ν12-ν22)wxy+ν1ν2(wyy-wxx).With (1.1), we also associate the initial conditions
(1.8)w(0)=w0,w′(0)=w1inΩ,w(s)=χ(s),-∞<s<0,
where
(1.9)(χ(s),χ′(s))∈Lc∞(-∞,0;H3×H2),
the symbol Lc∞ denotes the subspace of L∞ such that there exists a constant T such that the functions vanish as s<-T.This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials, to retain a memory of their past history. From the mathematical point of view, these damping effects are modeled by integro-differential operators. Therefore, the dynamics of viscoelastic materials are of great importance and interest as they have wide applications in natural sciences. From the physical point of view, the problem (1.1) describes the position w(x,y,t) of the material particle (x,y) at time t, which is clamped in the portion Γ0 of its boundary.Models of Petrovsky type are of interest in applications in various areas in mathematical physics, as well as in geophysics and ocean acoustics [1, 2]. The Petrovsky type models without memory were discussed in [3, 4]. Messaoudi [3] considered the initial-boundary value problem
(1.10)utt+Δ2u+|ut|m-2ut=|u|p-2u,x∈Ω,t>0,u(x,t)=∂νu(x,t)=0,x∈∂Ω,t≥0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
established an existence result for (1.10), and showed that the solution continues to exist globally if m≥p, however if m<p and the initial energy is negative, the solution blows up in finite time. Chen and Zhou [4] proved that the solution of (1.10) blows up with positive initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing initial energy under the condition m=2 by different method.The Petrovsky type equations with memory arouse the attention of mathematicians to study them. Alabau-Boussouira et al.   [5] discussed the initial-boundary value problem of linear Petrovsky equation related to a plate model with memory,
(1.11)utt+Δ2u-∫0t‍g(t-s)Δ2u(t,s)ds=0inΩ×(0,∞),u=∂νu=0on∂Ω×(0,∞),u|t=0=u0,ut|t=0=u1inΩ,
and showed that the solution decays exponentially or polynomially as t→+∞ if the initial data is sufficient small. Yang [6] considered the problem in N-dimensional space,
(1.12)utt+Δ2u+λut=∑i=1N‍∂∂xiσi(uxi)inΩ×(0,∞),u|∂Ω=0,∂u∂n|∂Ω=0on[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
and proved that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→∞ in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. Muñoz Rivera et al. [7] considered the initial-boundary problem for viscoelastic plate equation,
(1.13)utt-γΔutt+Δ2u(t)-∫0t‍g(t-τ)Δ2u(τ)dτ=0inΩ×(0,∞),u(x,y,0)=u0(x,y),ut(x,y,0)=u1(x,y)inΩ,u=∂νu=0onΓ0×(0,∞),ℬ1u(t)+ℬ1∫0∞‍g(s)ut(s)ds=0onΓ1×(0,∞),ℬ2u(t)-γ∂νu′′(t)+ℬ2∫0∞‍g(s)ut(s)ds=0onΓ1×(0,∞),
and proved that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel g satisfies
(1.14)g(t)≤-c0g1+(1/p)(t),g,g1+(1/p)∈L1(R)withp>2,
then the energy decays as 1/(1+t)p. On the recently related papers concerning the Petrovsky type models, the readers can see references [8–12].In [13–15], Li et al. proved the existence uniqueness, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system, respectively. To our best knowledge, we do not find the research report on the problem (1.1) which is considered in this paper.Motivated by the above work, we obtain the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity. The main contribution of this paper are as follows. (a) The problem considered in this paper is nonlinear equation with integral dissipation, to our knowledge this model has not been considered; (b) the hypothesis onh and initial data are weaker; (c) we naturally define the energy by simple computation and only define simple auxiliary functionals to prove our result by precise priori estimates.The outline of this paper is the following. In Section2, we present some material needed to be proved. Section 3 contains the statement and the proof of our results.


## 2. Notations and Preliminaries

In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue spaceLp(Ω) and Sobolev space Hm(Ω) and adopt the following notations
(2.1)(u,v):=(u,v)L2(Ω),〈u,v〉:=(u,v)L2(Γ1),∥u∥p:=∥u∥Lp(Ω),∥u∥:=∥u∥L2(Ω).
We denote
(2.2)|A(t)|2:=∑i,j‍aij2(t),ddtA(t):=A′(t)=(aij′(t)),A(t)·B(t):=∑i,j‍aij(t)bij(t),(A(t),B(t)):=∑i,j‍(aij(t),bij(t))
for any pair of tensors A(t)=(aij(t)) and B(t)=(bij(t)). We introduce the following space
(2.3)W(Ω)={w∈H2(Ω),w=∂νw=0onΓ0}.
Define the bilinear form a(·,·) as follows
(2.4)a(w,v)=∫Ω‍[wxxvxx+wyyvyy+μ(wxxvyy+wyyvxx)+2(1-μ)wxyvxy]dxdy.
