I wish to study existence/uniqueness of the solutions to a system (possibly coupled) of nonlinear PARABOLIC PDE arise from biology on a unit circle.
Could any one suggest me any references for studying existence uniqueness problem of parabolic (linear or nonlinear) on a unit circle? (I have found enough papers for the related problem given the domain as $R^n$ (bounded or unbounded) However, I am a bit lost what to do after I change my domain to a unit circle, that is the space $\mathbb{S}^1=\mathbb{R}/2\pi\mathbb{Z}$.
I cannot find any parabolic examples in Friedman's book for the domain as unit circle.
Appreciate any suggestions for reference.