Consider a second-order parabolic pde
$$ u_t = \Delta u + F(t,x,u,\nabla u), \quad (t,x) \in [0,+\infty) \times M $$
where $M$ is a compact, smooth manifold with boundary. We assume that $F(x,t,u,p)$ is smooth everywhere except at one interior point $x_0 \in M$ and we set a smooth (parabolic) boundary condition
\begin{align} u(t,x) &= 0 \quad (t,x) \in [0,+\infty) \times \partial M \\ u(0,x) &= u_0(0,x) \quad x \in \overline{M} \end{align}
for some $u_0 \in C^{\infty}(\overline{M})$.
If $F$ had no singularity, then $u$ would exist up to some time $T$ and be smooth on $(0,T) \times \overline{M}$ (see Ch.15, Prop. 3.2, p.289 in Taylor, "Partial Differential Equations III : Nonlinear Equations").
Now, I assume that with our singular $F$, a solution $u \in C^{\infty}((0,T) \times M) \cap C^0([0,T) \times \overline{M})$ exists. Can I prove that $u$ is smooth on $(0,T) \times \overline{M} \setminus \{x_0\}$, i.e. smooth on the boundary which is away from $x_0$ ?
I think this should be true by formulating locally a second-order parabolic pde away from $x_0$ with no singularity. But we need to be careful with the smoothness of the boundary condition for such a problem.
I'm stating the problem in greater generality but I am mostly interested in the one-dimensional case where $M$ is an open interval and everything is easier.