The generic gradient-like structure of certain asymptotically autonomous semilinear parabolic equations

TL;DR

This paper studies the structure of solutions to certain asymptotically autonomous semilinear parabolic equations, showing that generically solutions connect equilibria with Morse index relations, and are isolated under specific conditions.

Contribution

It demonstrates that solutions of these equations form connections between equilibria with Morse index relations and are isolated when indices are equal, under generic perturbations.

Findings

Solutions connect equilibria with Morse index relations.

Solutions are isolated when Morse indices are equal.

Generically, solutions form gradient-like structures.

Abstract

We consider asymptotically autonomous semilinear parabolic equations u_t + Au = f(t,u). Suppose that $f(t,.)\to f^\pm$ as $t\to\pm\infty$, where the semiflows induced by \label{eq:140602-1511} u_t + Au = f^\pm(u) \tag{*} are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation $g$ with $g(t)\to 0$ as $|t|\to\infty$, every solution of u_t + Au = f(t,u) + g(t) is a connection between equilibria $e^\pm$ of \eqref{eq:140602-1511} with $m(e^-)\geq m(e^+)$. Moreover, if the Morse indices satisfy $m(e^-) = m(e^+)$, then $u$ is isolated by linearization.