Construction of entire solutions for semilinear parabolic equations

Abstract

Entire solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one entire 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 entire solutions. This article shows how nontrivial entire solutions can be constructed for a semilinear equation that has at least two distinct equilibrium solutions. The resulting entire solution is a heteroclinic orbit which connects the two given equilibria.

Keywords

• Entire solution
• heteroclinic connection
• equilibrium
• semilinear parabolic equation