Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations

Abstract

Scalar one-dimensional parabolic equations with periodically time-dependent nonlinearities are considered. For each such equation, the associated discretetime dynamical system is shown to admit Morse decompositions of the global attractor whose Morse sets are contained in a given, arbitrarily small neighborhood of the set of fixed points. Existence of such Morse decompositions implies that the chain recurrent set coincides with the set of fixed points. In particular, the dynamical system has a gradient-like structure. As an application of these results, a description of the asymptotic behavior of solutions of asymptotically periodic equations is given: Any bounded solution approaches a set of periodic solutions of the limiting equation. Other possible applications to nonlocal equations and thin domain problems are discussed.