Refined Regularity of the Blow-Up Set Linked to Refined Asymptotic Behavior for the Semilinear Heat Equation

We consider [Formula: see text], a solution of [Formula: see text] which blows up at some time [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text]. Define [Formula: see text] to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an [Formula: see text]-dimensional continuum for some [Formula: see text], then S is in fact a [Formula: see text] manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, 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 [Formula: see text] and reach significant small terms in the polynomial order [Formula: see text] for some [Formula: see text]. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.