Long-time behavior for a class of weighted equations with degeneracy

Abstract

In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space \begin{document}$ H_0^{1,a}(\Omega) $\end{document} (see Definition 1.1) cannot be bounded embedded into \begin{document}$ L^2(\Omega) $\end{document} . We claim that the dimension of the global attractor is infinite by estimating its lower bound of \begin{document}$ Z_2 $\end{document} -index. Moreover, we prove that there is an infinite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.