# On connecting orbits of semilinear parabolic equations on $S_{1}$

### Yasuhito Miyamoto

Reseach Institute for Electronic Science Hokkaido University Kita 12 Nishi 6, kita-ku, Sapporo 060-0812, Japan

## 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 $u∈S$ and $v∈S$ are connected heteroclinically and which are not, under various conditions. We also show that

where ${r_{j}}_{j=1}$ is the set of the roots of $f(⋅,0)$ 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.

## Cite this article

Yasuhito Miyamoto, On connecting orbits of semilinear parabolic equations on $S_{1}$. Doc. Math. 9 (2004), pp. 435–469

DOI 10.4171/DM/173