If I have a PDE $$u_t = Au + f$$ with conditions $$u(0,x) = u(T,x)$$ then if it has a solution, why is the solution called periodic? Isn't it only true that $u(0) = u(T)$? It does not follow that $u(0+\epsilon) = u(T+\epsilon)$, which I would have thought is what periodic should be.
Is that all that is required for the solution to be called that?
Finally, is there any literature that address weak periodic solutions of parabolic PDE via Galerkin method?