# Existence of Global Attractors for the Nonlinear Plate Equation with Memory Term

Authors: Lifang Niu; Jianwen Zhang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

A two-dimensional nonlinear plate equation is revisited, which arises from the model of the viscoelastic thin rectangular plate with four edges supported. We establish that the system is  exponentially decayed if the memory  kernel satisfies the condition of the exponential decay. Furthermore, we show the existence of the global attractor by verifying the condition (C).


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## Body

## 1. Introduction

In this paper, we investigate the nonlinear plate equation with memory type:(1.1)ρutt+r1r2ut+ϕ(0)▵2u-(N1+β|ux|2)uxx-(N2+β|uy|2)uyy+∫0∞‍ϕ′(s)▵2u(t-s)ds=r1f(x,y),inΩ×R+,
verifying the initial conditions:
(1.2)u(x,y,0)=u0(x,y,t)
and the boundary conditions:
(1.3)u(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=0,uxx(0,y,t)=uxx(1,y,t)=uyy(x,0,t)=uyy(x,1,t),
where x,y∈[0,1],  t∈R,η,r1,r2,N1,N2,β are nonnegative constants, Ω=[0,1]×[0,1] is a bounded domain with boundary ∂Ω,ϕ(0),ϕ(∞)>0 and ϕ′(s)<0 for every s∈R+.The asymptotical behavior of solutions for the nonlinear plate equations had been studied by many authors [1–10]; of those, Santos and Junior [5] studied a kind of plate equation with memory type. Yang and Zhong [11] studies the plate equation:
(1.4)utt+α(x)g(ut)+Δ2u+λu+f(u)=h(x),x∈Ω,
where Ω⊂Rn is a bounded domain and proves the existence of a global attractor in the space H01(Ω)×L2(Ω). After Yang and Zhong [11], Yue and Zhong [12] obtained the existence of a global attractor about some equations similar to (1.4). Xiao [13] discusses the long-time behavior of the plate equation:
(1.5)εutt+Δ2u+λut+β(x)u=f(x,u),x∈Ω=Rn,t≥0
on the unbounded domain Rn and show that there exists a compact global attractor for the above equation satisfying certain initial-boundary data. Wang and Zhang [14] prove that the two-dimensional nonlinear equation
(1.6)ρutt+DΔ2u+εμut-(N1+T2∫Ω‍ux2dxdy)uxx-(N2+T2∫Ω‍uy2dxdy)uyy=0
has a global attractor in space (H01(Ω)⋂‍L2(Ω)).


