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 $f(t,\mathrm{.})\to f^{\pm }$ as $t\to \pm \mathrm{\infty }$, 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 $\mathrm{\mid }t\mathrm{\mid }\to \mathrm{\infty }$, every solution of u_t + Au = f(t,u) + g(t) is a connection between equilibria $e^{\pm }$ of \eqref{eq:140602-1511} with $m(e^{-})\ge m(e^{+})$. Moreover, if the Morse indices satisfy $m(e^{-})=m(e^{+})$, then $u$ is isolated by linearization.