# Periodic solution for linear parabolic equation - existence, regularity

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:

1. Are there some periodic Sobolev spaces that are studied somewhere? A special interest is on density and (compact) embeddings of such spaces.

2. 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.

• Periodic Sobolev means locally Sobolev on the whole space + periodicity. You can take anything that is true on a bounded domain, forget the boundary regularity, and that's your result on your periodic space. Oct 29, 2022 at 17:20
• Can you detail a little bit please? I do not see what is the link between periodicity and regularity on the boundary... Oct 30, 2022 at 3:10
• Every point is an interior point. If you look in Evan's PDE textbook, you'll notice that interior regularity is one thing, and the boundary regularity another. In your case, there is no boundary, so everything stems from interior regularity -- because of periodicity. Oct 30, 2022 at 17:06

For 1., if I am not mistaking you're searching for time-periodic functions enjoying Sobolev regularity in the space variable so the Sobolev regularity is not really linked with the periodicity: you're just working with $$L^p(\mathbf{T};X)$$ or $$\mathscr{C}^0(\mathbf{T};X)$$ where $$\mathbf{T}$$ is the flat-torus $$\mathbf{R}/\mathbf{Z}$$ and $$X$$ some Banach space.
For 2. I am a bit confused by your setting. Any reasonnable theory of solutions for your system should apply similarly to the case in which $$\Omega=\mathbf{T}^d$$ (that is : periodicity in the space variable also) and $$c=f=0$$. But in that case your time-periodicity conditions imposes the following condition on the (space) Fourier coefficients of $$y$$ : on the one hand you have $$y_k(T) = e^{-|k|^2 T}y_k(0)$$ (usual exponential decay from the heat kernel) and on the other one you want $$y_k(T)=y_k(0)$$ ... this does not leave much room for solution other than constants ...