Consider the following mixed problem for a radial, nonlinear, 2D-heat equation$$\begin{cases}u_t = u_{rr} + \dfrac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\u(0,r) = f(r) \in C^{\infty}([0,1]), \quad f(0) = 0 \\u|_{[0,+\infty) \times \{0,1\}} = 0\end{cases}$$where $F(t,r,p,q)$ is a smooth function.
It follows from Taylor, Partial Differential Equations III, Paragraph 15, Proposition 3.3 that a unique solution $u \in C^0([0,T) \times [0,1]) \cap C^{\infty}((0,T) \times [0,1])$ exists until a maximal time of existence $T > 0$ and if $T < +\infty$, then$$\lim \limits_{t\to T} \|u(t,\cdot)\|_{C^{1}_r[0,1]} = +\infty$$Assuming $u \in L^{\infty}([0,T] \times [0,1])$ (and maybe other assumptions such as a finite energy $u_r \in L^{\infty}_{[0,T]}L^2_{[0,1]}(rdr)$), is it possible to prove that the blow-up can only happen at $r = 0$, in the sense that$$\lim \limits_{t\to T, r \to 0} u_r(t,r) = +\infty, \quad \lim \limits_{t\to T} \|u_r(t,\cdot)\|_{L^{\infty}_r[\varepsilon,1]} < +\infty \quad \forall \varepsilon > 0\;?$$I think usual interior estimates should allow to bound $\|u_r\|_{L^{\infty}_r[\varepsilon,1-\varepsilon]}$ in terms of $\|u\|_{L^{\infty}}$ and $\varepsilon^{-1}$. But I don't think a blow-up could occur at $r = 1$ because the PDE is smooth at that point.