a Gradient-Like Nonautonomous Evolution Process

TL;DR: A class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood is given, where f is a dissipative smooth nonlinearity and the damping β : ℝ → (0, ∞) is a suitable function.

Abstract: In this paper we consider a dissipative damped wave equation with nonautonomous damping of the form \begin{eqnarray*} \hspace*{2.057in}u_{tt}+\beta(t)u_t=\Delta u+f(u) \hspace*{2.057in} (1) \end{eqnarray*} in a bounded smooth domain Ω ⊂ ℝn with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping β : ℝ → (0, ∞) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of nonautonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small nonautonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small nonautonomous perturbation of any autonomous gradient-like evolution proc...