Weak periodic solution of parabolic PDE 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?
Thanks
 A: Step I. Solve the initial value problem
$$
\left\{
\begin{array}{lll}
X_t(t)=-A(t)X(t), \\ X(s)=I,
\end{array}
\right.
$$
where $I : H\to H$ is the identity. Assume that $K(t,s)$ is the solution - this is a bounded operator from $H$ to $H$, for every $s,t$ - There is both theory and approximations for obtaining $K$, i.e.,
$$
K(t,s)=\lim_{n\to\infty}\prod_{k=1}^n \bigg(1-\frac{t-s}{n}A\Big(s+k\frac{t-s}{n}\Big)\bigg).
$$
Step II. The solution of 
$$
X_t+A(t)X=f(t), \quad X(0)=X^0,
$$ 
is then expressed as
$$
X(t)=K(t,0)\,X^0+\int_0^t K(t,s)\,f(s)\,ds.
$$
Step III. Check whether there exists a $X^0$, such that
$$
X(T)=X(0)
$$
or equivalently
$$
K(T,0)\,X(0)+\int_0^T K(T,s)\,f(s)\,ds=X(0).
$$
or
$$
\big(K(T,0)-I\big)X(0)=-\int_0^T K(T,s)\,f(s)\,ds.
$$
This only requires that the operator $K(T,0)-I :H\to H$ possesses an inverse or equivalently
$$
1\not\in \sigma\big(K(T,0)\big).
$$
A: The conditions are the "usual" ones, see e.g. Brezis, Functional analysis, 2010:
o) $V \subset H$ is dense, $H$ and $H^*$ are identified, $V$ is separable.
o) $A(t) : V \to V^*$ linear, bounded and coercive (i.e. $A + \mu I$ is $V$-elliptic for a $\mu > 0$ large enough) uniformly in $t$, and $t \mapsto (A(t) u, v)$ is measurable for all $u, v \in V$.
o) $f \in L^2(0, T; V^*)$.
o) $u(0) = u(T)$ is understood as equality in $H$.
Then, there exists a unique solution $u \in H^1(0,T; V^*) \cap L^2(0,T; V)$ and it depends linearly and continuously on $f$.
Periodicity of data is not required; it doesn't even make sense.
The proof by the Faedo-Galerkin approximation method can be applied exactly as in the non-periodic case; see e.g. Evans, Partial differential equations, 1998; or Wloka Partial differential equations, 1987; and of course, Lions/Magenes and Dautray/Lions.
