# Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

Authors: Jum-Ran Kang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain.


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## Body

## 1. Introduction

LetΩ be an open bounded domain of ℝ2 containing the origin and having C2 boundary. Let γ:[0,∞[→ℝ be a continuously differentiable function. Consider the family of subdomains {Ωt}0≤t<∞ of ℝ2 given by
(1)Ωt=T(Ω),T:y∈Ω⟶x=γ(t)y,
whose boundaries are denoted by Γt, and let Q^ be the noncylindrical domain of ℝ3 given by
(2)Q^=⋃0≤t<∞‍Ωt×{t}
with boundary
(3)Σ^=⋃0≤t<∞‍Γt×{t}.
In this paper, we consider the following Kirchhoff plates equations with viscoelastic boundary conditions:
(4)u′′+Δ2u=0inΩt×(0,∞),(5)u=∂u∂ν=0onΓ0,t×(0,∞),(6)-u+∫0t‍g1(t-s)ℬ2u(s)ds=0onΓ1,t×(0,∞),(7)∂u∂ν+∫0t‍g2(t-s)ℬ1u(s)ds=0onΓ1,t×(0,∞),(8)u(0,x)=u0(x),u′(0,x)=u1(x)inΩ0,
where ν=(ν1,ν2) is the unit normal at (σ,t)∈Σ^ directed towards the exterior of Q^. We divide the boundary into two parts:
(9)Γt=Γ0,t∪Γ1,twithΓ¯0,t∩Γ¯1,t=∅,Γ0,t≠∅.
We are denoting by ℬ1 and ℬ2 the following differential operators:
(10)ℬ1u=Δu+(1-μ)B1u,ℬ2u=∂Δu∂ν+(1-μ)∂B2u∂η,
where B1 and B2 are given by
(11)B1u=2ν1ν2ux1x2-ν12ux2x2-ν22ux1x1,B2u=(ν12-ν22)ux1x2+ν1ν2(ux2x2-ux1x1),
and the constant μ,0<μ<1/2, represents Poisson’s ratio. From the physics point of view, system (4) describes the small transversal vibrations of a thin plate with a moving boundary device. The integral equations (6) and (7) describe the memory effects which can be caused, for example, by the interaction with another viscoelastic element. The relaxation functions g1,g2∈C1(0,∞) are positive and nondecreasing.The uniform stabilization of plates equations with linear or nonlinear boundary feedback in cylindrical domain was investigated by several authors; see for example [1–3] among others. The uniform decay for viscoelastic plates with memory was studied by [4, 5] and the references therein. Santos et al. [6] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Santos and Junior [7] investigated the stability of solutions for Kirchhoff plate equations with boundary memory condition. Park and Kang [8] studied the exponential decay for the Kirchhoff plate equations with nonlinear dissipation and boundary memory condition. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation functions. But the existence of solutions and decay of energy for the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domain are not studied yet. In a moving domain, the transverse deflection u(x,t) of the thin plate which changes its configuration at each instant of time increases its deformation and hence increases its tension. Moreover, the horizontal movement of the boundary yields nonlinear terms involving derivatives in the space variables. To control these nonlinearities, we add in the boundary a memory type. This term will play an important role in the dissipative nature of the problem.In [9–17], the authors considered the global existence and the uniform decay of solution in noncylindrical domains. Dal Passo and Ughi [15] investigated a certain class of parabolic equations in noncylindrical domains. Benabidallah and Ferreira [9] proved the existence of solutions for the nonlinear beam equation in noncylindrical domains. Santos et al. [17] studied the global solvability and asymptotic behavior for the nonlinear coupled system of viscoelastic waves with memory in noncylindrical domains. Park and Kang [14] investigated the global existence and stability for von Karman equations with memory in noncylindrical domains. Motivated by these results, we prove the exponential decay of the energy to the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domains.This paper is organized as follows. In Section2, we recall notations and hypotheses. In Section 3, we prove the existence and uniqueness of solutions by employing Faedo-Galerkin’s method. In Section 4, we establish the exponential decay rate of the solution.


## 2. Notations and Hypotheses

We begin this section introducing notations and some hypotheses. Throughout this paper we use standard functional spaces and denote that∥·∥p,∥·∥p,t are Lp(Ω) norm and Lp(Ωt) norm. We define the inner product
(12)(u,v)=∫Ω‍u(x)v(x)dx,(u,v)t=∫Ωt‍u(x)v(x)dx.
Also, let us assume that there exists x0∈ℝ2 such that
(13)Γ0,t={x∈Γt:ν(x)·(x-x0)≤0},Γ1,t={x∈Γ:ν(x)·(x-x0)>0}.
The method used to prove the result of existence and uniqueness is based on the transformation of our problem into another initial boundary value problem defined over a cylindrical domain whose sections are not time dependent. This is done using a suitable change of variable. Then we show the existence and uniqueness for this new problem. Our existence result on noncylindrical domains will follow by using the inverse transformation. That is, by using the diffeomorphism
(14)τ:Q^⟶Q,(x,t)∈Ωt×{t}⟶(y,t)=(xγ(t),t)
and τ-1:Q→Q^ defined by
(15)τ-1(y,t)=(x,t)=(γ(t)y,t).
