July  2001, 7(3): 593-641. doi: 10.3934/dcds.2001.7.593

The attractor for a nonlinear hyperbolic equation in the unbounded domain

1. 

Institute for Problems of Transmission Information, Russian Academy of Sciences, Bolshoi Karetniĭ 19, Moscow 101 447, GSP-4, Russian Federation

Received  July 2000 Revised  October 2000 Published  April 2001

We study the long-time behavior of solutions for damped nonlinear hyperbolic equations in the unbounded domains. It is proved that under the natural assumptions these equations possess the locally compact attractors which may have the infinite Hausdorff and fractal dimension. That is why we obtain the upper and lower bounds for the Kolmogorov's entropy of these attractors.
Moreover, we study the particular cases of these equations where the attractors occurred to be finite dimensional. For such particular cases we establish that the attractors consist of finite collections of finite dimensional unstable manifolds and every solution stabilizes to one of the finite number of equilibria points.
Citation: S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593
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