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The attractor for a nonlinear hyperbolic equation in the unbounded domain

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  • 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.
    Mathematics Subject Classification: 35B40, 35B45.


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