For each function u we denote by v the function
(16)v(y,t)=u∘τ-1(y,t)=u(x,t),
the initial boundary value problem (4)–(8) becomes
(17)v′′+γ-4Δ2v+A(t)v+b(y,t)·∇v+c(y,t)·∇v′=0v′′+γ-4Δ2v+A(t)v+b(y,t)·∇v+cninΩ×(0,∞),(18)v=∂v∂ν=0onΓ0×(0,∞),(19)-v+∫0t‍g1(t-s)γ-2(s)ℬ2v(s)ds=0onΓ1×(0,∞),(20)∂v∂ν+∫0t‍g2(t-s)γ-2(s)ℬ1v(s)ds=0onΓ1×(0,∞),(21)v(y,0)=v0(y),v′(y,0)=v1(y)inΩ,
where
(22)A(t)v=∑i,j=12‍∂yi(aij∂yjv),aij=(γ′γ-1)2yiyj(i,j=1,2),b(y,t)=-γ-2(γ′′γ+(γ′)2)y,c(y,t)=-2γ′γ-1y.
The above method was introduced by Dal Passo and Ughi [15] for studying a certain class of parabolic equations in noncylindrical domains. This idea was used in [11, 13, 14, 16, 17].We will use (19) and (20) to estimate the values ℬ1 and ℬ2 on Γ1. Denoting by
(23)(g*v)(t)=∫0t‍g(t-s)v(s)ds
the convolution product operator and differentiating (19) and (20) we arrive at the following Volterra equations:
(24)ℬ2vγ2+1g1(0)g1′*ℬ2vγ2=1g1(0)v′,ℬ1vγ2+1g2(0)g2′*ℬ1vγ2=-1g2(0)∂v′∂ν.
Applying Volterra’s inverse operator, we get
(25)ℬ2vγ2=1g1(0){v′+k1*v′},ℬ1vγ2=-1g2(0){∂v′∂ν+k2*∂v′∂ν},
where the resolvent kernels of -gi′/gi(0) satisfy
(26)ki+1gi(0)gi′*ki=-1gi(0)gi′,∀i=1,2.
Denoting by τ1=1/g1(0) and τ2=1/g2(0), we obtain
(27)ℬ2vγ2=τ1{v′+k1(0)v-k1(t)v0+k1′*v},(28)ℬ1vγ2=-τ2{∂v′∂ν+k2(0)∂v∂ν-k2(t)∂v0∂ν+k2′*∂v∂ν}.
Therefore, we use (27) and (28) instead of the boundary conditions (19) and (20).Let us define the bilinear forma(·,·) as follows:
(29)a(w,v)=wx1x1vx1x1+wx2x2vx2x2+μ(wx1x1vx2x2+wx2x2vx1x1)+2(1-μ)wx1x2vx1x2.SinceΓ0≠∅ we know that ∫Ω‍a(v,v)dy is equivalent to the H2(Ω) norm, that is,
(30)c0∥v∥H2(Ω)2≤∫Ω‍a(v,v)dy≤C0∥v∥H2(Ω)2,
where c0 and C0 are generic positive constants.Let us denote that(31)(g∘v)(t):=∫0t‍g(t-s)(v(t)-v(s))ds,(g□v)(t):=∫0t‍g(t-s)|v(t)-v(s)|2ds.
The following lemma states an important property of the convolution operator.Lemma 1.
Forg,v∈C1([0,∞):ℝ) one has
(32)(g*v)v′=-12g(t)|v(t)|2+12g′□v-12ddt[g□v-(∫0t‍g(s)ds)|v|2].The proof of this lemma follows by differentiating the termg□v.We state the following lemma which will be useful in what follows.Lemma 2 (see [7]).
Letw and v be functions in H4(Ω)∩H02(Ω). Then one has
(33)∫Ω‍(Δ2w)vdy=∫Ω‍a(w,v)dy+∫Γ1‍{(ℬ2w)v-(ℬ1w)∂v∂ν}dΓ.Lemma 3 (see [18]).
Suppose thatf∈L2(Ω), g∈H1/2(Γ1), and h∈H3/2(Γ1); then, any solution of
(34)∫Ω‍a(v,w)dy=∫Ω‍fwdy+∫Γ1‍gwdΓ+∫Γ1‍h∂w∂νdΓ,∀w∈H02(Ω)
satisfies v∈H4(Ω) and also
(35)Δ2v=f,v=∂v∂ν=0onΓ0,ℬ1v=h,ℬ2v=gonΓ1.To show the existence of solution, we will use the following hypotheses:(36)γ′≤0,γ∈L∞(0,∞),inf0≤t<∞γ(t)=γ0>0,(37)γ′∈W2,∞(0,∞)∩W2,1(0,∞),(38)0<max0≤t<∞|γ′(t)|γ(t)≤12c1c0-1Md,
where d=diam(Ω),  M=meas(Ω), and c0 is a positive imbedding constant such that ∥∇v∥2≤c1∥Δv∥2, for all v∈H02(Ω).


## 3. Existence and Regularity

In this section we will study the existence and regularity of solutions for system (4)–(8).The well posedness of system (17)–(21) is given by the following theorem.Theorem 4.
Letki∈C2(ℝ+) be such that
(39)ki,-ki′,ki′′≥0.
The function γ satisfies that
(40)|γ′(t)|γ-1(t)<min{1,-ki′(t)2}.
