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


Authors: Wen Xian Shen and Yingfei Yi
Journal: Trans. Amer. Math. Soc. 347 (1995), 4413-4431
MSC: Primary 58F39; Secondary 34C27, 35B40, 35K55, 54H20, 58F27
DOI: https://doi.org/10.1090/S0002-9947-1995-1311916-9
MathSciNet review: 1311916
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Abstract: Skew-product semi-flow $ {\Pi _t}:X \times Y \to X \times Y$ which is generated by

$\displaystyle \left\{ \begin{gathered}{u_t} = {u_{xx}} + f(y \cdot \,t,x,u,{u_x... ...D\;{\text{or }}N\;{\text{boundary conditions}} \hfill \\ \end{gathered} \right.$

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\vert\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.

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DOI: https://doi.org/10.1090/S0002-9947-1995-1311916-9
Article copyright: © Copyright 1995 American Mathematical Society

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