## 2. Preliminaries

We denote byH=L2(Ω),V=H02(Ω) endowed with the scalar product and the norm on H and V(·,·),|·|,((·,·)),∥·∥, respectively, where (u,v)=∫Ω‍u(x,y)v(x,y)dxdy,  ((u,v))=∫Ω‍Δu(x,y)Δv(x,y)dxdy. Define D(A)={v∈V,Av∈H}, where A=Δ2. For the operator A, we assume that A:D(A)→H are isomorphism, and there exists α>0 such that (Au,u)≥α∥u∥2, for all u∈V. We also define the power As of A for s∈R which operates on the spaces D(As), and we write V2s=D(As),s∈R. This is a Hilbert space with the inner product and norm defined
(2.1)(u,v)2s=(Asu,Asv),∥u∥2s=((u,v)2s)1/2,∀u,v∈D(As),
and Ar is an isomorphism from D(As) onto D(As-r), for all s,r∈R. In particular, D(A0)=H,D(A1/2)=V,D(A-1/2)=V*,D(A)⊂V⊂H=H*⊂V*, where  H*,V* are the dual space, respectively, and each space is dense in the following one and the injections are continuous. Using the Poincáre inequality we have
(2.2)∥v∥≥λ1|v|,∀v∈V,
where λ1 denotes the first eigenvalue of A1/2.Letμ(s)=-ϕ′(s),ϕ(∞)=1 and assume that the memory kernel μ is required to satisfy the following assumptions:(h1)
μ∈C1(R+)⋂‍L1(R+),μ'(s)≤0, for all s∈R+;(h2)
∫0∞‍μ(s)ds=M>0;(h3)
μ′(s)+αμ(s)≤0,for all s∈R+,α>0.In view ofh1, let Lμ2(R+,H02) be the Hilbert space of H02-valued functions on R+, endowed with the following inner product and the norm:
(2.3)(ϕ,ψ)μ,V=∫0∞‍μ(s)(Δϕ(s),Δψ(s))ds
and |ϕ|μ,V2=(ϕ,ϕ)μ,V=∫0∞‍μ(s)∥ϕ∥2ds. Finally, we introduce the following Hilbert spaces:
(2.4)H0=V×H×Lμ2(R+,V),H1=D(A)×V×Lμ2(R+,D(A)).We define(2.5)ηt(x,y,s)=u(x,y,t)-u(x,y,t-s)=^η(x,y,s),
and (1.1) is transformed into the system
(2.6)ρutt+r1r2ut+▵2u-(N1+β|ux|2)uxx-(N2+β|uy|2)uyy+∫0∞‍μ(s)▵2ηds=r1f(x,y),ηtt+ηst=ut,
where the second equation is obtained by differentiating (2.5). The corresponding initial-boundary value conditions are then given by
(2.7)u(0,y,t)=u(1,y,t)=u(x,0,t)=u(x,1,t)=0,uxx(0,y,t)=uxx(1,y,t)=uyy(x,0,t)=uyy(x,1,t),ηt(0,y,s)=ηt(1,y,s)=ηt(x,0,s)=ηt(x,1,s)=0,ηxxt(0,y,s)=ηxxt(1,y,s)=ηyyt(x,0,s)=ηyyt(x,1,s)=0,u(x,y,0)=u1(x,y),ut(x,y,0)=u2(x,y),η0(x,y,s)=u0(x,y,0)-u0(x,y,-s),
where u1(x,y)=u0(x,y,t),u2(x,y)=∂tu0(x,y,t)|t=0,x,y∈[0,1],t≥0,s∈R+.According to the classical Faedo-Galerkin method it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value, so we omit it and only give the following theorem.Theorem 2.1 (see [15, 16]).
Let(h1) hold and f(x,y)∈L2(Ω). Then given any time interval I=[0,1], problems (2.6)-(2.7) have a unique solution (u,ut,η) in I with initial data (u1,u2,η0)∈H0, and the mapping (u1,u2,η0)→(u(t),ut(t),ηst) is continuous in H0.Thus, it admits to define aC0 semigroup
(2.8)S(t):{u1,u2,η0}→{u(t),ut(t),ηst},t∈R+,
and they map H0 into themselves.In addition, the following abstract results will be used in our consideration.Theorem 2.2 (see [17]).
AC0 semigroup {S(t)}t≥0 in a Banach space X is said to satisfy condition (C) which arised by [17] if for any ε>0 and for any bounded set B of X, there exists t(B)>0 and a finite dimensional subspace X1 of X, such that {∥PS(t)x∥X,x∈B,t≥t(B)} is bounded and
(2.9)∥(I-P)S(t)x∥X<Xε,
where P:X→X1 is a bounded projector.Lemma 2.3 (see [17]).
Let{S(t)}t≥0 be a C0 semigroup in a Hilbert space M. Then {S(t)}t≥0 has a global attractor if and only if (1)
{S(t)}t≥0 satisfies the condition C;(2)
there exists a bounded absorbing subsetB of M.


