The attractor for a nonlinear hyperbolic equation in the unbounded domain

Abstract

We study the long-time behavior of solutions for damped nonlinear hyperbolicequations in the unbounded domains. It is proved that under the natural assumptions theseequations possess the locally compact attractors which may have the infinite Hausdorff andfractal dimension. That is why we obtain the upper and lower bounds for the Kolmogorov'sentropy of these attractors.Moreover, we study the particular cases of these equations where the attractors occurredto be finite dimensional. For such particular cases we establish that the attractors consistof finite collections of finite dimensional unstable manifolds and every solution stabilizes toone of the finite number of equilibria points.