ISRN.APPLIED.MATHEMATICS ISRN Applied Mathematics 2090-5572 Hindawi Publishing Corporation 372507 10.1155/2013/372507 372507 Research Article A Remark on the Global Attractors of the Nonlinear Evolution Equations Ya-ya Chang Qiao-zhen Ma Homeier H. Kou J. 1 School of Mathematics and Statistics Northwest Normal University Lanzhou, Gansu 730070 China nwnu.edu.cn 2013 10 12 2013 2013 18 09 2013 13 11 2013 2013 Copyright © 2013 Chang Ya-ya and Ma Qiao-zhen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence of global attractor of the nonlinear elastic rod oscillation equation when the forcing term belongs only to H1(); furthermore, we prove that the fractal dimension of global attractor is finite.

1. Introduction

Let Ω be an open bounded set of 3 with smooth boundary Ω. We consider the following equation: (1)utt-Δu-Δut-ωΔutt+f(u)=g(x),-ωutt+f(u)=g(x,t)Ω×+,ut=0=u0,utt=0=u1,xΩ,uΩ=0,t0, where ω>0 and gH-1(Ω). The nonlinear term fC1(,), f(0)=0, and satisfies the following: (2)liminf|s|f(s)s>-λ1,(3)|f(s)|C(1+|s|4),s, 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 , we know that (1) has a unique solution uC([0,T];H01(Ω)), utC([0,T];H01(Ω)), for gH-1(Ω). The proof has no essential difference between g(x)L2(Ω) and g(x)H-1(Ω), so we omit it; see .

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 . For (1), when g(x)L2(Ω), in , 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 denote H=L2(Ω), V=H01(Ω), and H*, V* is, respectively, the dual space of H, V. Write 1=H01(Ω)×H01(Ω). Let A=-Δ and D(A)=H2(Ω)H01(Ω); we define D(A(s/2)); s 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 [<xref ref-type="bibr" rid="B8">8</xref>]).

Assume that f satisfies (2) and (3), gH-1(Ω); then, the solution semigroup {S(t)}t0 has a bounded absorbing set 0 in 1; that is, for any bounded subset 1, there exists T=T() such that (5)S(t)0,tT.

Lemma 2.

Let Ω3 be a bounded domain with smooth boundary, and one assumes that f satisfies (2) and (3), gH-1(Ω); then, the semigroup {S(t)}t0 possesses a global attractor 𝒜0 on 1.

Proof.

Since L2(Ω)H-1(Ω) is dense, for any η>0, there exists g(x)L2(Ω) such that (6)g(x)-gη(x)H-1η,for  any  gH-1(Ω). The remained proof of Lemma 2 is similar to that of , so we omit it.

Lemma 3 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let B be a bounded subset in Hilbert space X, the mapping V:BX, such that BV(B), and satisfy (7)V(v)-V(v~)Xlv-v~X,v,v~B,QNV(v)-QNV(v~)Xδv-v~X,(0<δ<1), where QN:XXN 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.

Let Ω3 be a bounded domain with smooth boundary; one assumes that f satisfies (2) and (3), gH-1(Ω); then, the fractal dimension of the global attractor 𝒜0 of the semigroup {S(t)}t0 is finite.

Proof.

According to Lemma 2, 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)v0𝒜0. 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(wt2+A(1/2)w2+ωA(1/2)wt2)+A(1/2)wt2+(f(u)-f(v),wt)=0. As the global attractor 𝒜0 is bounded in 1(Ω), so there exists M0>0 such that (13)maxxΩu,maxxΩv,A(1/2)u,A(1/2)vM0. 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.C1Ω(1+|u|4+|v|4)·|w|·|wt|dxC2Ω|w||wt|dxC22λ1A(1/2)w2+C22wt2.

By (12) and (14), we obtain (15)ddt(wt2+A(1/2)w2+ωA(1/2)wt2)2C3(wt2+A(1/2)w2+A(1/2)wt2), where Ci(i=1,2,3) is constant independent of ω; by Gronwall's inequality, we get (16)wt2+A(1/2)w2+ωA(1/2)wt2(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(QNwt2+A(1/2)QNw2+ωA(1/2)QNwt2)+A(1/2)QNwt2+(f(u)-f(v),QNwt)=0. Similar to the estimates of (13)-(14), we obtain (18)Ω|(f(u)-f(v))·QNwt|dxΩ|f(θu+(1-θ)v)|·|w|·|QNwt|dxC4w·QNwtC4w·A(1/2)QNwtλN+1-(1/2)C42λN+1-(1/2)w2+C42λN+1-(1/2)QNA(1/2)wt2C42λ1λN+1-(1/2)A(1/2)w2+C42λN+1-(1/2)QNA(1/2)wt2. Then, (19)12ddt(QNwt2+A(1/2)QNw2+ωA(1/2)QNwt2)+A(1/2)QNwt2C42λ1λN+1-(1/2)A(1/2)w2+C42λN+1-(1/2)A(1/2)QNwt2, where λN is the Nth eigenvalue of problem (11), so we have (20)ddt(QNwt2+A(1/2)QNw2+ωA(1/2)QNwt2)+λN+1(1/2)QNwt2+λN+1-1A(1/2)QNw2+(1-C4λN+1-(1/2))A(1/2)QNwt2(C4λ1λN+1-(1/2)+λN+1-1)A(1/2)w2. We let (21)y(t)=QNwt2+A(1/2)QNw2+ωA(1/2)QNwt2. 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)QNw12y(t)w012×(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)e2C3t1δ<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)}t0 generated by the problem (1) has a finite fractal dimension.

Acknowledgments

This work was partly supported by the NSFC (11101334) and the NSF of Gansu Province (1107RJZA223), and by the Fundamental Research Funds for the Gansu Universities.

Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics 1997 68 2nd New York, NY, USA Springer Applied Mathematical Sciences MR1441312 Shang Y. D. Initial-boundary value problem for the equation utt-Δu-Δut-Δutt=f(u) Acta Mathematicae Applicatae Sinica 2000 23 3 385 393 MR1797635 Bogolubky I. L. Some examples of inelastic solution interaction Computer Physics Communications 1977 13 Zhu W. G. Nonlinear waves in elastic rods Acta Solid Mechanica Sinica 1980 1 2 247 253 Seyler C. E. Fanstermacher D. L. Asymmetric regularized long wave equation Physics of Fluids 1984 27 1 58 66 Clarkson P. A. LeVeque R. J. Saxton R. Solitary-wave interactions in elastic rods Studies in Applied Mathematics 1986 75 2 95 121 MR859173 ZBL0606.73028 Xie Y. Zhong C. The existence of global attractors for a class nonlinear evolution equation Journal of Mathematical Analysis and Applications 2007 336 1 54 69 10.1016/j.jmaa.2006.03.086 MR2348490 ZBL1132.35019 Xie Y. Zhong C. Asymptotic behavior of a class of nonlinear evolution equations Nonlinear Analysis. Theory, Methods & Applications 2009 71 11 5095 5105 10.1016/j.na.2009.03.086 MR2560179 ZBL1180.35114 Çelebi A. O. Kalantarov V. K. Polat M. Attractors for the generalized Benjamin-Bona-Mahony equation Journal of Differential Equations 1999 157 2 439 451 10.1006/jdeq.1999.3634 MR1713267 ZBL0934.35151