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We consider $u(x,t)$, a solution of $\partial_tu = \Delta u + |u|^{p-1}u$which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to\mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to bethe blow-up set of $u$, that is the set of all blow-up points. Under suitablenondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensionalcontinuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a$\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptoticbehavior of $u$ near blow-up. In order to obtain such a refined behavior, wehave to abandon the explicit profile function as a first order approximationand take a non-explicit function as a first order description of the singularbehavior. This way we escape logarithmic scales of the variable $(T-t)$ andreach significant small terms in the polynomial order $(T-t)^\mu$ for some $\mu> 0$. The refined asymptotic behavior yields geometric constraints of theblow-up set, leading to more regularity on $S$.

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