For simplicity, we denote a(w,w) by a(w). For the relaxation function h(t), we assume that(A1)
h:[0,+∞)→(0,+∞) is a C2 function satisfying
(2.5)h(t)>0,h′(t)<0,h′′(t)≥0.(A2)
Bothh∞:=h(∞) and h∞′:=h'(∞) exist, and
(2.6)h∞>0,h∞′=0,h(0)=1.Hypotheses (A1) assure that the viscoelastic energy (defined below) is nonincreasing and the assumption (A2) means that the material behaves like an viscoelastic solid at t=+∞ (cf. [16]).Our results are based on the following existence theorem.Theorem 2.1.
One assumes thath(t) satisfies conditions (A1)-(A2). For any initial data
(2.7)(w0,w1)∈H3(Ω)×H2(Ω)
subject to the compatibility conditions satisfied on the boundary Γ,
(2.8)∇w0=∇w1=0,w1=w0=0onΓ0,ℬ1(w0+∫0∞‍h′(s)χ(-s)ds)=0onΓ1.
Then for any T>0, there exists a unique global solution to (1.1)–(1.10) satisfying
(2.9)(w,w′,w′′)∈L∞(0,T;H3(Ω)×H2(Ω)×H1(Ω)).Proof.
We can use the method used in [13, 17] to show the existence and uniqueness as well as the regularity of the global solution.At First, the energy of the system must be properly defined. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression of energy functional, using assumption (A2), we rewrite system (1.1) and (1.5) in the form
(2.10)w′′(t)-γΔw′′(t)+h∞Δ2w(t)+Δ2∫0∞‍h′(s)[wt(s)-w(t)]ds=div[𝒞(σ(t))∇w(t)]inΩ×(0,∞),(2.11)h∞ℬ1w(t)+ℬ1∫0∞‍h′(s)[wt(s)-w(t)]ds=0onΓ1×(0,∞),h∞ℬ2w(t)-γ∂νw′′(t)+ℬ2∫0∞‍h′(s)[wt(s)-w(t)]ds+𝒞(σ(t))ν·∇w(t)=0onΓ1×(0,∞).In order to define the energy functional, we give the following lemma.Lemma 2.2.
Letw and v be the functions in H4(Ω)∩W. Then, one has
(2.12)∫Ω‍(Δ2w)vdxdy=a(w,v)+∫Γ1‍[(ℬ2w)v-(ℬ1w)∂νv]dΓ.Proof..
The definition ofa(w,v) gives
(2.13)∫Ω‍ΔwΔvdxdy=a(w,v)+∫Ω‍[(1-μ)(wxxvyy+wyyvxx)-2(1-μ)wxyvxy]dxdy.
Using Green’s formula, we see
(2.14)∫Ω‍(Δ2w)vdxdy=∫Γ1‍(∂νΔw)vdΓ-∫Γ1‍Δw∂νvdΓ+∫Ω‍ΔwΔvdxdy=∫Γ1‍(∂νΔw)vdΓ-∫Γ1‍Δw∂νvdΓ+a(w,v)+∫Ω‍[(1-μ)(wxxvyy+wyyvxx)-2(1-μ)wxyvxy]dxdy,(2.15)∫Ω‍[(wxxvyy+wyyvxx)-2wxyvxy]dxdy=∫Γ1‍(wxxvyν2+wyyvxν1)dΓ-∫Ω‍(wxxyvy+wxyyvx)dxdy-∫Γ1‍(wxyvyν1+wxyvxν2)dΓ+∫Ω‍(wxxyvy+wxyyvx)dxdy=∫Γ1‍(wxxvyν2+wyyvxν1)dΓ-∫Γ1‍(wxyvyν1+wxyvxν2)dΓ.
Using
(2.16)∂νv=vxν1+vyν2,∂τv=-vxν2+vyν1,
we get
(2.17)vx=∂νvν1-∂τvν2,vy=∂νvν2+∂τvν1.
Inserting (2.14) and (2.15) into (2.17) and noting
(2.18)∫Γ1‍∂τ(wv)dΓ=∫Γ‍∂τ(wv)dΓ=∫Γ‍(wv)xdx+(wv)ydy=∫Ω‍[(wv)yx-(wv)xy]dxdy=0,
we have
(2.19)∫Γ1‍w∂τvdΓ=-∫Γ1‍v∂τwdΓ,
we obtain the conclusion.In order to define the energy functionE(t) of the problem (1.1)–(1.8), we give the following computations. Multiplying equation (2.10) by w′, integrating the result over Ω and adding Green’s formula, we get from Lemma 2.2 and (1.4) that
(2.20)∫Ω‍w′′(t)w′(t)dxdy+γ∫Ω‍∇w′′(t)·∇w′(t)dxdy+h∞a(w(t),w′(t))+∫Γ1‍[w′(t)(h∞ℬ2w(t)-γ∂νw′′(t))-h∞∂νw′(t)ℬ1w(t)]dΓ+∫0∞‍h′(s)(Δ2wt(s)-Δ2w(t),w′(t))ds=∫Γ1‍𝒞(σ(t))∇w(t)·νdΓ-∫Ω‍𝒞(σ(t))∇w(t)·∇w′(t)dxdy=∫Γ1‍𝒞(σ(t))∇w(t)·νdΓ-∫Ω‍𝒞(σ(t))·∇w′(t)⊗∇w(t)dxdy.
