On global dynamics of reaction--diffusion systems at resonance

TL;DR

This paper investigates the global behavior of reaction-diffusion systems at resonance using homotopy invariants, establishing conditions for solution existence and connecting stationary points.

Contribution

It introduces Landesman-Lazer type conditions at resonance and applies homotopy invariants to analyze the system's global dynamics.

Findings

Established Landesman-Lazer conditions at resonance

Expressed the Rybakowski-Conley index for bounded solutions

Proved existence of connecting solutions between stationary points

Abstract

In this paper we use the homotopy invariants methods to study the global dynamics of the reaction-diffusion systems that are at resonance at infinity. Considering degrees of the resonance for the nonlinear perturbation we establish Landesman-Lazer type conditions and use them to express the Rybakowski-Conley index of the invariant set consisting of all bounded solutions. Obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations.