Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$

$$\partial_t p + b(t)\partial_x p + D(t,x)\partial^2_{xx}p=0,$$

where $b$, $D$ are nice enough functions. I look for the continuity of the derivatives $\partial_t p$, $\partial_x p$ of the solution. It is known by Nash's paper (Continuity of Solutions of Parabolic and Elliptic Equations) that, under very reasonable conditions on $b$, $D$, we have the Hölder-type continuity of $p$. Is there any work concerning such continuity analysis of $\partial_t p$, $\partial_x p$?

PS: My idea is to consider $q\mathrel{:=}\partial_x p$. Then

$$\partial_t q + b(t)\partial_xq + D(t,x)\partial^2_{xx}q=-\partial_x D(t,x)\partial_{xx}p$$

is a similar parabolic equation for $q$ with an additional source. But the term $\partial_x D(t,x)\partial_{xx}p$ contains $\partial_{xx}p$, which makes the estimation even harder….