I'm considering the following parabolic PDE \begin{equation} \frac{\partial u}{\partial t}+\frac{1}{2} \sum_{i, j=1}^d a^{i j}(t, x) \frac{\partial u}{\partial x^i \partial x^j}+\sum_{i=1}^d b^i(t, x) \frac{\partial u}{\partial x^i}+c(t, x) u+g(t, x)=0, \end{equation} with terminal condition \begin{equation} u(T,x) =\varphi(x), \end{equation} and Robin boundary condition \begin{equation} \frac{\partial u}{\partial \nu}+\gamma(t, x) u=\psi(t, x), \end{equation} where $\nu(x)$ is inner normal vector. I'm very confused about the condition $\gamma(t,x) \leq 0$ will be necessary for the well-posedness of above equation?
For example , it is necessary for heat equation with Robin boundary condition to have $\gamma(t,x) \leq 0$.
But I learned from this paper with the assumptions 2.1-2.2 and 3.1-3.4 therein, it seems that no need to have $\gamma(t,x) \leq 0$.
I quite appreciate any help.