Connecting orbits for nonlinear differential equations at resonance

TL;DR

This paper investigates the existence of connecting orbits in nonlinear differential equations at resonance, deriving an index formula that generalizes known conditions and applies to heat equations at resonance.

Contribution

It introduces a new index formula for semiflows at resonance, generalizing Landesman-Lazer and strong resonance conditions, and applies it to heat equations.

Findings

Derived an index formula relating Conley index to geometrical assumptions.

Generalized classical resonance conditions in the context of differential equations.

Provided criteria for existence of connecting orbits at resonance for heat equations.

Abstract

We study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the Conley index of associated semiflow with respect to appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalization of well known in literature Landesman-Lazer and strong resonance conditions. Obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity.