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

We consider $$ , a solution of $$ which blows up at some time $$ , where $$ , $$ and $$ . Define $$ to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an $$ -dimensional continuum for some $$ , then S is in fact a $$ 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 $$ and reach significant small terms in the polynomial order $$ for some $$ . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.