# Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation

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….

The equation for $$q = p_x$$ can be written in divergence form as $$q_t + b(t)q_x + (D(t,\,x)q_x)_x = 0,$$ so Nash's theorem (which applies to divergence-form equations) implies that $$p_x$$ is Holder continuous under mild hypotheses on the coefficients (boundedness and measurability).
If the coefficients are more regular (e.g. in Holder classes) then parabolic Schauder estimates (e.g. in the book of Lieberman) give higher regularity of $$p$$, the general principle being that $$p$$ has twice as many spatial derivatives as temporal ones (by the scaling of the equation).
• Thank you so much for your quick reply. While with this transformation, is $v$ still a fundamental solution? Here $v(f^{-1}(s),x)=\delta_{y+h\circ f^{-1}(s)}(x)$. Do you think this matters for the estimation? I do appreciate if you could write an answer (even an outline) to my other question Commented Mar 3, 2022 at 19:46
• Yes, that is the book. Regarding your other question, I don't know. However, if $u_t + b(t)u_x + \sigma^2/2(1+\alpha(t))^2u_{xx} = 0$ then after the change of variable $u(t,x) = v(f(t),x-h(t))$ with $h' = b$ and $f' = 1/2(1+\alpha)^2$ we see that $v_t + \sigma^2(f^{-1}(t), x+h(f^{-1}(t))v_{xx} = 0$, which simplifies the structure of the equations being considered and reduces the problem to understanding small perturbations of a single Lipschitz coefficient. Commented Mar 3, 2022 at 19:46
• Here you may assume that $b,\sigma$ are as nice as possible, and $\alpha$ should belong to the class of non-increasing Holder continuous functions taking values in $[0,1]$. The motivation to study this stability is to show the uniqueness at mathoverflow.net/questions/417113/… Commented Mar 3, 2022 at 19:47