solve a parabolic PDE backwards in time

Question

I am trying to solve the following equations: \begin{equation*} \begin{aligned} -\frac{\partial u}{\partial t}&=\Delta u+Cu+Z, (x,t)\in\Omega_T\\ \frac{\partial u}{\partial x}&=0, x\in \partial \Omega_T \\ u(T,x)&=0, \forall x\in\Omega. \end{aligned} \end{equation*} where I employ the BTCS(backward in time and central space) method. This kind of problem is usually not well-posed so I make a time transformation such that $s=T-t$ which makes it as an initial problem \begin{equation*} \begin{aligned} \frac{\partial u}{\partial s}&=\Delta u+Cu+Z, (x,s)\in\Omega_T\\ \frac{\partial u}{\partial x}&=0, x\in \partial \Omega_T \\ u(0,x)&=0, \forall x\in\Omega. \end{aligned} \end{equation*} However, I still encountered the solution blows up(goes to huge numbers)... is there any reference to such problems?