Clearly
(2.21)(Δ2wt(s)-Δ2w(t),w′(t))=-(Δ2wt(s)-Δ2w(t),w′t(s)-w′(t))+(Δ2w′t(s)-Δ2w(t),w′t(s)).
By applying Lemma 2.2, the first term on the right-hand side of (2.21) equals
(2.22)-(Δ2wt(s)-Δ2w(t),w′t(s)-w′(t))=-12∂ta(wt(s)-w(t))-∫Γ1‍[(w′t(s)-w′(t))ℬ2(wt(s)-w(t))-∂ν(w′t(s)-w′(t))ℬ1(wt(s)-w(t))]dΓ.
In the same way, the second term on the right-hand side of (2.21) equals
(2.23)(Δ2w′t(s)-Δ2w(t),w′t(s))=-12∂sa(wt(s)-w(t))+∫Γ1‍[w′t(s)ℬ2(wt(s)-w(t))-∂ν(w′t(s))ℬ1(wt(s)-w(t))]dΓ.
Therefore, from (2.21)–(2.23), we have
(2.24)(Δ2wt(s)-Δ2w(t),w′(t))=-12∂ta(wt(s)-w(t))-12∂sa(wt(s)-w(t))+∫Γ1‍[w′(t)ℬ2(wt(s)-w(t))-∂νw′(t)ℬ1(wt(s)-w(t))]dΓ.
Note
(2.25)-12∫0∞‍h′(s)∂ta(wt(s)-w(t))ds=-12ddt∫0∞‍h′(s)a(wt(s)-w(t))ds.
and (by h∞′=0)
(2.26)-12∫0∞‍h′(s)∂sa(wt(s)-w(t))ds=-h′(s)a(wt(s)-w(t))|0∞+12∫0∞‍h′′(s)a(wt(s)-w(t))ds=12∫0∞‍h′′(s)a(wt(s)-w(t))ds.
Consequently, we conclude form (2.24)–(2.26) that
(2.27)∫0∞‍h′(s)(Δ2wt(s)-Δ2w(t),w′(t))ds=-12ddt∫0∞‍h′(s)a(wt(s)-w(t))ds+12∫0∞‍h′′(s)a(wt(s)-w(t))ds+∫0∞‍h′(s)∫Γ1‍[w′(t)ℬ2(wt(s)-w(t))+12∫0∞‍h′′(s)∫0∞‍-∂νw′(t)ℬ1(wt(s)-w(t))]dΓds.
Summing (2.20) and (2.27), using the boundary conditions (2.11) together with the symmetric of σ(t) yields
(2.28)12ddt(∥w′(t)∥2+γ∥∇w′(t)∥2+h∞a(w(t))-∫0∞‍h′(s)a(wt(s)-w(t))ds)+∫Ω‍𝒞(σ(t))·∇w′(t)⊗∇w(t)dxdy+12∫0∞‍h′′(s)a(wt(s)-w(t))ds=0.
Note the relation
(2.29)∫Ω‍𝒞(σ(t))·∇w′(t)⊗∇w(t)dxdy=14ddt∫Ω‍𝒞(σ(t))·σ(t)dxdy.
In fact
(2.30)∫Ω‍𝒞(σ(t))·∇w′(t)⊗∇w(t)dxdy=α∫Ω‍(wxwx+μwywy(1-μ)wxwy(1-μ)wywxwywy+μwxwx)·(wx′wxwx′wywy′wxwy′wy)dxdy=α∫Ω‍[(wx)3wx′+(wy)3wy′+(wy)2wxwx′+(wx)2wywy′]dxdy
with α=E/d(1-μ2). Denote σ:=(σ11σ12σ21σ22). Hence
(2.31)𝒞(σ)=α(σ11+μσ22(1-μ)σ12(1-μ)σ21σ22+μσ11),[𝒞(σ(t))]′·σ(t)=α[σ11′σ11+σ22′σ22+μ(σ11′σ22+σ22′σ11)+(1-μ)(σ12′σ12+σ21′σ21)]=𝒞(σ(t))·[σ(t)]′,
which implies that
(2.32)ddt∫Ω‍𝒞(σ(t))·σ(t)dxdy=∫Ω‍[𝒞(σ(t))]′·σ(t)dxdy+∫Ω‍𝒞(σ(t))·σ′(t)dxdy=2∫Ω‍[𝒞(σ(t))]′·σ(t)dxdy=4α∫Ω‍[(wx)3wx′+(wy)3wy′+(wy)2wxwx′+(wx)2wywy′]dxdy.
From (2.30) and (2.32), we conclude that (2.29) holds. Combining (2.28) and (2.29), we conclude that
(2.33)12ddt[∫0∞‍∥w′(t)∥2+γ∥∇w′(t)∥2+h∞a(w(t))-∫0∞‍h′(s)a(wt(s)-w(t))ds+12∫Ω‍𝒞(σ(t))·σ(t)dxdy]=-12∫0∞‍h′′(s)a(wt(s)-w(t))ds.
The relation (2.33) inspires us to define energy functional as the following
(2.34)E(t)=12(∥w′(t)∥2+γ∥∇w′(t)∥2)+12(h∞a(w(t))-∫0∞‍h′(s)a(wt(s)-w(t))ds+12∫Ω‍𝒞(σ(t))·σ(t)dxdy).From (2.33) and (2.34), we obtain the following lemma.Lemma 2.3.
