PDE with periodic solution with Galerkin method

Can anyone give me a reference to where I cacn find existence of solutions to parabolic PDE (eg. a standard linear PDE) with initial and end conditions $u(0) = u(T) = u_0$ so that periodic solutions are obtained. And are there any other conditions under which the solution is periodic?

I want to see this with specifically the Galerkin method.

Thanks

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