If (v0,v1)∈(H4(Ω)∩H02(Ω))×H02(Ω) satisfy the compatibility condition
(41)ℬ2v0-τ1γ2(0)v1=0,ℬ1v0+τ2γ2(0)∂v1∂νonΓ1
then there exists only one solution for system (17)–(21) satisfying
(42)v∈L∞(0,T;H4(Ω)∩H02(Ω)),v′∈L∞(0,T;H02(Ω)),v′′∈L∞(0,T;L2(Ω)).Proof.
The main idea is to use the Galerkin method. To do this let us denote byB the operator
(43)Bw=Δ2w,D(B)=H02(Ω)∩H4(Ω).
It is well known thatB is a positive self-adjoint operator in the Hilbert space L2(Ω) for which there exist sequences {wn}n∈ℕ and {λn}n∈ℕ of eigenfunctions and eigenvalues of B such that the set of linear combinations of {wn}n∈ℕ is dense in D(B) and λ1<λ2≤⋯≤λn→∞ as n→∞. Let us define
(44)v0m=∑j=1m‍(v0,wj)wj,v1m=∑j=1m‍(v1,wj)wj
Note that for any (v0,v1)∈D(B)×H02(Ω), we have v0m→v0 strong in D(B) and v1m→v1 strong in H02(Ω).
Let us denote byVm the space generated by w1,w2,…,wm. Standard results on ordinary differential equations guarantee that there exists only one local solution
(45)vm(t)=∑j=1m‍gjm(t)wj,
of the approximate system
(46)∫Ω‍vm′′wjdy+γ-4∫Ω‍a(vm,wj)dy+∫Ω‍A(t)vmwjdy+∫Ω‍c(y,t)·∇vm′wjdy+∫Ω‍b(y,t)·∇vmwjdy=-τ1γ-2∫Γ1‍{vm′+k1(0)vm-k1(t)v0m+k1′*vm}wjdΓ-τ2γ-2∫Γ1‍{∂vm′∂ν+k2(0)∂vm∂ν-k2(t)∂v0m∂ν+k2′*∂vm∂ν∂vm′∂ν}wj∂νdΓ(j=1,2,…,m),(47)vm(x,0)=v0m,vm′(x,0)=v1m.
By standard methods for differential equations, we prove the existence of solutions to the approximate equation (46) on some interval [0,tm). Then, this solution can be extended to the whole interval [0,T], for all T>0, by using the following first estimate.
The First Estimate. Multiplying (46) by gjm′(t), summing up the product result j=1,2,…,m, and making some calculations using Lemma 1, we get
(48)12ddt[∫Γ1‍(k2(t)|∂vm∂ν|2-k2′∂vm∂ν)∫Ω‍|vm′|2dy+γ-4∫Ω‍a(vm,vm)dy12ddtc+τ1γ-2∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓ12ddtc+τ2γ-2∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ]+2γ-5γ′∫Ω‍a(vm,vm)dy+τ1γ-3γ′∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓ+τ2γ-3γ′∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ=-∫Ω‍A(t)vmvm′dy-∫Ω‍c(y,t)·∇vm′vm′dy-∫Ω‍b(y,t)·∇vmvm′dy-τ1γ-2∫Γ1‍(|vm′|2-k1(t)v0mvm′-12k1′(t)|vm|2+12k1′′□vm)dΓ-τ2γ-2∫Γ1‍(|∂vm′∂ν|2-k2(t)∂v0m∂ν∂vm′∂ν-12k2′(t)|∂vm∂ν|2+12k2′′□∂vm∂ν|∂vm′∂ν|2)dΓ.
Now we will estimate terms of the right-hand side of (48). From the hypotheses on γ and Green’s formula, we get
(49)-∫Ω‍A(t)vmvm′dy=-∫Ω‍∑i,j=12‍∂yi(aij∂yjvm)vm′dy=∫Ω‍∑i,j=12‍(aij∂yjvm)∂yivm′dy=∫Ω‍∑i,j=12‍(γ′γ-1)2yiyj∂yjvm∂yivm′dy=ddt∫Ω‍12(γ′γ-1)2|∇vm·y|2dy-(γ′γ-1)[γ′′γ-1-(γ′γ-1)2]∥∇vm·y∥22,∫Ω‍c(y,t)·∇vm′vm′dy=-∫Ω‍2γ′γ-1y·∇vm′vm′dy=-∫Ω‍γ′γ-1y·∇|vm′|2dy=2γ′γ-1∥vm′∥22,-∫Ω‍b(y,t)·∇vmvm′dy=∫Ω‍γ-2(γ′′γ+(γ′)2)y·∇vmvm′dy≤(|γ′′γ-1|+|γ′γ-1|22)(∥y·∇vm∥22+∥vm′∥22)≤C1(∥∇vm∥22+∥vm′∥22).
Young’s inequality yields(50)∫Γ1‍k1(t)v0mvm′dΓ≤12∫Γ1‍|vm′|2dΓ+k12(t)2∫Γ1‍|v0m|2dΓ,∫Γ1‍k2(t)∂v0m∂ν∂vm′∂νdΓ≤12∫Γ1‍|∂vm′∂ν|2dΓ+k22(t)2∫Γ1‍|∂v0m∂ν|2dΓ.
