I am interested in proving the existence and regularity of solution to the following problem:

$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial y}{\partial\nu}(t,x)=0, & (t,x)\in (0,T)\times\Omega \\ y(0,x)=y(T,x), & x\in\Omega\end{cases}$$

where $\Omega\subseteq\mathbb{R}^N$ is a bounded domain with smooth boundary, $c$ is a bounded and $T$ - periodic function on $Q_T=(0,T)\times\Omega$, and $f\in L^2(Q_T)$ being $T$ periodic too.

I know that this type of problem is treated in the literature: for example in *P.Hess - Periodic Parabolic Boundary Value Problems and Positivity, 1991*, but in the setting of Holder spaces, not that of Sobolev spaces.

If the problem were an initial value problem then the existence will follow via Banach contraction theorem, like in *Pazy - Semigroups of linear operators and applications to Partial Differential Equations* (see Chapter 6), by considering the operator $T:C([0,T], L^2(\Omega))\to C([0,T],L^2(\Omega))$ defined by the right hand side of the mild formulation of our problem.

My questions are:

Are there some

**periodic Sobolev spaces**that are studied somewhere? A special interest is on density and (compact) embeddings of such spaces.What space should I consider in order to obtain a solution for less regularity of $f$, like $f\in L^2([0,T]\times\Omega)$ (generalizing the technique with Banach contraction principle)?

I found some ideas in *Bagyrov - On the Existence of a Positive Solution
of a Nonlinear Second-Order Parabolic Equation
with Time-Periodic Coefficients, 2005.*