## 3. Global AttractorΛ in H0

Theorem 3.1.
Assume(h1)–(h3) hold. Then the ball of H0,B0=BH0(0,R1), centered at 0 with R1=c/c0ρ0, is a bounded absorbing set in H0 for the semigroup {S(t)}t≥0.Proof.
We fixedδ and take δ∈(0,δ0), where δ0=min(r1r2/4ρ,λ12/2r1r2).
First, taking the inner product of the first equation of (2.6) with v=ut+δu, after computation we conclude
(3.1)12ddt{ρ|v|2+|Δu|2+N1|ux|2+N2|uy|2+β2|ux|4+β2|uy|4}+I1+N1δ|ux|2+N2δ|uy|2+βδ|ux|4+βδ|uy|4+(η,v)μ,V=r1(f,v),
where I1=-δρ|v|2+r1r2|v|2+δ|Δu|2-(r1r2δ-ρδ2)(u,v), and we can easily obtain
(3.2)I1≥δ2|Δu|2+12r1r2|v|2.
Combining with the second equation of (2.6) we have
(3.3)(η,v)μ,V=12ddt|η|μ,V2+(η,ηs)μ,V+δ(η,v)μ,V.
According to (h2)–(h3), we conclude
(3.4)(η,ηs)μ,V=12∫0∞‍μ(s)dds|Δηt(s)|2ds=-12∫0∞‍μ′(s)dds|Δηt(s)|2ds≥α2|η|μ,V2,δ(η,v)μ,V=δ∫0∞‍μ(s)(Δηt(s),Δu)ds≥-δ(∫0∞‍μ(s)|Δηt(s)|2ds)1/2·(∫0∞‍μ(s)|Δu|2ds)1/2≥-α4∫0∞‍μ(s)ds-δ2α∫0∞‍μ(s)|Δu|2ds≥-α4|η|μ,v2-Mδ2α|Δu|2.
Integrating with (3.4), from (3.3), entails
(3.5)(η,v)μ,V≥12ddt|η|μ,v2+α4|η|μ,v2-Mδ2α|Δu|2.
Write E(t)={ρ|v|2+|Δu|2+N1|ux|2+N2|uy|2+(β/2)|ux|4+(β/2)|uy|4+|η|μ,V2}, from (2.7), (3.2), and (3.5), we obtain
(3.6)ddtE(t)+δ(1-2Mδα)|Δu|2+r1r22|v|2+α2|η|μ,V2+2δN1|ux|2+2δN2|uy|2+δβ|ux|4+δβ|uy|4≤2r1r2|f|2.
Take δ small enough, such that 1-(2Mδ/α)>1/2. Write c=(2r1/r2)|f|2,  c0=min{δ/2,r1r2/2ρ,α/2}, thus in line with (3.6), we have
(3.7)ddtE(t)+c0E(t)≤C.
By the Gronwall Lemma, we conclude
(3.8)E(t)≤E(0)exp(-c0t)+cc0(1-exp(c0)),∀t≥0.
Due to (3.8), write ρ0=min(ρ,1), we have
(3.9)|Δu|2∣+|v|2∣+|η|μ,V2≤1ρ0E(0)exp(-c0t)+cc0ρ0(1-exp(c0)),∀t≥0.
Write R1=c/c0ρ0, we end up with
(3.10)limsup{∥u∥2∣+|v|2∣+|η|μ,V2}≤R12,∀t≥0.Theorem 3.2.
Supposef∈L2(Ω)  and  conditions (h1)–(h3) are hold. Then the solution semigroup {S(t)}t≥0 associated with system (2.6) and (2.7) has a global attractor Λ in H0, and it attracts all bounded subsets of H0, in the norm of H0.Proof.
Applying Lemma2.3, we only to prove that the condition (C) holds in H0.
Letwi~ be an orthonormal basis of D(A) which consists of eigenvectors of A. It is also an orthonormal basis of H,V, respectively. The corresponding eigenvalues are denoted by
(3.11)0<λ1~<λ2~<λ3~<⋯,λi~→∞,asi→∞
with Awi~=λi~wi~, for all i∈N. We write Hm=span{w1~,w2~,…,wm~}. For any (u,ut,η)∈H0, we decompose that (u,ut,η)=(u1,u1t,η1)+(u2,u2t,η2), where (u1,u1t,η1)=(Pmu,Pmut,Pmη), and Pm:H→Hm is the orthogonal projector. Since f∈H, for any ε>0, there exists some m, such that
(3.12)|(I-Pm)f|<ε4.
Taking the scalar product of the first equation of (2.6) in H with v2=u2t+Au2, combining with the second equation and using the same way with Theorem 3.1, we find
(3.13)ddtE(t)~+C0E(t)~≤2r1r2|f|2≤r1ε28r2,
where E(t)~={ρ|v2|2+|Δu2|2+N1|u2x|2+N2|u2y|2+(β/2)|u2x|4+(β/2)|u2y|4+|η2|μ,V2. According to Theorem 3.1, some M0 and t0  exist, for any t≥t0, such that E(t0)≤M0. By the Gronwall Lemma, we conclude
(3.14)E(t)~≤E(t0)exp(-c0(t-t0))+r1ε28c0r2≤M0exp(-c0(t-t0))+r1ε28c0r2,∀t≥t0.
Take t1 large enough, such that t1-t0≥(1/c0)lnM0/ε2, so we conclude
(3.15)|v2|2+|Δu2|2+|η2|μ,V2≤1ρ0(1+r18c0r2)ε2,fort≥t0.
Thus S{t}t≥0 satisfies the condition (C).



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*Source: 658052-2012-10-14.xml*