Construction of eternal solutions for a semilinear parabolic equation

TL;DR

This paper demonstrates how to construct nontrivial eternal solutions, specifically heteroclinic orbits, for a class of semilinear parabolic equations with multiple equilibria, expanding understanding of their long-term behavior.

Contribution

It introduces a method to construct nontrivial eternal solutions connecting two equilibria in semilinear parabolic equations.

Findings

Existence of nontrivial eternal solutions for certain semilinear equations.

Construction of heteroclinic orbits connecting equilibrium solutions.

Extension of the class of equations known to support eternal solutions.

Abstract

Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution -- the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singularities in both backward and forward time. However, semilinear parabolic equations can also support nontrivial eternal solutions. This article shows how nontrivial eternal solutions can be constructed for a semilinear equation that has at least two distinct equilibrium solutions. The resulting eternal solution is a heteroclinic orbit which connects the two given equilibria.