# A Remark on the Global Attractors of the Nonlinear Evolution Equations

Authors: Chang Ya-ya; Ma Qiao-zhen
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

We study the existence of global attractor of the nonlinear elastic rod oscillation equation when the forcing term belongs only toH−1(Ω); furthermore, we prove that the fractal dimension of global attractor is finite.


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## Body

## 1. Introduction

LetΩ be an open bounded set of ℝ3 with smooth boundary ∂Ω. We consider the following equation:
(1)utt-Δu-Δut-ωΔutt+f(u)=g(x),-ω△utt+f(u)=g(x,t)∈Ω×ℝ+,u∣t=0=u0,ut∣t=0=u1,∀x∈Ω,u∣∂Ω=0,∀t≥0,
where ω>0 and g∈H-1(Ω). The nonlinear term f∈C1(ℝ,ℝ), f(0)=0, and satisfies the following:
(2)liminf|s|→∞f(s)s>-λ1,(3)|f′(s)|⩽C(1+|s|4),∀s∈ℝ,
where λ1 is the first eigenvalue of -Δ in H01(Ω) and C is a positive constant.In line with the Galerkin methods introduced in [1], we know that (1) has a unique solution u∈C([0,T];H01(Ω)), ut∈C([0,T];H01(Ω)), for g∈H-1(Ω). The proof has no essential difference between g(x)∈L2(Ω) and g(x)∈H-1(Ω), so we omit it; see [2].Equation (1), which appears as a class of nonlinear evolution equations, like the strain solitary wave equation and dispersive-dissipative wave equation, is used to represent the propagation problems of a lengthwise wave in nonlinear elastic rods and lon-sonic of space transformation by weak nonlinear effect; see [3–6]. For (1), when g(x)∈L2(Ω), in [2], the author has discussed the existence of global strong solutions in (H2(Ω)∩H01(Ω))×(H2(Ω)∩H01(Ω)); in [7, 8], the authors have obtained the existence of global attractors in the weak topological space and the strong topology space, respectively. Recently, existence of the uniform compact attractors has been proved about the nonautonomous case of (1); that is, g(x)=g(x,t). In this paper, we prove existence of global attractor and its fractal dimension for (1) under the condition that g(x) only satisfies the lower regularity.


