I want to solve the following PDE:
$$\frac{\partial h}{\partial u}=\frac{1}{2}\frac{\partial^{2} h}{\partial y^{2}}+\mathbf{1}_{\{u\leq v-t\}}yh-\beta\mathbf{1}_{\{u\leq v-t\}}h$$ such that $(u,y)\in[0,v)\times[0,\infty), h(0,y)=f(y)\in C^{\infty}$.
I would like to know if it is an ill-posed problem or not. From my understanding, the PDE is not an ill-posed problem: when $u$ travels from 0 to $v-t$, we have a smooth function for the solution, then starting from $v-t$ to $v$, we can have another PDE evolution starting from $v-t$ with evolution equation the term $$\frac{\partial h}{\partial u}=\frac{1}{2}\frac{\partial^{2} h}{\partial y^{2}}$$. It should not be an ill-posed problem. If it is, I would like to know the sufficient conditions to ensure that it is a well-solved problem and why it is an ill-posed problem. Must the coefficients be continuous in $(u,y)$, or simply the coefficients to be continuous in $y$?
Of course, if the coefficients is not continuous on the space variable, say $y\mathbf{1}_{\{y>a\}}$, the problem is clearly ill-posed.