Let $\Omega_x:= (-\infty,\infty)$ and $\Omega_t:= [0,\infty)$ be the domains respect the spatial $x$ and time $t$ variable. I'm trying to solve the following partial differential equation for $u:\Omega_x\times\Omega_t\to \mathbb{R}$ $$\frac{\partial\,u(x,t)}{\partial\,t} = f(x)\left( \frac{\partial^2\,u(x,t)}{\partial\,x^2} - \frac{\partial\,u(x,t)}{\partial\,x} \right),$$ with boundary conditions $u(x,0) = g(x)$, $u(0,t) = 0$. Where $f\in C^\infty(\Omega_x)$ monotonic increasing and $g \in C(\Omega_x)$ non decreasing (Not necessarily smooth).
Many thanks.