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

We consider asymptotically autonomous semilinear parabolic equationsut+Au=f(t,u). Suppose that f(t,.)→f± as t→±∞, where the semiflows induced by(*)ut+Au=f±(u) are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation g with g(t)→0 as |t|→∞, every solution ofut+Au=f(t,u)+g(t) is a connection between equilibria e± of (*) with m(e−)≥m(e+). Moreover, if the Morse indices satisfy m(e−)=m(e+), then u is isolated by linearization.