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

Abstract

We consider u⁢(x,t)${u(x,t)}$, a solution of ∂t⁡u=Δ⁢u+|u|p-1⁢u${\partial_{t}u=\Delta u+|u|^{p-1}u}$ which blows up at some time T>0${T>0}$, where u:ℝN×[0,T)→ℝ${u:\mathbb{R}^{N}\times[0,T)\to\mathbb{R}}$, p>1${p>1}$ and (N-2)⁢p0${\mu>0}$. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.

Keywords

• blow-up solution
• blow-up set
• blow-up profile
• regularity
• semilinear heat equation
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