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Dimension of the global attractor for damped nonlinear wave equations

Author(s): Zhou Shengfan
Journal: Proc. Amer. Math. Soc. 127 (1999), 3623-3631.
MSC (1991): Primary 35B40, 35L70
Posted: May 17, 1999
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Abstract: An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.

References:

1.: J. K. Hale, Asymptotic Behavior Of Dissipative Systems, Amer. Math. Soc., Providence, Rhode Island, 1988. MR 89g:58059
2.: R.Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Science 68, MR 89m:58056
3.: G. Wang and S. Zhu, On dimension of the global attractor for damped sine-Gordon equation, Preprint, to appear in J. Math. Phys.
4.: A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., Vol., Springer-Verlag, 1983. MR 85g:47061

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Additional Information:

Zhou Shengfan
Affiliation: Department of Mathematics, Sichuan Union University, Chengdu, 610064, People's Republic of China
Email: nic2601@scuu.edu.cn

DOI: 10.1090/S0002-9939-99-05121-7
PII: S 0002-9939(99)05121-7
Keywords: Wave equation, global attractor, Hausdorff dimension
Received by editor(s): February 19, 1998
Posted: May 17, 1999
Additional Notes: This research was supported by the National Natural Science Foundation of China
Communicated by: Michael Handel
Copyright of article: Copyright 1999, American Mathematical Society


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