Homotopy invariants methods in the global dynamics of strongly damped wave equation

Abstract

We are interested in the following differential equation $\ddot{u}(t)=-Au(t)-cA\dot{u}(t)+\lambda u(t)+F(u(t))$ 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 determine the Conley index of the set $K_{\infty }$, that is the union of the bounded orbits of this equation.

arXiv comment23 pages. arXiv admin note: substantial text overlap with arXiv:1310.6794