Under the above notations and assumptions (A1) and (A2), one has that
(2.35)ddtE(t)=-12∫0∞‍h′′(s)a(wt(s)-w(t))ds≤0.


## 3. Main Results

In this section,c,c¯,c~, and ci denote some positive constants. Here, we need to point out that δ denotes the small enough different positive constant and Cδ denotes the different positive constant depending on δ in a different place, respectively. The main goal of this paper is to show, respectively, that the solution decays exponentially or polynomially to zero as the time goes to infinity under suitable conditions. Our main results are formulated below.Theorem 3.1.
Letw be the global solution of the problem (1.1)–(1.8) with the conditions
(3.1)h′(t)≤ch′′′(t),-c3h′(t)≤h′′(t)≤-c4h′(t).
Then the energy functional E(t) decays exponentially to zero as the time goes to infinity.To prove our main result, we will give some important preliminaries.Lemma 3.2.
Assumew∈W(Ω), functional a(w) and tensor σ(t) are defined as above. Then(1)
c1∥∇2w∥2≤a(w)≤c2∥∇2w∥2(2)
∥∇w∥Lq2≤c3a(w),q≥2(3)
∥w∥2≤c4a(w)(4)
c5|σ(t)|2≤𝒞(σ(t))·σ(t)≤c6|σ(t)|2.Proof.
(
1
) In fact
(3.2)a(w)=∫Ω‍{(wxx)2+(wyy)2+2μwxxwyy+2(1-μ)(wxy)2}dxdy≤∫Ω‍{(wxx)2+(wyy)2+μ[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy=∫Ω‍{(1+μ)[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy,(3.3)a(w)=∫Ω‍{(wxx)2+(wyy)2+2μwxxwyy+2(1-μ)(wxy)2}dxdy≥∫Ω‍{(wxx)2+(wyy)2-μ[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy=∫Ω‍{(1-μ)[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy,(3.4)∥∇2w∥2=(∇2w,∇2w)=∫Ω‍[(wxx)2+(wyy)2+2(wxy)2]dxdy.
Combining (3.2)–(3.4) and noting 0<μ<1/2, we finish the desire conclusion (1).
(
2
) By H1(Ω)↪Lq(Ω)(q≥2), the definition of W(Ω), and Poincaré inquality, we know
(3.5)∥∇w(t)∥Lq(Ω)2≤c∥w(t)∥H1(Ω)2≤c∥w(t)∥W(Ω)2≤c3a(w).
(
3
) Using Poincaré inequality, we get
(3.6)∥w(t)∥≤C*∥∇w(t)∥,
which with conclusion (2) shows the desire result.
(
4
) Denote σ:=(σ11σ12σ21σ22). Hence
(3.7)𝒞(σ)=α(σ11+μσ22(1-μ)σ12(1-μ)σ21σ22+μσ11),
with α=E/d(1-μ2)>0, and
(3.8)𝒞(σ)·σ=α[σ112+σ222+2μσ11σ22+(1-μ)(σ122+σ212)],
which imply
(3.9)α[(1-μ)(σ112+σ222)+(1-μ)(σ122+σ212)]≤𝒞(σ)·σ≤α[(1+μ)(σ112+σ222)+(1-μ)(σ122+σ212)].
From (3.9) and noting 0<μ<1/2, we obtain the desire conclusion (4).
The proof is completed.Lemma 3.3.
Supposew,v∈W and a(w,v) is defined as above, then
(3.10)a(w,v)≤δa(w,w)+94δa(v,v).Proof.
In fact, forη>0 small enough,
(3.11)|a(w,v)|≤∫Ω‍|wxxvxx+wyyvyy+μ(wxxvyy+wyyvxx)+2(1-μ)wxyvxy|dxdy≤η∫Ω‍{(1+μ)[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy+14η∫Ω‍{(1+μ)[(vxx)2+(vyy)2]+2(1-μ)(vxy)2}dxdy≤1+μ1-μη∫Ω‍{(1-μ)[(wxx)2+(wyy)2]+2(1-μ)(wxy)2}dxdy+1+μ1-μ14η∫Ω‍{(1-μ)[(vxx)2+(vyy)2]+2(1-μ)(vxy)2}dxdy.
Denote δ=((1+μ)/(1-μ))η, then ((1+μ)/(1-μ))(1/4η)=((1+μ)/(1-μ))2(1/4δ). Owing to 1<(1+μ)/(1-μ)<3, from (3.3) and (3.11), we obtain the conclusion.Lemma 3.4.
Define the functional(3.12)φ(t):=12∫Ω‍[w(t)w′(t)+γ∇w′(t)·∇w(t)]dxdy,
then
(3.13)ddtφ(t)≤12(∥w′(t)∥2+γ∥∇w′(t)∥2)-12(1-δ+∫0∞‍h′(s)ds)a(w)-∫Ω‍𝒞(σ(t))·σ(t)dxdy+98δ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.Proof.
