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

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.