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

Abstract

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