Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 139, pp. 1-8.
Title: Construction of entire solutions for semilinear parabolic equations
Author: Michael Robinson (Univ. of Pennsylvania, Philadelphia, PA, USA)
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.
Submitted September 26, 2008. Published October 16, 2008.
Math Subject Classifications: 35B40, 35K55.
Key Words: Entire solution; heteroclinic connection; equilibrium;
semilinear parabolic equation.