Multiplying (1.1) with w and integrating the result over Ω, we obtain
(3.14)∫Ω‍(w′′(t)-γΔw′′(t)+Δ2w+Δ2∫0∞‍h′(s)wt(s)ds-div[𝒞(σ(t))∇w(t)])wdxdy=0,
which with Lemma 2.2 and (1.5) implies
(3.15)12ddt∫Ω‍[w(t)w′(t)+γ∇w′(t)·∇w(t)]dxdy=12(∥w′(t)∥2+γ∥∇w′(t)∥2)+12∫Ω‍[w(t)w′′(t)+γ∇w(t)·∇w′′(t)]dxdy=12(∥w′(t)∥2+γ∥∇w′(t)∥2)+12∫Ω‍w(t)(w′′(t)-γΔw′′(t))dxdy+12∫Γ1‍w∂νw′′(t)dΓ=12(∥w′(t)∥2+γ∥∇w′(t)∥2)-12∫Ω‍w(Δ2w+Δ2∫0∞‍h′(s)wt(s)ds-div[𝒞(σ(t))∇w(t)]∫0∞‍)dxdy+12∫Γ1‍w∂νw′′(t)dΓ=12(∥w′(t)∥2+γ∥∇w′(t)∥2)-12∫Ω‍𝒞(σ(t))·∇w(t)⊗∇w(t)dxdy-12a(w)-12a(∫0∞‍h′(s)wt(s)ds,w)-12∫Γ1‍w(t)ℬ2(w+∫0∞‍h′(s)wt(s))dΓ+12∫Γ1‍∂νw(t)ℬ1(w+∫0∞‍h′(s)wt(s))dΓ+12∫Γ1‍[𝒞(σ(t))ν·∇w]wdΓ+12∫Γ1‍w∂νw′′(t)dΓ=12(∥w′(t)∥2+γ∥∇w′(t)∥2)-12∫Ω‍𝒞(σ(t))·∇w(t)⊗∇w(t)dxdy-12a(w)-12a(∫0∞‍h′(s)wt(s)ds,w)=12(∥w′(t)∥2+γ∥∇w′(t)∥2)-∫Ω‍𝒞(σ(t))·σ(t)dxdy-12(1+∫0∞‍h′(s)ds)a(w)-12a(w,∫0∞‍h′(s)(wt(s)-w(t))ds).
By Lemma 3.3, we know
(3.16)|12a(w(t),∫0∞‍h′(s)(wt(s)-w(t))ds)|≤12δa(w(t))+98δ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.
Using (3.15)-(3.16), we get the conclusion.Lemma 3.5.
Supposew is the solution of (1.1)–(1.8) and (A1), (A2), and (3.1) hold, then(1)
(∫0∞‍h′(s)wt(s)ds)′=h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds=∫0∞‍h′′(s)[wt(s)-w(t)]ds;(2)
h∞′′=0;(3)
(∫0∞‍h′′(s)wt(s)ds)′=h′′(0)w(t)+∫0∞‍h′′′(s)wt(s)ds=∫0∞‍h′′′(s)[wt(s)-w(t)]ds.Proof.
(1) By the formula of integral by part and (A2), we deduce
(3.17)(∫0∞‍h′(s)wt(s)ds)′=∫0∞‍h′(s)∂twt(s)ds=-∫0∞‍h′(s)∂swt(s)ds=-h′(s)wt(s)|0∞+∫0∞‍h′′(s)wt(s)ds=h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds=∫0∞‍h′′(s)[wt(s)-w(t)]ds.
(2) Using (A2) and (3.1), we infer the conclusion.
(3) Byh′′∞=0 and the formula of integral by part, we get the conclusion (3).
The proof is completed.Lemma 3.6.
Define the functional(3.18)ψ(t)=γ∫Γ1‍∂νw′(t)(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dΓ+∫Ω‍(w′(t)-γΔw′(t))(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+12a(∫0∞‍h′(s)wt(s)ds).
then for some δ(δ>0) small enough, there exist Cδ>0 and T(T>0) as t>T the following inequality holds
(3.19)ddtψ(t)≤h′(0)2(∥w′(t)∥2+γ∥∇w′(t)∥2)+δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.Proof.
Multiplying (1.1) with h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds and integrating the result over Ω, we have
(3.20)∫Ω‍(w′′(t)-γΔw′′(t)+Δ2w+Δ2∫0∞‍h′(s)wt(s)ds-div[𝒞(σ(t))∇w(t)])·(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy=0,
which with Lemma 3.5 gives
(3.21)ddt∫Ω‍(w′(t)-γΔw′(t))(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy=∫Ω‍(w′′(t)-γΔw′′(t))(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+∫Ω‍(w′(t)-γΔw′(t))[h′(0)w′(t)+(∫0∞‍h′′(s)wt(s)ds)′]dxdy=-∫Ω‍(Δ2w+∫0∞‍h′(s)Δ2wt(s)ds-div[𝒞(σ(t))∇w(t)])×(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+∫Ω‍(w′(t)-γΔw′(t))(h′(0)w′(t)+h′′(0)w(t)+∫0∞‍h′′′(s)wt(s)ds)dxdy.
