Here is the PDE $$ \begin{cases} \lim\limits_{t\rightarrow \infty}\partial_x f(t,x)=e^{-5x}\\ \lim\limits_{t\rightarrow \infty}\partial_t f(t,x)=-rxe^{-5x}\\ f(t,0)=0\\ f(t,\infty)=1 \end{cases} $$ where $r>0$ is a postive constant.
Can it be solved? Does a solution exist?
