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Estimates on the dimension of an attractor for a nonclassical hyperbolic equation

Author(s): Delin Wu; Chengkui Zhong
Journal: Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 63-70.
MSC (2000): Primary 35K57, 35B40, 35B41
Posted: June 16, 2006
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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.

References:

[1]
N. U. Ahmed and H. Harbi, Mathematical analysis of dynamic models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874. MR 1616611 (99d:73050)

[2]
F. Balibrea and J. Valero, On dimension of attractors of differential inclusions and reaction-diffusion equations, Discrete Contin. Dynam. Systems, 5 (1999), 515-528. MR 1696326 (2000k:37122)

[3]
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. MR 2026186 (2004j:37151)

[4]
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. MR 0768102 (87g:35186)

[5]
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. MR 1180900 (93f:35200)

[6]
J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. MR 0913856 (89m:35141)

[7]
N. I. Karachalios and N. M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on $ R^{N}$, J. Differential Equations, 157 (1999), 183-205. MR 1710020 (2000e:35151)

[8]
N. I. Karachalios and N. M. Stavrakakis, Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $ R^{N}$, Discrete Contin. Dynam. Systems, 4 (2002), 939-951. MR 1920653 (2003e:35199)

[9]
A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review, 32 (1990), 537-578. MR 1084570 (92g:73059)

[10]
J. C. Robinson, Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors, Cambridge University Press, Cambridge, 2001. MR 1881888 (2003f:37001a)

[11]
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York, 1997. MR 1441312 (98b:58056)

[12]
S. F. Zhou, On dimension of the global attractor for damped nonlinear wave equations, J. Math. Phys., 40 (1999), 1432-1438. MR 1674622 (2000g:35155)

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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

DOI: 10.1090/S1079-6762-06-00162-4
PII: S 1079-6762(06)00162-4
Keywords: Dynamical system, attractor, nonclassical hyperbolic equation, Hausdorff and fractal dimensions
Received by editor(s): May 26, 2005
Posted: 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
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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Delin Wu, Chengkui Zhong , Estimates on the dimension of attractor for a nonclassical hyperbolic equation , Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 63-70.

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