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?