Structure of ω-limit sets for almost-periodic parabolic equations on S1 with reflection symmetry

The structure of the ω-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equationut=uxx+f(t,u,ux),t>0,x∈S1=R/2πZ, where f is uniformly almost periodic in t   and satisfies f(t,u,ux)=f(t,u,−ux)f(t,u,ux)=f(t,u,−ux). We show that any ω-limit set Ω contains at most two minimal sets. Moreover, any hyperbolic ω  -limit set Ω is a spatially-homogeneous 1-cover of hull H(f)H(f). When dim⁡Vc(Ω)=1dim⁡Vc(Ω)=1 (Vc(Ω)Vc(Ω) is the center space associated with Ω), it is proved that either Ω is a spatially-homogeneous, or Ω is a spatially-inhomogeneous 1-cover of H(f)H(f).