Lifting properties of minimal sets for parabolic equations on $S^1$ with reflection symmetry
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Abstract:
We consider the skew-product semiflow generated by the following parabolic equation: \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}), t>0, x\in S^{1}=\mathbb {R}/2\pi \mathbb {Z}, \end{equation*} where $f(t,u,u_x)=f(t,u,-u_x)$. It is proved that the flow on uniquely ergodic minimal set $M$ is topologically conjugate to a subflow on $\mathbb {R}\times H(f)$ and $M$ is uniquely ergodic if and only if the set consisting of $1$-cover points of $H(f)$ has full measure. It is further proved that any minimal set $M$ is almost automorphic provided that $f$ is uniformly almost automorphic. Moreover, for any almost automorphic solution $u(t,x)$ contained in $M$, the frequency module $\mathcal {M}(u(t,x))$ is contained in the frequency module of $f$.References
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Additional Information
- Dun Zhou
- Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
- MR Author ID: 1112837
- Received by editor(s): January 2, 2016
- Received by editor(s) in revised form: May 6, 2016
- Published electronically: September 15, 2016
- Additional Notes: The author was partially supported by NSF of China No.11371338, 11471305, Wu Wen-Tsun Key Laboratory and the Fundamental Research Funds for the Central Universities
- Communicated by: Yingfei Yi
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1175-1185
- MSC (2010): Primary 37B55, 35K58; Secondary 35B15, 37D10, 37L30, 37A39
- DOI: https://doi.org/10.1090/proc/13283
- MathSciNet review: 3589317