Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation

We consider $u(x,t)$, a solution of ${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ which blows up at some time $T>0$, where $u:$\mathbb{R}$^N \times[0,T) \to $\mathbb{R}$$,$p > 1$and$(N-2)p < N+2$\mathrm{.}Define$S \subset $\mathbb{R}$^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\mathrm{\ell })$-dimensional continuum for some $\mathrm{\ell }\in \{1,\dots ,N-1\}$, then $S$ is in fact a ${\mathcal{C}}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined behavior, we have to abandon the explicit profile function as a first order approximation and take a non-explicit function as a first order description of the singular behavior. This way we escape logarithmic scales of the variable $(T-t)$ and reach significant small terms in the polynomial order $(T-t{)}^{\mu }$ for some $\mu

0$. The refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on $S$.