I have the following PDE $$\partial_t u(t, x) = (ax^2 + b) \partial_{xx}u(t, x) + (cx-d)\partial_x u(t, x) + e u(t,x), (t, x) \in (0, T) \times (-B, B),$$ with $u(t, B) = u(t, -B) = 0$, for all $t \geq 0$, and $u(0, x) = f(x)$, where $a, b > 0$ and $c, d, e \in \mathbb{R}$ are constants and $B > 0$ represents the boundary. The function $f$ can be chosen as smooth or integrable as required.
The equation is of parabolic type and degenerate (since $cx-d$ can vanish a priori). I am trying find a reference for the existence and uniqueness of a (classical) solution of such problems. I have found the following paper that deals with the case of an unbounded domain only:
Igari, Katsuju - Degenerate parabolic differential equations
Thank you !
Actually, this equation is not degenerate. A degenerate equation changes its type, e.g. the coefficient in front of the second derivative would vanish in which case the equation would not be of parabolic type anymore. Here, the equation is actually (uniformly) parabolic and thus the standard results of Evans, Partial Differential Equations (Chapter 7) should apply.