×

zbMATH — the first resource for mathematics

The attractor for a nonlinear hyperbolic equation in the unbounded domain. (English) Zbl 1153.35311
Summary: We study the long-time behavior of solutions for damped nonlinear hyperbolic equations in unbounded domains \(\partial_ tu-\Delta u+f(u)+\lambda_ 0 u=g(t)\), \(x\in\Omega\), \(t>0\); \(u| _ {t=0}=u_ 0(x)\), \(u| _ {\partial\Omega}=0\). 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.

MSC:
35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35K57 Reaction-diffusion equations
PDF BibTeX Cite
Full Text: DOI