On Connecting Orbits of Semilinear Parabolic Equations on $S^1$

It is well-known that any bounded orbit of semilinear parabolic equations of the form \[ u_t=u_{xx}+f(u,u_x),\ \ x\in S^1={\mathbb R}/{\mathbb Z},\ \ t>0, \] 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 {\it clusters} and the {\it 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 $\sharp S\geq N+\sum_{j=1}^N [[\sqrt{(f_u(r_j,0))_+}/(2\pi )]]$ 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 {\it structure} of $S$ completely determines the heteroclinic connections.

2000 Mathematics Subject Classification: 35B41, 34C29

Keywords and Phrases: global attractor, heteroclinic orbit, zero number, semilinear parabolic equation.

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