Replacing the above calculations in (48) and using our assumptions ki,-ki′,ki′′≥0 and (30), we have
(51)12ddt[∫Γ1‍(k2(t)|∂vm∂ν|2-k2′∂vm∂ν)∥vm′∥22+γ-4∫Ω‍a(vm,vm)dy-(γ′γ-1)2∥∇vm·y∥2212ddtc+τ1γ-2∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓ12ddtc+τ2γ-2∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ]≤C2[∫Γ1‍(k2(t)|∂vm∂ν|2-k2′∂vm∂ν)∥vm′∥22+∫Ω‍a(vm,vm)dyC2c+∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓC2c2+∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ]+τ1γ-22k12(t)∫Γ1‍|v0m|2dΓ+τ2γ-22k22(t)∫Γ1‍|∂v0m∂ν|2dΓ.
From our choice ofv0m and v1m and integrating (51) over (0,t) with t∈(0,tm), we obtain
(52)∥vm′∥22+γ-4∫Ω‍a(vm,vm)dy-(γ′γ-1)2∥∇vm·y∥22+τ1γ-2∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓ+τ2γ-2∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ≤C3∫0t‍[(k2(s)|∂vm(s)∂ν|2-(k2′∂vm∂ν)(s))∥vm′(s)∥22+∫Ω‍a(vm(s),vm(s))dyC3∫0t‍c+∫Γ1‍(k1(s)|vm(s)|2-(k1′□vm)(s))dΓC3∫0t‍c+∫Γ1‍(k2(s)|∂vm(s)∂ν|2-(k2′□∂vm∂ν)(s))dΓ]ds+C4.
We observe that, from (30) and (38),
(53)(γ′γ-1)2∥∇vm·y∥22≤(γ′γ-1)2Md2∥∇vm∥22≤(γ′γ-1)2c1c0-1Md2∫Ω‍a(vm,vm)dy≤γ-42∫Ω‍a(vm,vm)dy,
for all t≥0. Hence, by Gronwall’s lemma we get
(54)∥vm′∥22+∫Ω‍a(vm,vm)dy+∫Γ1‍(k1(t)|vm|2-k1′□vm)dΓ+∫Γ1‍(k2(t)|∂vm∂ν|2-k2′□∂vm∂ν)dΓ≤C5,
where C5 is a positive constant which is independent of m and t.
The Second Estimate. First of all, we are going to estimate vm′′(0) in L2(Ω)-norm. Letting t→0+ in (46), multiplying the result by gjm′′(0), and using the compatibility condition (41), we have
(55)∥vm′′(0)∥22≤C6.
Now, differentiating (46) with respect to t, we obtain
(56)∫Ω‍vm′′′wjdy+γ-4∫Ω‍a(vm′,wj)dy-4γ-5γ′∫Ω‍a(vm,wj)dy=-∫Ω‍ddt[A(t)vm]wjdy-∫Ω‍ddt[c(y,t)·∇vm′]wjdy-∫Ω‍ddt[b(y,t)·∇vm]wjdy-τ1γ-2∫Γ1‍{vm′′+k1(0)vm′+k1′*vm′}wjdΓ-τ2γ-2∫Γ1‍{∂vm′′∂ν+k2(0)∂vm′∂ν+k2′*∂vm′∂ν}wj∂νdΓ+2τ1γ-3γ′∫Γ1‍{vm′+k1(0)vm-k1(t)v0m+k1′*vm}wjdΓ+2τ2γ-3γ′∫Γ1‍{∂vm′∂ν+k2(0)∂vm∂ν-k2(t)∂v0m∂ν+k2′*∂vm∂ν}wj∂νdΓ.
Multiplying (56) by gjm′′(t), summing up the product result in j, and using Lemma 1, we have
(57)12ddt[∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′∂vm′∂ν)∥vm′′∥22+γ-4∫Ω‍a(vm′,vm′)dy12ddtc-8γ-5γ′∫Ω‍a(vm,vm′)dy12ddtc+τ1γ-2∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓ12ddtc+τ2γ-2∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ]+6γ-5γ′∫Ω‍a(vm′,vm′)dy+4γ-4(γ′′γ-1-5(γ′γ-1)2)∫Ω‍a(vm,vm′)dy+τ1γ-3γ′∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓ+τ2γ-3γ′∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ=-∫Ω‍ddt[A(t)vm]vm′′dy-∫Ω‍ddt[c(y,t)·∇vm′]vm′′dy-∫Ω‍ddt[b(y,t)·∇vm]vm′′dy-τ1γ-2∫Γ1‍(|vm′′|2+12k1' '□vm′-12k1′(t)|vm′|2)dΓ-τ2γ-2∫Γ1‍(|∂vm′′∂ν|2+12k2′′□∂vm′∂ν-12k2′(t)|∂vm′∂ν|2)dΓ+2τ1γ-3γ′∫Γ1‍{vm′+k1(0)vm-k1(t)v0m+k1′*vm}vm′′dΓ+2τ2γ-3γ′∫Γ1‍{∂vm′∂ν+k2(0)∂vm∂ν-k2(t)∂v0m∂ν+k2′*∂vm∂ν∂vm′∂ν}vm′′∂νdΓ.
