Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of isolated compact invariant sets. As the frequency of the oscillations tends to infinity, we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation.