Research Article | Open Access
Chang Ya-ya, Ma Qiao-zhen, "A Remark on the Global Attractors of the Nonlinear Evolution Equations", International Scholarly Research Notices, vol. 2013, Article ID 372507, 3 pages, 2013. https://doi.org/10.1155/2013/372507
A Remark on the Global Attractors of the Nonlinear Evolution Equations
We study the existence of global attractor of the nonlinear elastic rod oscillation equation when the forcing term belongs only to ; furthermore, we prove that the fractal dimension of global attractor is finite.
Let be an open bounded set of with smooth boundary . We consider the following equation: where and . The nonlinear term , , and satisfies the following: where is the first eigenvalue of in and is a positive constant.
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 , in , the author has discussed the existence of global strong solutions in ; 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, . In this paper, we prove existence of global attractor and its fractal dimension for (1) under the condition that only satisfies the lower regularity.
2. The Main Results
Without loss of generality, we denote , , and , is, respectively, the dual space of , . Write . Let and ; we define ; is Hilbert space family, and its inner product and norm are
The following results will be used later.
Lemma 3 (see ). Let be a bounded subset in Hilbert space , the mapping , such that , and satisfy
where is orthogonal mapping and is spanned subspace by the former Nth eigenvector of ; then, the fractal dimension of satisfies
where is the Gaussian constant.
Our main result is as follows.
Proof. According to Lemma 2, provided
Let satisfy the following equation:
Taking the scalar product of (11) in with , we obtain that
As the global attractor is bounded in , so there exists such that
Therefore, using (3) and (13), it follows that
By (12) and (14), we obtain where is constant independent of ; by Gronwall's inequality, we get For some , define ; hence, we prove that the first inequality in Lemma 3 holds true.
Taking the inner product of (11) in with , we commute the operator with the projection to get Similar to the estimates of (13)-(14), we obtain Then, where is the th eigenvalue of problem (11), so we have We let Choosing large enough that and setting , integrating with (16) and (20), we get Gronwall's inequality implies that So, we have Take a proper and large enough such that Therefore, for , satisfies the condition of Lemma 3; then, the fractal dimension of the global attractor satisfies This implies that the global attractor for semigroup generated by the problem (1) has a finite fractal dimension.
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.
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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.