I am attempting to solve the following PDE: for $y >0$ and $t>0$,
$$V_t(y,t) + a V_{yy}(y,t) - b V_y(y,t) = f(y,t), \ V(y,0) = 0 . $$
By some changes of variables, I covert the above Euler-type equation to the following backward heat equation: for $x\in \mathbb{R}$ and $t\geq 0$:
$$U_t(x,t) + U_{xx}(x,t) = g(x,t), \ U(x,0)=0 .$$
Since $g$ is not a very nice function, I am trying to solve it by convoluting it with a Green's function. I want to see if the above equation indeed has a solution (or could I modify the non-homogeneous function $f$ or $g$, or the initial condition so that it has one). Uniqueness is not my concern at the moment. Any discussions are appreciated.