Now we will estimate terms of the right-hand side of (57). From the hypotheses on γ and Green’s formula, we get
(58)-∫Ω‍ddt[A(t)vm]vm′′dy=-∫Ω‍ddt[∑i,j=12‍∂yi((γ′γ-1)2yiyj∂yjvm)]vm′′dy=-∫Ω‍[∑i,j=12‍∂yi(2γ′γ-1(γ′′γ-1-(γ′γ-1)2)yiyj∂yjvm-∫Ω‍cccccccccc+(γ′γ-1)2yiyj∂yjvm′)∑i,j=12‍]vm′′dy=-∫Ω‍[∑i,j=12‍∂yi(∂yj2γ′γ-1(γ′′γ-1-(γ′γ-1)2)-∫Ω‍ccccccccc×yiyj∂yjvm)∑i,j=12‍]vm′′dy+∫Ω‍∑i,j=12‍(γ′γ-1)2yiyj∂yjvm′∂yivm′′dy=-∫Ω‍[∑i,j=12‍∂yi(2γ′γ-1(γ′′γ-1-(γ′γ-1)2)yiyj∂yjvm)]vm′′dy+ddt∫Ω‍12(γ′γ-1)2|∇vm′·y|2dy-(γ′γ-1)[γ′′γ-1-(γ′γ-1)2]∥∇vm′·y∥22,(59)-∫Ω‍ddt[c(y,t)·∇vm′]vm′′dy=∫Ω‍ddt[2γ′γ-1y·∇vm′]vm′′dy=∫Ω‍[2(γ′′γ-1-(γ′γ-1)2)y·∇vm′+2γ′γ-1y·∇vm′′]vm′′dy=∫Ω‍[2(γ′′γ-1-(γ′γ-1)2)y·∇vm′]vm′′dy+∫Ω‍γ′γ-1y·∇|vm′′|2dy≤(|γ′′γ-1|+|γ′γ-1|2)(∥y·∇vm′∥22+∥vm′′∥22)-2γ′γ-1∥vm′′∥22,(60)-∫Ω‍ddt[b(y,t)·∇vm]vm′′dy=∫Ω‍ddt[(γ′′γ-1+(γ′γ-1)2)y·∇vm]vm′′dy=∫Ω‍ddt[γ′′γ-1+(γ′γ-1)2]y·∇vmvm′′dy+∫Ω‍(γ′′γ-1+(γ′γ-1)2)y·∇vm′vm′′dy≤C7(∥y·∇vm∥22+∥y·∇vm′∥22+∥vm′′∥22).
We know that
(61)(k1′*vm)(t)=∫0t‍k1′(t-s)(vm(s)-vm(t))ds+k1(t)vm(t)-k1(0)vm(t).
By using Hölder’s inequality and our assumption k1′≤0, we note that
(62)∥∫0t‍k1′(t-s)(v(t)-v(s))ds∥Γ12≤(∫0t‍k1′(s)ds)∫Γ1‍∫0t‍k1′(t-s)(v(t)-v(s))2dsdΓ≤∫Γ1‍k1(0)|k1′|□vdΓ
and, hence, by applying Young’s inequality, we obtain
(63)|2τ1γ-3γ′∫Γ1‍{vm′+k1(0)vm-k1(t)v0m+k1′*vm}vm′′dΓ|≤τ1γ-3|γ′|∫Γ1‍|vm′′|2dΓ+τ1γ-3|γ′|×∫Γ1‍(|vm′|2+k12(t)|v0m|2+k1(0)|k1′|□vm·∫Γ1‍c+k12(t)|vm|2)dΓ.
By the same argument of (63), we can obtain the similar estimate
(64)|2τ2γ-3γ′∫Γ1‍{∂vm′∂ν+k2(0)∂vm∂ν-k2(t)∂v0m∂ν+k2′*∂vm∂ν∂vm′∂ν}vm′′∂νdΓ|≤τ2γ-3|γ′|∫Γ1‍|vm′′∂ν|2dΓ+τ2γ-3|γ′|×∫Γ1‍(|vm′∂ν|2+k22(t)|∂v0m∂ν|2·∫Γ1‍c+k2(0)|k2′|□∂vm∂ν+k22(t)|∂vm∂ν|2)dΓ.
Applying (58)–(64) to (57) and using the first estimate (54) and our assumptions ki,-ki′,ki′′≥0 and |γ′|γ-1<min{1,-(ki′/2)}, we have
(65)12ddt[(|∂vm′∂ν|2)∥vm′′∥22+γ-4∫Ω‍a(vm′,vm′)dy-(γ′γ-1)2∥∇vm′·y∥2212ddtc-8γ-5γ′∫Ω‍a(vm,vm′)dy12ddtc+τ1γ-2∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓ12ddtc+τ2γ-2∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ]≤C8[(|∂vm′∂ν|2)∥vm′′∥22+∫Ω‍a(vm′,vm′)dy+∫Ω‍a(vm,vm′)dyC8c+∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓC8c+∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ]+τ1γ-3|γ′|∫Γ1‍k12(t)|v0m|2dΓ+τ2γ-3|γ′|∫Γ1‍k22(t)|∂v0m∂ν|2dΓ+C9.
From (55) and our choice of v0m and v1m and integrating (65) over (0,t) with t∈(0,tm), we obtain
(66)∥vm′′∥22+γ-4∫Ω‍a(vm′,vm′)dy-(γ′γ-1)2∥∇vm′·y∥22-8γ-5γ′∫Ω‍a(vm,vm′)dy+τ1γ-2∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓ+τ2γ-2∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ≤2C8∫0t‍[(k2(s)|∂vm′(s)∂ν|2-(k2′∂vm′∂ν)(s))∥vm′′(s)∥22+∫Ω‍a(vm′(s),vm′(s))dy2C8∫0t‍c+∫Ω‍a(vm(s),vm′(s))dy2C8∫0t‍c+∫Γ1‍(k1(s)|vm′(s)|2-(k1′□vm′)(s))dΓ2C8∫0t‍c+∫Γ1‍(k2(s)|∂vm′(s)∂ν|2-(k2′□∂vm′∂ν)(s))dΓ]ds+C10.
Using the same arguments as for (53), we get
(67)(γ′γ-1)2∥∇vm′·y∥22<γ-42∫Ω‍a(vm′,vm′)dy,
for all t≥0. Therefore, by Gronwall’s lemma, we obtain
(68)∥vm′′∥22+∫Ω‍a(vm′,vm′)dy+∫Γ1‍(k1(t)|vm′|2-k1′□vm′)dΓ+∫Γ1‍(k2(t)|∂vm′∂ν|2-k2′□∂vm′∂ν)dΓ≤C11,
where C11 is a positive constant which is independent of m and t.
According to (54) and (68), we get
(69){vm}isboundedinL∞(0,T;H02(Ω)),(70){vm′}isboundedinL∞(0,T;H02(Ω)),(71){vm′′}isboundedinL∞(0,T;L2(Ω)).
From (69) to (71), there exists a subsequence of {vm}, which we still denote by {vm}, such that
(72)vm⟶vweakstarinL∞(0,T;H02(Ω)),(73)vm′⟶v′weakstarinL∞(0,T;H02(Ω)),(74)vm′′⟶v′′weakstarinL∞(0,T;L2(Ω)).
Letting m→∞ in (46) and using (72)–(74), we obtain
(75)∫Ω‍a(v,w)dy=-γ4∫Ω‍v′′wdy-γ4∫Ω‍A(t)vwdy-γ4∫Ω‍c(y,t)·∇v′wdy-γ4∫Ω‍b(y,t)·∇vwdy-τ1γ2∫Γ1‍{v′+k1(0)v-k1(t)v0+k1′*v}wdΓ-τ2γ2∫Γ1‍{∂v′∂ν+k2(0)∂v∂ν-k2(t)∂v0∂ν+k2′*∂v∂ν}∂w∂νdΓ
for any w∈H02(Ω). From Lemma 3 we obtain that v∈L∞(0,T;H4(Ω)). The uniqueness of solutions follows by using standard arguments.Theorem 5.
Under the hypotheses of Theorem4, let u0∈H02(Ω0)∩H4(Ω0),  u1∈H02(Ω0). Then there exists a unique solution u of the problem (4)–(8) satisfying
(76)u∈L∞(0,∞;H02(Ωt)∩H4(Ωt)),u′∈L∞(0,∞;H02(Ωt)),u′′∈L∞(0,∞;L2(Ωt)).Proof.
This idea was used in [11, 13, 14, 16, 17]. To show the existence in noncylindrical domains, we return to our original problem in the noncylindrical domains by using the change variable given in (14) by (y,t)=τ(x,t),  (x,t)∈Q^. Let v be the solution obtained from Theorem 4 and u defined by (16); then u belongs to the class
(77)u∈L∞(0,∞;H02(Ωt)∩H4(Ωt)),u′∈L∞(0,∞;H02(Ωt)),u′′∈L∞(0,∞;L2(Ωt)).
Denoting by
(78)u(x,t)=v(y,t)=(v∘τ)(x,t),
then from (15) it is easy to see that u satisfies (4)–(8) in the sense of L∞(0,∞;L2(Ωt)). If u1, u2 are two solutions obtained through the diffeomorphism τ given by (14), then v1=v2, so u1=u2. Thus the proof of Theorem 5 is completed.


## 4. Exponential Decay

In this section, we show that the solution of system (4)–(8) decays exponentially. First of all, we introduce the useful lemma for a noncylindrical domain.Lemma 6 (see [11, 12]).
LetG(·,·) be the smooth function defined in Ωt×[0,∞[. Then
(79)ddt∫Ωt‍G(x,t)dx=∫Ωt‍ddtG(x,t)dx+γ′γ-1∫Γt‍G(x,t)(x·ν¯)dΓ,
where ν¯ is the x-component of the unit normal exterior ν.By the same argument of (27) and (28), it can be written as
(80)ℬ2u=τ1{u′+k1(0)u-k1(t)u0+k1′*u},(81)ℬ1u=-τ2{∂u′∂ν+k2(0)∂u∂ν-k2(t)∂u0∂ν+k2′*∂u∂ν}.
We use (80) and (81) instead of the boundary conditions (6) and (7).We will use the following lemma.Lemma 7 (see [4]).
For everyu∈H4(Ω) and for every μ∈ℝ, one has
(82)∫Ωt‍(m·∇u)Δ2udx=∫Ωt‍a(u,u)dx+12∫Γt‍(m·ν)a(u,u)dΓ+∫Γt‍[(ℬ2u)(m·∇u)-(ℬ1u)∂∂ν(m·∇u)]dΓ.Now, we define the energy of problem (4)–(8) by
(83)E(t)=12[∥u′∥2,t2+∫Ωt‍a(u,u)dx+τ1∫Γ1,t‍(k1(t)|u|2-k1′□u)dΓ+τ2∫Γ1,t‍(k2(t)|∂u∂ν|2-k2′□∂u∂ν)dΓ].