## 2. The Main Results

Without loss of generality, we denoteH=L2(Ω), V=H01(Ω), and H*, V* is, respectively, the dual space of H, V. Write ℋ1=H01(Ω)×H01(Ω). Let A=-Δ and D(A)=H2(Ω)∩H01(Ω); we define D(A(s/2)); s∈ℝ is Hilbert space family, and its inner product and norm are
(4)(·,·)D(A(s/2))=(A(s/2)·,A(s/2)·),∥·∥D(A(s/2))=∥A(s/2)·∥.The following results will be used later.Lemma 1 (see [8]).
Assume thatf satisfies (2) and (3), g∈H-1(Ω); then, the solution semigroup {S(t)}t≥0 has a bounded absorbing set ℬ0 in ℋ1; that is, for any bounded subset ℬ⊂ℋ1, there exists T=T(ℬ) such that
(5)S(t)ℬ⊂ℬ0,∀t≥T.Lemma 2.
LetΩ⊂ℝ3 be a bounded domain with smooth boundary, and one assumes that f satisfies (2) and (3), g∈H-1(Ω); then, the semigroup {S(t)}t≥0 possesses a global attractor 𝒜0 on ℋ1.Proof.
SinceL2(Ω)↪H-1(Ω) is dense, for any η>0, there exists g(x)∈L2(Ω) such that
(6)∥g(x)-gη(x)∥H-1≤η,foranyg∈H-1(Ω).
The remained proof of Lemma 2 is similar to that of [7], so we omit it.Lemma 3 (see [9]).
LetB be a bounded subset in Hilbert space X, the mapping V:B→X, such that B⊆V(B), and satisfy
(7)∥V(v)-V(v~)∥X≤l∥v-v~∥X,∀v,v~∈B,∥QNV(v)-QNV(v~)∥X≤δ∥v-v~∥X,(0<δ<1),
where QN:X→XN⊥ is orthogonal mapping and XN is spanned subspace by the former Nth eigenvector of X; then, the fractal dimension of B satisfies
(8)dF(B)≤Nln(8k2l2/(1-δ2))ln(2/(1-δ2)),
where k is the Gaussian constant.
Our main result is as follows.Theorem 4.
LetΩ⊂ℝ3 be a bounded domain with smooth boundary; one assumes that f satisfies (2) and (3), g∈H-1(Ω); then, the fractal dimension of the global attractor 𝒜0 of the semigroup {S(t)}t≥0 is finite.Proof.
According to Lemma2, provided
(9)u(x,0)=u0(x),v(x,0)=v0(x)∈𝒜0;
then,
(10)u(x,t)=S(t)u0,v(x,t)=S(t)v0∈𝒜0.
Let w=u-v satisfy the following equation:
(11)wtt-Δw-Δwt-ωΔwtt+f(u)-f(v)=0.
Taking the scalar product of (11) in L2(Ω) with wt, we obtain that
(12)12ddt(∥wt∥2+∥A(1/2)w∥2+ω∥A(1/2)wt∥2)+∥A(1/2)wt∥2+(f(u)-f(v),wt)=0.
As the global attractor 𝒜0 is bounded in ℋ1(Ω), so there exists M0>0 such that
(13)maxx∈Ω∥u∥,maxx∈Ω∥v∥,∥A(1/2)u∥,∥A(1/2)v∥≤M0.
Therefore, using (3) and (13), it follows that
(14)(f(u)-f(v),wt)=|∫01(f′(θu+(1-θ)v)dθ·w,wt)|≤∫Ω|∫01f′(θu+(1-θ)v)dθ|·|w|·|wt|dx.≤C1∫Ω(1+|u|4+|v|4)·|w|·|wt|dx≤C2∫Ω|w||wt|dx≤C22λ1∥A(1/2)w∥2+C22∥wt∥2.
By (12) and (14), we obtain
(15)ddt(∥wt∥2+∥A(1/2)w∥2+ω∥A(1/2)wt∥2)≤2C3(∥wt∥2+∥A(1/2)w∥2+∥A(1/2)wt∥2),
where Ci(i=1,2,3) is constant independent of ω; by Gronwall's inequality, we get
(16)∥wt∥2+∥A(1/2)w∥2+ω∥A(1/2)wt∥2≤(∥wt(0)∥2+∥A(1/2)w(0)∥2+∥A(1/2)wt(0)∥2)e2C3t.
For some t1>0, define l=e2C3t1; hence, we prove that the first inequality in Lemma 3 holds true.
Taking the inner product of (11) in L2(Ω) with QNwt(x,t), we commute the operator A with the projection QN to get
(17)12ddt(∥QNwt∥2+∥A(1/2)QNw∥2+ω∥A(1/2)QNwt∥2)+∥A(1/2)QNwt∥2+(f(u)-f(v),QNwt)=0.
Similar to the estimates of (13)-(14), we obtain
(18)∫Ω‍|(f(u)-f(v))·QNwt|dx≤∫Ω‍|f′(θu+(1-θ)v)|·|w|·|QNwt|dx≤C4∥w∥·∥QNwt∥≤C4∥w∥·∥A(1/2)QNwt∥λN+1-(1/2)≤C42λN+1-(1/2)∥w∥2+C42λN+1-(1/2)∥QNA(1/2)wt∥2≤C42λ1λN+1-(1/2)∥A(1/2)w∥2+C42λN+1-(1/2)∥QNA(1/2)wt∥2.
Then,
(19)12ddt(∥QNwt∥2+∥A(1/2)QNw∥2+ω∥A(1/2)QNwt∥2)+∥A(1/2)QNwt∥2≤C42λ1λN+1-(1/2)∥A(1/2)w∥2+C42λN+1-(1/2)∥A(1/2)QNwt∥2,
where λN is the Nth eigenvalue of problem (11), so we have
(20)ddt(∥QNwt∥2+∥A(1/2)QNw∥2+ω∥A(1/2)QNwt∥2)+λN+1(1/2)∥QNwt∥2+λN+1-1∥A(1/2)QNw∥2+(1-C4λN+1-(1/2))∥A(1/2)QNwt∥2≤(C4λ1λN+1-(1/2)+λN+1-1)∥A(1/2)w∥2.
We let
(21)y(t)=∥QNwt∥2+∥A(1/2)QNw∥2+ω∥A(1/2)QNwt∥2.
Choosing N large enough that 1-C4λN+1-(1/2)>0 and setting α=min{λN+1-1,λN+1(1/2),1-C4λN+1-(1/2)}, integrating with (16) and (20), we get
(22)y′(t)+αy(t)≤C4λ1λN+1-(1/2)×(∥wt(0)∥2+∥A(1/2)w(0)∥2+∥A(1/2)wt(0)∥2)e2C3t.
Gronwall's inequality implies that
(23)y(t)≤y(0)e-αt+12C3+α(C4λ1λN+1-(1/2)+λN+1-1)×(∥wt(0)∥2+∥A(1/2)w(0)∥2+∥A(1/2)wt(0)∥2)e2C3t.≤(∥wt(0)∥2+∥A(1/2)w(0)∥2+∥A(1/2)wt(0)∥2)×(e-αt+12C3+α(C4λ1λN+1-(1/2)+λN+1-1)e2C3t).
So, we have
(24)∥QNw∥ℋ12≤y(t)≤∥w0∥ℋ12×(e-αt+12C3+α(C4λ1λN+1-(1/2)+λN+1-1)e2C3t).
Take a proper t1>0 and N large enough such that
(25)e-αt1+12C3+α(C4λ1λN+1-(1/2)+λN+1-1)e2C3t1≤δ<1.
Therefore, for t=t1, S(t1) satisfies the condition of Lemma 3; then, the fractal dimension of the global attractor 𝒜0 satisfies
(26)dF(B)≤Nln(8k2l2/(1-δ2))ln(2/(1-δ2)).
This implies that the global attractor for semigroup {S(t)}t≥0 generated by the problem (1) has a finite fractal dimension.



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