Almost automorphically-forced flows on S1 or ℝ in one-dimensional almost periodic semilinear heat equations

In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation: ut=uxx+f(t,u,ux),t>0,0<x<L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${u_t} = {u_{xx}} + f(t,u,{u_x}),\,\,\,\,\,t > 0,\,\,\,\,\,0 < x < L$$\end{document} with the periodic boundary condition u(t,0)=u(t,L),ux(t,0)=ux(t,L),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(t,0) = u(t,L),\,\,\,\,\,\,\,{u_x}(t,0) = {u_x}(t,L),$$\end{document} where f is uniformly almost periodic in t. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we refer to it as a minimal set) can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S1 = ℝ/Lℤ (see Definition 2.4 for “residually embedded”). Particularly, if f (t,u,p) = f(t,u, −p), then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on ℝ (see Definition 2.4 for “embedded”). Moreover, it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation. In addition, we further consider the asymptotic dynamics of the skew-product semiflow generated by (0.1) with the Neumann boundary condition ux(t, 0) = ux(t, L) = 0 or the Dirichlet boundary condition u(t, 0) = u(t, L) = 0. For such a system, it has been known that the ω-limit set of any bounded orbit contains at most two minimal sets. By applying the new results for (0.1) + (0.2), under certain direct assumptions on f, we prove in this paper that the flow on any minimal set of (0.1) with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on ℝ. Finally, a counterexample is given to show that even for quasi-periodically-forced equations, the results we obtain here cannot be further improved in general.