We observe that E(t) is a positive function. Using Lemmas 6 and 1, we have
(84)E′(t)≤γ′γ-12∫Γ1,t[|u′|2+a(u,u)](x·ν¯)dΓ-τ12∫Γ1,t‍|u′|2dΓ+τ12k12(t)∫Γ1,t|u0|2dΓ+τ12k1′(t)∫Γ1,t‍|u|2dΓ-τ12∫Γ1,t‍k1′′□udΓ-τ22∫Γ1,t‍|∂u′∂ν|2dΓ+τ22k22(t)∫Γ1,t|∂u0∂ν|2dΓ+τ22k2′(t)∫Γ1,t‍|∂u∂ν|2dΓ-τ22∫Γ1,t‍k2′′□∂u∂νdΓ.
Let us consider the following functional:
(85)ψ(t)=∫Ωt‍(m·∇u)u′dx.The following lemma plays an important role for the construction of the Lyapunov functional.Lemma 8.
Let one suppose that the initial data(u0,u1)∈(H4(Ω0)∩H02(Ω0))×H02(Ω0) and satisfies the compatibility condition (41). Then the solution of system (4)–(8) satisfies
(86)ψ′(t)≤12∫Γ1,t‍(m·ν)|u′|2dΓ-∫Ωt‍|u′|2dx-∫Ωt‍a(u,u)dx-12∫Γ1,t‍(m·ν)a(u,u)dΓ-∫Γ1,t‍[(ℬ2u)(m·∇u)-(ℬ1u)∂∂ν(m·∇u)]dΓ+γ′γ-1∫Γ1,t‍(m·∇u)u′(x·ν¯)dΓ.Proof.
Differentiatingψ and using (4) and Lemmas 6 and 7, we get
(87)ψ′(t)=∫Ωt‍(m·∇u′)u′dx+∫Ωt‍(m·∇u)u′′dx+γ′γ-1∫Γ1,t‍(m·∇u)u′(x·ν¯)dΓ=12∫Γ1,t‍(m·ν)|u′|2dΓ-∫Ωt‍|u′|2dx-∫Ωt‍a(u,u)dx-12∫Γt‍(m·ν)a(u,u)dΓ-∫Γt‍[(ℬ2u)(m·∇u)-(ℬ1u)∂∂ν(m·∇u)]dΓ+γ′γ-1∫Γt‍(m·∇u)u′(x·ν¯)dΓ.
Let us next examine the integrals over Γ0,t in (87). Since u=∂u/∂ν=0 on Γ0,t, we have
(88)B1u=B2u=∇u=0onΓ0,t,ux1=∂u∂νν1,ux2=∂u∂νν2,
and hence
(89)∫Γ0,t‍(ℬ1u)∂∂ν(m·∇u)dΓ=∫Γ0,t‍Δu(m·ν)∂2u∂ν2dΓ=∫Γ0,t‍(m·ν)|Δu|2dΓ,(90)∫Γ0,t‍(m·ν)a(u,u)dΓ=∫Γ0,t‍(m·ν)|Δu|2dΓ.
Therefore, from (87)–(90) we have
(91)ψ′(t)=12∫Γ1,t‍(m·ν)|u′|2dΓ-∫Ωt‍|u′|2dx-∫Ωt‍a(u,u)dx+12∫Γ0,t‍(m·ν)|Δu|2dΓ-12∫Γ1,t‍(m·ν)a(u,u)dΓ-∫Γ1,t‍(ℬ2u)(m·∇u)dΓ+∫Γ1,t‍(ℬ1u)∂∂ν(m·∇u)dΓ+γ′γ-1∫Γ1,t‍(m·∇u)u′(x·ν¯)dΓ.
Noting that m·ν≤0 on Γ0,t follows from (91), we have the conclusion of the lemma.Let us introduce the Lyapunov functional(92)ℒ(t)=NE(t)+ψ(t),
with N>0. Using Young’s inequality and choosing N>0 sufficiently large, we see that
(93)q0E(t)≤ℒ(t)≤q1E(t)
for q0 and q1 are positive constants. We will show later that the functional ℒ satisfies the inequality of the following result.Lemma 9 (see [7]).
Letf be a real positive function of class C1. If there exist positive constants p0, p1, and p2 such that
(94)f′(t)≤-p0f(t)+p1e-p2t
then there exist positive constants p and c such that
(95)f(t)≤(f(0)+c)e-pt.Finally, we will show the main result of this section.Theorem 10.
Assume that there exist positive constantsβ1 and β2 such that
(96)ki(0)>0,ki′(t)≤-β1ki(t),ki′′(t)≥-β2ki′(t),i=1,2.
If (u0,u1)∈H02(Ω0)×L2(Ω0) then there exist constants ω,C>0 such that
(97)E(t)≤CE(0)e-ωt,∀t≥0.Proof.