By Green formula, we have
(3.22)∫Ω‍(w′(t)-γΔw′(t))(h′(0)w′(t)+h′′(0)w(t)+∫0∞‍h′′′(s)wt(s)ds)dxdy=h′(0)(∥w′(t)∥2+γ∥∇w′(t)∥2)-γ∫Γ1‍∂νw′(t)×(h′(0)w′(t)+h′′(0)w(t)+∫0∞‍h′′′(s)wt(s)ds)dΓ+γ∫Ω‍∇w′(t)·(h′′(0)∇w(t)+∫0∞‍h′′′(s)∇wt(s)ds)dxdy+∫Ω‍w′(t)(h′′(0)w+∫0∞‍h′′′(s)wt(s)ds)dxdy.
Clearly
(3.23)γ∫Γ1‍∂νw′(t)(h′(0)w′(t)+h′′(0)w(t)+∫0∞‍h′′′(s)wt(s)ds)dΓ=γddt∫Γ1‍∂νw′(t)(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dΓ-γ∫Γ1‍∂νw′′(t)(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dΓ.
Inserting (3.22) and (3.23) into (3.21) yields
(3.24)ddt∫Ω‍(w′(t)-γΔw′(t))(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+γddt∫Γ1‍∂νw′(t)(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dΓ=-∫Ω‍(Δ2w+∫0∞‍h′(s)Δ2wt(s)ds-div[𝒞(σ(t))∇w(t)])×(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+h′(0)(∥w′(t)∥2+γ∥∇w′(t)∥2)+γ∫Γ1‍∂νw′′(t)(∫0∞h′′(s)‍[wt(s)-w(t)])dΓ+γ∫Ω‍∇w′(t)·(h′′(0)∇w(t)+∫0∞‍h′′′(s)∇wt(s)ds)dxdy+∫Ω‍w′(t)(h′′(0)w+∫0∞‍h′′′(s)wt(s)ds)dxdy.
Applying Lemma 2.2, Lemma 3.5 and noting a(w,v)=a(v,w), we have
(3.25)-∫Ω‍(Δ2w+Δ2∫0∞‍h′(s)wt(s)ds-div[𝒞(σ(t))∇w(t)])×(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy=-a(w,∫0∞‍h′′(s)[wt(s)-w(t)]ds)-12ddta(∫0∞‍h′(s)wt(s)ds,∫0∞‍h′(s)wt(s)ds)-∫Ω‍𝒞(σ(t))∇w(t)·(∫0∞‍h′′(s)[∇wt(s)-∇w(t)]ds)dxdy-∫Γ1‍[(ℬ2w)(∫0∞‍h′′(s)[wt(s)-w(t)]ds)(ℬ1w)∂ν-(ℬ1w)∂ν(∫0∞‍h′′(s)[wt(s)-w(t)]ds)]dΓ-∫Γ1‍(ℬ2∫0∞‍h′(s)wt(s)ds-(C(σ(t))ν·∇w(t)))(∫0∞‍h′′(s)[wt(s)-w(t)]ds)dΓ-∫Γ1‍(ℬ1∫0∞‍h′(s)wt(s)ds)∂ν(∫0∞‍h′′(s)[wt(s)-w(t)]ds)dΓ.
Inserting (3.25) into (3.24) and owing to (1.5) yield
(3.26)ddtψ(t)=ddt∫Ω‍(w′(t)-γΔw′(t))(h′(0)w(t)+∫0∞‍h′′(s)wt(s)ds)dxdy+12ddta(∫0∞‍h′(s)wt(s)ds)+γddt∫Γ1‍∂νw′(t)(h′(0)w(t)+∫0∞‍h′′wt(s)ds)dΓ=h′(0)(∥w′(t)∥2+γ∥∇w′(t)∥2)-∫Ω‍𝒞(σ(t))∇w(t)·(∫0∞‍h′′(s)[∇wt(s)-∇w(t)]ds)dxdy-a(w,∫0∞‍h′′(s)[wt(s)-w(t)]ds)+γ∫Ω‍∇w′(t)·(∫0∞‍h′′′(s)[∇wt(s)-∇w(t)]ds)dxdy+∫Ω‍w′(t)(∫0∞‍h′′′(s)[∇wt(s)-∇w(t)]ds)dxdy.
Now, we estimate some terms on the right-hand side of (3.26). The Lemma 3.3 and (3.1) imply
(3.27)|a(w,∫0∞‍h′′(s)[wt(s)-w(t)]ds)|≤δa(w,w)+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.
Using Cauchy inequality, Hölder inequality, and Lemma 3.2 and noting (3.1), we obtain
(3.28)|γ∫Ω‍∇w′(t)·(∫0∞‍h′′′(s)[∇wt(s)-∇w(t)]ds)dxdy|≤δγ∥∇w′(t)∥2+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds,|∫Ω‍w′(t)(∫0∞‍h′′′(s)[∇wt(s)-∇w(t)]ds)dxdy|≤δ∥w′(t)∥2+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.
Applying Hölder inequality, Lemma 3.2, and Lemma 2.3, we conclude
(3.29)|∫Ω‍𝒞(σ(t))∇w(t)·(∫0∞‍h′′(s)[∇wt(s)-∇w(t)]ds)dxdy|=|∫Ω‍𝒞(σ(t))·(∫0∞‍h′′(s)∇w(t)⊗[∇wt(s)-∇w(t)]ds)dxdy|≤δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+Cδ∫0∞‍|h′(s)|∥∇w(t)∥2∥∇wt(s)-∇w(t)∥2ds≤δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+a(w(t))Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds≤δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+1h∞E(0)C¯δ∫0∞‍|h′(s)|a(wt(s)-w(t))ds=δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.
