Structure of $\omega$-limit Sets for Almost-periodic Parabolic Equations on $S^1$ with Reflection Symmetry

Abstract

The structure of the $\omega$-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion 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 $f$ is uniformly almost periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any $\omega$-limit set $\Omega$ contains at most two minimal sets. Moreover, any hyperbolic $\omega$-limit set $\Omega$ is a spatially-homogeneous $1$-cover of hull $H(f)$. When $\dim V^c(\Omega)=1$ ($V^c(\Omega)$ is the center space associated with $\Omega$), it is proved that either $\Omega$ is a spatially-homogeneous, or $\Omega$ is a spatially-inhomogeneous $1$-cover of $H(f)$.