On convergence of solutions to equilibria for quasilinear parabolic problems

TL;DR

This paper proves that solutions to certain quasilinear parabolic equations converge to equilibrium states, even when the set of equilibria forms a finite-dimensional normally hyperbolic manifold, without relying on Lyapunov functionals.

Contribution

It establishes convergence results for quasilinear parabolic problems with non-discrete equilibrium sets that form a normally hyperbolic manifold, independent of Lyapunov functionals.

Findings

Solutions converge to equilibria on normally hyperbolic manifolds.

Results are local and do not depend on Lyapunov functionals.

Applicable to quasilinear parabolic evolution equations.

Abstract

We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the \L ojasiewicz-Simon approach, but are of local nature.