Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE $$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, x) = 0, \forall x\in \mathbb R.$$ It is standard that there is a classical solution. Assuming $\partial_{xxx} v$ exists, by taking $\partial_x$ to the equation, with $\hat v = \partial_x v$, we obtain $$\partial_t \hat v = b \partial_x \hat v + \partial_{xx} \hat v + \partial_x \ell + \hat v \partial_x b, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad \hat v(0, x) = 0, \forall x\in \mathbb R.$$ Therefore, by uniqueness of the solution, we conclude that

- The solution $\hat u$ of $$(P2) \quad \partial_t \hat u = b \partial_x \hat u + \partial_{xx} \hat u + \partial_x \ell + \hat u \partial_x b, \ \hbox{ on } \mathbb R^+\times \mathbb R; \quad \hat u(0, x) = 0, \hbox{ on } \mathbb R,$$ satisfies $\hat u = \partial_x v$.

Now, I want to know if the above conclusion satisfies without assuming $v\in C^{1,3}$, that is

- Let $b, \ell \in C_b^{1,2}(\mathbb R^2)$, do $\hat u$ of (P2) and $v$ of (P1) satisfy $\hat u = \partial_x v$?