The question is as in the title.

Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r.t $t$.

Next, assume that $\frac{\partial^2 f}{\partial t \,\partial x}(t,x)$ has a version defined as a continuous function everywhere on $[0,1]^2$.

Lastly, suppose that \begin{equation} \frac{\partial f}{\partial t}(t,x)= -\Delta_x f(t,x)+ 3f(x,t)\frac{\partial f}{\partial x}(t,x) \end{equation} holds almost everywhere on $[0,1]^2$.

Then, I wonder if we can conclude that $\frac{\partial f}{\partial t}(t,x)$ can have a version defined as a continuous function everywhere on $[0,1]^2$. Moreover, can we even proceed further and conclude that $f(x,t)$ is $C^\infty$ "jointly" in $x$ and $t$, by iteration argument?

This seems like a subtle issue to me.. Could anyone please help?