Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

TL;DR

This paper derives a precise asymptotic profile for the gradient of solutions to the semilinear heat equation near blowup, refining previous construction techniques and confirming the gradient's behavior matches the known solution profile.

Contribution

The paper introduces a refined construction method to determine the blowup profile of the gradient for the semilinear heat equation, filling a gap in the existing literature.

Findings

Derived the asymptotic profile of the gradient near blowup point

Confirmed the gradient profile matches the known solution profile

Refined previous construction techniques for blowup analysis

Abstract

In this paper, we consider the standard semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1}u, \quad p >1. \end{eqnarray*} The determination of the (believed to be) generic blowup profile is well-established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of Bricmont-Kupiainen 1994 and Merle-Zaag 1997, and derive the following profile for the gradient: %and derive construct a blowup solution to the above equation with the gradient's asymptotic $$ \nabla u(x,T) \sim - \frac{\sqrt{2b}}{p-1} \frac{x}{|x| \sqrt{ |\ln|x||}} \left[\frac{b|x|^2}{2|\ln|x||} \right]^{-\frac{p+1}{2(p-1)}} \text{ as } x \to 0, $$ where $ b =\frac{(p-1)^2}{4p}$, which is as expected the gradient of the well-known blowup profile of the solution.