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.
Submitted September 26, 2008. Published October 16, 2008.
Math Subject Classifications: 35B40, 35K55.
Key Words: Entire solution; heteroclinic connection; equilibrium; semilinear parabolic equation.
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| Michael Robinson |
Department of Mathematics
University of Pennsylvania
209 South 33rd Street
Philadelphia, PA 19104, USA
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