Estimates on the dimension of an attractor for a nonclassical hyperbolic equation
Authors:
Delin Wu and Chengkui Zhong
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 63-70
MSC (2000):
Primary 35K57, 35B40, 35B41
DOI:
https://doi.org/10.1090/S1079-6762-06-00162-4
Published electronically:
June 16, 2006
MathSciNet review:
2226526
Full-text PDF Free Access
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Abstract: In this paper, we estimate the dimension of a global attractor for a nonclassical hyperbolic equation with a viscoelastic damping term in Hilbert spaces $H_{0}^{2}\times L^{2}$ and $D(A)\times H_{0}^{2}$, where $D(A)=\{v\in H_{0}^{2}\mid Av\in L^{2}\}$ and $A=\Delta ^{2}$. We obtain an explicit formula of the upper bound of the dimension of the attractor. The obtained dimension decreases as damping grows and is uniformly bounded for large damping, which conforms to physical intuition.
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[1]a N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853–874.
[2]b F. Balibrea and J. Valero, On dimension of attractors of differential inclusions and reaction-diffusion equations, Discrete Contin. Dynam. Systems, 5 (1999), 515–528.
[3]c V. V. Chepyzhov and A. A. Ilyin, On the fractal dimension of invariant sets: applications to Navier-Stokes equations, Discrete Contin. Dynam. Systems, 10 (2004), 117–135.
[4]d P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for $2$D Navier-Stokes equations, Comm. Pure Appl. Math., 38 (1985), 1–27.
[5]e A. Eden, A. J. Milani, and B. Nicolaenko, Finite-dimensional exponential attractors for semilinear wave equations with damping, J. Math. Anal. Appl., 169 (1992), 408–419.
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[12]l S. F. Zhou, On dimension of the global attractor for damped nonlinear wave equations, J. Math. Phys., 40 (1999), 1432–1438.
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Additional Information
Delin Wu
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China
Email:
wudelin03@st.lzu.edu.cn
Chengkui Zhong
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China
Email:
ckzhong@lzu.edu.cn
Keywords:
Dynamical system,
attractor,
nonclassical hyperbolic equation,
Hausdorff and fractal dimensions
Received by editor(s):
May 26, 2005
Published electronically:
June 16, 2006
Additional Notes:
Supported in part by the NSFC Grant (19971036) and Trans-Century Training Programme Foundation for the Talents by the State Education Commission.
Communicated by:
Boris Hasselblatt
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
