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- W1590765362 abstract "It is well-known that any bounded orbit of semilinear parabolic equations of the form ut = uxx + f(u,ux), x 2 S 1 = R/Z, t > 0, converges to steady states or rotating waves (non-constant solutions of the form U(x i 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 2 S and v 2 S are connected heteroclinically and which are not, under various conditions. We also show that )S ¸ N + P N=1 (( p (fu(rj,0))+/(2¼))) where {rj} N=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." @default.
- W1590765362 created "2016-06-24" @default.
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- W1590765362 date "2004-01-01" @default.
- W1590765362 modified "2023-10-18" @default.
- W1590765362 title "On connecting orbits of semilinear parabolic equations on $S^1$" @default.
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- W1590765362 doi "https://doi.org/10.4171/dm/173" @default.
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