A cell complex structure for the space of heteroclines for a semilinear parabolic equation

TL;DR

This paper demonstrates that for a certain semilinear parabolic equation on an unbounded domain, the space of heteroclinic orbits forms a finite-dimensional cell complex, unifying known results about attractors and equilibria.

Contribution

It establishes a finite-dimensional cell complex structure for the space of heteroclinic orbits in a specific unbounded domain parabolic equation, linking attractor and equilibrium theories.

Findings

The space of heteroclinic orbits has a cell complex structure.

The structure depends on spatial dimension and nonlinearity degree.

Results unify attractor and equilibrium properties for the equation.

Abstract

It is well known that for many semilinear parabolic equations there is a global attractor which has a cell complex structure with finite dimensional cells. Additionally, many semilinear parabolic equations have equilibria with finite dimensional unstable manifolds. In this article, these results are unified to show that for a specific parabolic equation on an unbounded domain, the space of heteroclinic orbits has a cell complex structure with finite dimensional cells. The result depends crucially on the choice of spatial dimension and the degree of the nonlinearity in the parabolic equation, and thereby requires some delicate treatment.