Localized and Expanding Entire Solutions of Reaction-Diffusion Equations

TL;DR

This paper investigates the long-term behavior of bounded solutions to reaction-diffusion equations in multiple dimensions, showing conditions under which solutions stabilize, connect steady states, or spread, with applications to bistable reactions.

Contribution

It provides new conditions for the classification of entire solutions as steady states or heteroclinic connections in reaction-diffusion systems.

Findings

Solutions are either time-independent or heteroclinic under certain conditions.

Solutions either remain localized or converge to steady states over time.

Application to bistable reactions demonstrates the theoretical results.

Abstract

This paper is concerned with the spatio-temporal dynamics of nonnegative bounded entire solutions of some reaction-diffusion equations in R N in any space dimension N. The solutions are assumed to be localized in the past. Under certain conditions on the reaction term, the solutions are then proved to be time-independent or heteroclinic connections between different steady states. Furthermore, either they are localized uniformly in time, or they converge to a constant steady state and spread at large time. This result is then applied to some specific bistable-type reactions.