Combining (3.26)–(3.29), we conclude that
(3.30)ddtψ(t)≤h′(0)2(∥w′(t)∥2+γ∥∇w′(t)∥2)+δ∫Ω‍𝒞(σ(t))·σ(t)dxdy+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds.
The proof is completed.proof of Theorem3.1.
Let(3.31)F(t)=PE(t)+Qφ(t)+Rψ(t).
By Lemma 2.3, Lemma 3.4, and Lemma 3.6 and noting (3.1), we have
(3.32)F′(t)≤-P12∫0∞‍h′′(s)a(wt(s)-w(t))ds+Q12(∥w′(t)∥2+γ∥∇w′(t)∥2)-Q12(1-δ+∫0∞‍h′(s)ds)a(w(t))-Q∫Ω‍𝒞(σ(t))·σ(t)dxdy+Q98δ∫0∞‍|h′(s)|a(wt(s)-w(t))ds+Rh′(0)2(∥w′(t)∥2+γ∥∇w′(t)∥2)+Rδ∫Ω‍𝒞(σ(t))·σ(t)dxdy+RCδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds≤K1(∥w′(t)∥2+γ∥∇w′(t)∥2)+K2a(w(t))+K3∫Ω‍𝒞(σ(t))×σ(t)dxdy+K4∫0∞‍|h′(s)|a(wt(s)-w(t))ds,
with
(3.33)K1=12Rh′(0)+12Q,K2=-12Q(1-δ+∫0∞‍h′(s)ds)=-12Q(h∞-δ),K3=-Q+Rδ,K4=-12Pc3+9Q8δ+RCδ.
Fixing R=1,δ=(1/2)min{h∞,-h'(0)}>0,Q=(1/2)(δ-h′(0))>0,P>9Q/4δc3+2(Cδ/c3)>0, we obtain Ki<0,i=1,2,3,4. Under the conditions, we obtain from (2.34) and (3.32) that there exists a positive α>0 such that
(3.34)ddtF(t)≤-αE(t).
Fixing a large enough positive constant P, we prove F(t) and E(t) are equivalent, that is,
(3.35)M1E(t)≤F(t)≤M2E(t)
with M1,M2>0. Indeed, using Cauchy inequality, Lemma 3.2, and (2.34), we have
(3.36)|Rφ(t)|=|Q2∫Ω‍(w(t)w′(t)+γ∇w′(t)·∇w(t))dxdy|≤Q4[∥w(t)∥2+γ∥∇w(t)∥2+∥w′(t)∥2+γ∥∇w′(t)∥2]≤c¯a(w(t))+c~E(t)≤cE(t).
Using the definition of ψ(t) and Green’s formula, we infer
(3.37)ψ(t)=∫Ω‍w′(t)(∫0∞‍h′′(s)[wt(s)-w(t)]ds)dxdy+∫Ω‍γ∇w′(t)·(∫0∞‍h′′(s)[∇wt(s)-∇w(t)]ds)dxdy+12a(∫0∞‍h′(s)wt(s)ds).
Applying Cauchy inequality, Lemma 3.2 and noting (3.1), we conclude that
(3.38)∫Ω‍w′(t)(∫0∞‍h′′(s)[wt(s)-w(t)]ds)dxdy+∫Ω‍γ∇w′(t)·(∫0∞‍h′′(s)[∇wt(s)-∇w(t)]ds)dxdy≤δ∥w′(t)∥2+γδ∥∇w′(t)∥2+Cδ∫0∞‍|h′(s)|a(wt(s)-w(t))ds,(3.39)12a(∫0∞‍h′(s)wt(s)ds)≤C∥∇2(∫0∞‍(h(0)-h∞)w(t)ds+∫0∞‍h′(s)[wt(s)-w(t)])∥2≤ca(w(t))+c∫0∞‍|h′(s)|(wt(s)-w(t))ds.
Relations (3.37)-(3.39) imply that
(3.40)|ψ(t)|≤cE(t).
Fixing a large enough positive constant P, inequalities (3.36) and (3.40) show (3.35) hold. Relations (3.34) and (3.35) imply that there exists a positive constant β>0 such that
(3.41)ddtF(t)≤-βF(t),
which with Gronwall’s inequality gives
(3.42)F(t)≤F(0)e-βt.
Combining (3.35) and (3.42), we finish the proof.Theorem 3.7.
Assume thatw is the global solution of (1.1)–(1.8) with condition C|h′(t)|1+(1/p)≤h′′t),p>2. then for some T>0 there exists a positive constant C, such that
(3.43)E(t)≤C(1+t)p/2,∀t≥T.In order to prove Theorem3.7, we quote the following technical lemma.Lemma 3.8.
Suppose thatw∈L∞(0,∞;H2(Ω)) and g is a continuous function and there exists 0<θ<1, such that
(3.44)∫0∞‍g1-θ(s)ds<∞.
Then, there exists C>0 such that
(3.45)∫0∞‍g(s)a(wt(s)-w(t))ds≤C{(∫0∞‍g1-θ(s)ds)∥v∥L∞(0,T;H2(Ω))2}1/(θp+1)×{∫0∞‍g1+(1/p)(s)a(wt(s)-w(t))dxdy}θp/(θp+1).Proof.
Applying Hölder inequality, we obtain(3.46)∫0∞‍g(s)a(wt(s)-w(t))ds≤{∫0∞‍|g(s)|1-θa(wt(s)-w(t))ds}1/(θp+1)×{∫0t‍|g(s)|1+(1/p)a(wt(s)-w(t))ds}θp/(θp+1)≤C{(∫0∞‍g1-θ(s)ds)∥w∥L∞(0,T;H2(Ω))2}1/(θp+1)×{∫0∞‍g1+(1/p)(s)a(wt(s)-w(t))ds}θp/(θp+1).
This completes the proof of Lemma3.8.Proof of Theorem3.7.
From the hypothesisc|h′(t)|1+(1/p)≤h′′(t), we have [|h′|-1/p(t)]′≥c/p. Integrating the above inequality over [0,t], we get
(3.47)|h′|(t)≤c7(1+t)-p,c7>0.
Fixing θ=1/2 in Lemma 3.8, then (1-θ)p=p/2>1, hence we obtain
(3.48)∫0∞‍|h′|1-θ(s)ds≤c8∫0∞‍1(1+s)p/2ds<∞.
Using this estimate into (3.46), noting (2.34) and Lemma 3.2, we get
(3.49)∫0∞‍|h′|(s)a(wt(s)-w(t))ds≤c9E2/(p+2)(0)(∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds)p/(p+2).
Denote G(t)=Qφ(t)+ψ(t). Applying (3.32) and (3.49), we arrive at
(3.50)ddtG(t)≤-α1E(t)+α2∫0∞‍|h′|(s)a(wt(s)-w(t))ds≤-α1E(t)+α2c9E2/(p+2)(0)(∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds)p/(p+2).
Since |G(t)|≤cE(t), applying Young inequality and from Lemma 3.2, (3.50), we deduce
(3.51)ddt[E2/p(t)G(t)]=2pG(t)E(2/p)-1(t)ddtE(t)+E2/p(t)ddtG(t)≤-2pcE2/p(t)ddtE(t)+E2/p(t)ddtG(t)≤-c10ddtE1+(2/p)(t)-α1E1+(2/p)(t)+α2C2E2/(p+2)(0)E2/p(t)(∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds)p/(p+2)≤-c10ddtE1+(2/p)(t)-α1E1+(2/p)(t)+α2C2E2/(p+2)(0)εE1+(2/p)(t)+α2C2E2/(p+2)(0)Cε∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds.
Since c|h'|1+(1/p)(t)≤h′′(t), then
(3.52)∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds≤1c∫0∞‍h′′(s)a(wt(s)-w(t))ds.
By Lemma 3.2, we have
(3.53)∫0∞‍h′′(s)a(wt(s)-w(t))ds=-2ddtE(t).
Hence, we get
(3.54)∫0∞‍|h′|1+(1/p)a(wt(s)-w(t))ds≤-c11ddtE(t).
Taking ε small enough, from (3.51) and (3.54), we get
(3.55)ddt[E2/p(t)G(t)]≤-c10ddtE1+(2/p)(t)-α12E1+(2/p)(t)-c12ddtE(t).
Thus, we have
(3.56)ddt[E2/p(t)(G(t)+c10E(t))]≤-c12ddtE(t)-α12E1+(2/p)(t).
Letλ>0 be a positive constant and define
(3.57)H(t)=λE(t)+E2/p(t)[G(t)+c10E(t)].
Since |G(t)|≤cE(t),(d/dt)E(t)≤0, for λ large enough, we get
(3.58)λ2E(t)≤H(t)≤2λE(t),∀t≥0.
From Lemma 3.2 and (3.56), taking λ large enough, we obtain
(3.59)ddtH(t)=λddtE(t)+ddt[E2/p(t)(G(t)+C0E(t))]≤λddtE(t)-c12ddtE(t)-α12E1+(2/p)(t)=(λ-c12)ddtE(t)-α12E1+(2/p)(t)≤-α12E1+(2/p)(t).
From (3.58) and (3.59), we have
(3.60)ddtH(t)≤-c13H1+(2/p)(t).
Applying Gronwall inequality, we get
(3.61)H(t)≤c14(1+t)p/2.
Hence, we obtain
(3.62)E(t)≤c15(1+t)p/2.
This completes the proof of Theorem3.7.


## 4. Comment

In this paper, the key point is to construct suitable energy functionalE(t), find the appropriate auxiliary functionals φ(t) and ψ(t) by multiplier method, and prove that F(t) and E(t) are equivalent and F(t) satisfies F'(t)≤-cF(t) by precise apriori estimates. Thanks to Lemma 3.8 and auxiliary functional H(t), we show the polynomial decay result under suitable conditions on relaxation h.



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*Source: 419717-2012-10-18.xml*