Existence and properties of the solution of a type of PDE

Question

In doing optimal control of Parabolic PDE's we often have to solve a problem like this: $$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial p}{\partial t}(t,x)+d\Delta p(t,x)=g(y(t,x),p(t,x)) & (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial y}{\partial\nu}(t,x)=\dfrac{\partial p}{\partial \nu}(t,x)=0 & (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y_0(x),\ p(T,x)=p_T(x) & x\in\Omega\end{cases}$$ where $\Omega\subseteq\mathbb{R}^2$ or $\mathbb{R}^3$ is a bounded connected set with smooth boundary, and $y,p:[0,T]\times\overline{\Omega}\to \mathbb{R}$ are the primal and the dual state. Here $f$, $g$ are two smooth functions from $\mathbb{R}^2$ to $\mathbb{R}$. Also $y_0$ and $p_T$ are two given smooth functions.

How can we deduce the existence and the properties of the solution?

The difficulty is that if we denote $q(t,x)=p(T-t,x)$ (to transform the problem into an INITIAL VALUE PROBLEM) we come across the following system: $$\begin{cases} \dfrac{\partial y}{\partial t}-d\Delta y(t,x)=f(y(t,x),q(T-t,x)) & (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial q}{\partial t}(t,x)-d\Delta q(t,x)=-g(y(T-t,x),q(t,x)) & (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial y}{\partial\nu}(t,x)=\dfrac{\partial q}{\partial \nu}(t,x)=0 & (t,x)\in (0,T)\times\partial\Omega \\ y(0,x)=y_0(x),\ q(0,x)=p_T(x) & x\in\Omega\end{cases}$$ which looks like a DELAYED or FORWARD-BACKWARD PDE. If the terms with $T-t$ had not appeared then the existence would have been insured by the classical results in A. Pazy - Semigroups of Linear Operators and Applications to Partial Differential Equations.

The problem can be implemented numerically via finite differences for example resulting in a system of equations, but I wonder if this type of problem has been studied in a theoretical fashion.

I will be thankful for any reference or advice!