From (84) and Lemma 8 we have
(98)ℒ′(t)≤γ′γ-1N2∫Γ1,t‍[|u′|2+a(u,u)](x·ν¯)dΓ-τ1N2∫Γ1,t‍|u′|2dΓ+τ1N2k12(t)∫Γ1,t‍|u0|2dΓ+τ1N2k1′(t)∫Γ1,t‍|u|2dΓ-τ1N2∫Γ1,t‍k1′′□udΓ-τ2N2∫Γ1,t‍|∂u′∂ν|2dΓ+τ2N2k22(t)∫Γ1,t‍|∂u0∂ν|2dΓ+τ2N2k2′(t)∫Γ1,t‍|∂u∂ν|2dΓ-τ2N2∫Γ1,t‍k2′′□∂u∂νdΓ+12∫Γ1,t‍(m·ν)|u′|2dΓ-∫Ωt‍|u′|2dx-∫Ωt‍a(u,u)dx-12∫Γ1,t‍(m·ν)a(u,u)dΓ-∫Γ1,t‍[(ℬ2u)(m·∇u)-(ℬ1u)∂∂ν(m·∇u)]dΓ+γ′γ-1∫Γ1,t‍(m·∇u)u′(x·ν¯)dΓ.
Since the boundary conditions (80) and (81) can be written as
(99)ℬ2u=τ1{u′+k1(t)u-k1(t)u0-k1′∘u},ℬ1u=-τ2{∂u′∂ν+k2(t)∂u∂ν-k2(t)∂u0∂ν-k2′∘∂u∂ν},
by using Young’s inequality we obtain
(100)|-∫Γ1,t‍(ℬ2u)(m·∇u)dΓ|≤τ12ϵ∫Γ1,t‍|u′|2dΓ+τ12ϵk12(t)∫Γ1,t‍|u|2dΓ+τ12ϵk12(t)∫Γ1,t‍|u0|2dΓ+τ12ϵ∫Γ1,t‍k1(0)|k1′|□udΓ+ϵ2∫Γ1,t‍|m·∇u|2dΓ,(101)|∫Γ1,t‍(ℬ1u)∂∂ν(m·∇u)dΓ|≤τ22ϵ∫Γ1,t‍|∂u′∂ν|2dΓ+τ22ϵk22(t)∫Γ1,t‍|∂u∂ν|2dΓ+τ22ϵk22(t)∫Γ1,t‍|∂u0∂ν|2dΓ+τ22ϵ∫Γ1,t‍k2(0)|k2′|□∂u∂νdΓ+ϵ2∫Γ1,t‍|∂∂ν(m·∇u)|2dΓ,
where ϵ is a positive constant. Since the bilinear form a(u,u) is strictly coercive, using the trace theory and the fact m·ν≥δ0 on Γ1,t, we get
(102)∫Γ1,t‍|m·∇u|2dΓ+∫Γ1,t‍|∂∂ν(m·∇u)|2dΓ≤λ0∫Ωt‍a(u,u)dx+λ0δ0∫Γ1,t‍(m·ν)a(u,u)dΓ,
where λ0 is a constant depending on Ω and μ. Substituting inequalities (100)–(102) into (98) we have
(103)ℒ′(t)≤γ′γ-1N2∫Γ1,t‍[|u′|2+a(u,u)](x·ν¯)dΓ-τ1N2∫Γ1,t‍|u′|2dΓ+τ1N2k12(t)∫Γ1,t‍|u0|2dΓ-τ1β1N2k1(t)∫Γ1,t‍|u|2dΓ+τ1β2N2∫Γ1,t‍k1′□udΓ-τ2N2∫Γ1,t‍|∂u′∂ν|2dΓ+τ2N2k22(t)∫Γ1,t‍|∂u0∂ν|2dΓ-τ2β1N2k2(t)∫Γ1,t‍|∂u∂ν|2dΓ+τ2β2N2∫Γ1,t‍k2′□∂u∂νdΓ-∫Ωt‍|u′|2dx-(1-ϵλ02)∫Ωt‍a(u,u)dx-(12-ϵλ02δ0)∫Γ1,t‍(m·ν)a(u,u)dΓ+12∫Γ1,t‍(m·ν)|u′|2dΓ+τ12ϵ∫Γ1,t‍|u′|2dΓ+τ12ϵk12(t)∫Γ1,t‍|u|2dΓ+τ12ϵk12(t)∫Γ1,t‍|u0|2dΓ+τ12ϵ∫Γ1,t‍k1(0)|k1′|□udΓ+τ22ϵ∫Γ1,t‍|∂u′∂ν|2dΓ+τ22ϵk22(t)∫Γ1,t‍|∂u∂ν|2dΓ+τ22ϵk22(t)∫Γ1,t‍|∂u0∂ν|2dΓ+τ22ϵ∫Γ1,t‍k2(0)|k2′|□∂u∂νdΓ+γ′γ-1∫Γ1,t‍(m·∇u)u′(x·ν¯)dΓ.
First, choose ϵ>0 sufficiently small such that
(104)1-ϵλ02>0,12-ϵλ02δ0>0.
Then, choosing N large enough, we have
(105)ℒ′(t)≤-c2E(t)+c3K2(t)E(0),
where c2,c3>0 and K(t)=k1(t)+k2(t). From (93), (96), and (105), we obtain
(106)ℒ′(t)≤-c2q1ℒ(t)+c4c3E(0)e-2β1tforsomec4>0.
By Lemma 9, there exist positive constants c5 and c6 such that
(107)ℒ(t)≤(ℒ(0)+c5E(0))e-c6t,∀t≥0.
Using (93), we conclude that
(108)E(t)≤CE(0)e-ωt,∀t≥0
for some positive constants C and ω.



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