Topological properties of invariant sets for strongly damped wave equation at resonance

Piotr Kokocki

Published 2014-04-13, updated 2015-04-30Version 2

We are interested in the following differential equation $\ddot u(t) = -A u(t) - c A \dot u(t) + \lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and use the Conley index methods to prove that the set, which is the union of the bounded orbits of this equation, is compact and non-empty.