On minimal sets of scalar parabolic equations with skew-product structures

Abstract:

Skew-product semi-flow ${\Pi _t}:X \times Y \to X \times Y$ which is generated by \[ \left \{ \begin {gathered} {u_t} = {u_{xx}} + f(y \cdot t,x,u,{u_x}),\qquad t > 0,\;0 < x < 1,\;y \in Y, \hfill \\ D\;{\text {or }}N\;{\text {boundary conditions}} \hfill \\ \end {gathered} \right .\] is considered, where $X$ is an appropriate subspace of ${H^2}(0,1),\;(Y, \mathbb {R})$ is a minimal flow with compact phase space. It is shown that a minimal set $E \subset X \times Y$ of ${\Pi _t}$ is an almost $1{\text { - }}1$ extension of $Y$, that is, set ${Y_0} = \{ y \in Y|\operatorname {card} (E \subset {P^{ - 1}}(y)) = 1\}$ is a residual subset of $Y$, where $P:X \times Y \to Y$ is the natural projection. Consequently, if $(Y,\mathbb {R})$ is almost periodic minimal, then any minimal set $E \subset X \times Y$ of ${\Pi _t}$ is an almost automorphic minimal set. It is also proved that dynamics of ${\Pi _t}$ is closed in the category of almost automorphy, that is, a minimal set $E \subset X \times Y$ of ${\Pi _t}$ is almost automorphic minimal if and only if $(Y,\mathbb {R})$ is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.