On connecting orbits of semilinear parabolic equations on $S^1$

Abstract

It is well-known that any bounded orbit of semilinear parabolic equations of the form

converges to steady states or rotating waves (non-constant solutions of the form $U(x-ct))$ under suitable conditions on $f$. Let $S$ be the set of steady states and rotating waves (up to shift). Introducing new concepts – the clusters and the structure of $S$ –, we clarify, to a large extent, the heteroclinic connections within $S$; that is, we study which $u\in S$ and $v\in S$ are connected heteroclinically and which are not, under various conditions. We also show that

where $\{r_j{\}}_{j=1}^N$ is the set of the roots of $f(\cdot ,0)$ and $[[y]]$ denotes the largest integer that is strictly smaller than $y$. In paticular, if the above equality holds or if $f$ depends only on $u$, the structure of $S$ completely determines the heteroclinic connections.