Lifting properties of minimal sets for parabolic equations on 𝑆¹ with reflection symmetry

Zhou, Dun

doi:10.1090/proc/13283

Lifting properties of minimal sets for parabolic equations on $S^1$ with reflection symmetry
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Proc. Amer. Math. Soc. 145 (2017), 1175-1185 Request permission

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 $f$.

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