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Lifting properties of minimal sets for parabolic equations on $ S^1$ with reflection symmetry


Author: Dun Zhou
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
Published electronically: September 15, 2016
MathSciNet review: 3589317
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Abstract: We consider the skew-product semiflow generated by the following parabolic equation:

$\displaystyle u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z},$    

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$.

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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

DOI: https://doi.org/10.1090/proc/13283
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
Article copyright: © Copyright 2016 American Mathematical Society