Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. (there is slight abuse of notation in the equality but never mind)
Under what conditions does a solution to this problem exist? By solution I mean $u \in L^2(0,T;V)$ with $u_t \in L^2(0,T;V^*)$ (or $u_t \in L^2(0,T;H)$ if data is smooth enough). Apart from requiring maybe $A(0) = A(T)$ and $f(0) = f(T)$.
How does one prove this via the Galerkin approach?
I know of two ways: 1) To use a maximal monotone approach as in Roubicek's book. This seems like overkill though for such a simple PDE 2) To use a fixed point method to show that the Galerkin approximations $u_n$ are also periodic ($u_n(0) = u_n(T)$). However this requires initial data to be of appropriate size so that the fixed point map maps into a compact set and Brouwer's FP theorem can be used.
So I guess there must be a simple way but I can't find it. I posted this on the StackExchange site too (https://math.stackexchange.com/questions/651194/weak-periodic-solution-of-parabolic-pde) but I didn't get any answer. Thanks.
