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

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.