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 ∂ t ⁡ u = Δ â¢ u + | u | p - 1 ⁢ u which blows up at some time T > 0 , where u : ℝ N × [ 0 , T ) → ℝ , p > 1 and ( N - 2 ) ⁢ p < N + 2 . Define S ⊂ ℝ N 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 ( N - ℓ ) -dimensional continuum for some ℓ ∈ { 1 , … , N - 1 } , then S is in fact a 𝒞 2 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 ( T - t ) and reach significant small terms in the polynomial order ( T - t ) µ for some µ > 0 . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.