Invariant sets and connecting orbits for nonlinear evolution equations at resonance

Abstract

We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman- Lazer and strong resonance conditions. Obtained index formulas are used to derive criteria determining the existence of orbits connecting stationary points.