Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation

Abstract

We consider $u(x,t)$, a solution of $u_t=\Delta u+\left|u\right|{}^{p-1}u$ which blows up at some time $T>0$, where $u:{\mathbb{R}}^N\times [0,T)\to \mathbb{R}$, $p>1$, and $(N-2)p<N+2$. Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set $S$ is continuous and $(N-1)$-dimensional implies that it is $C^2$. In particular, we compute the $N-1$ principal curvatures and directions of $S$. Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for $S$ and refined singular behavior of $u$ near $S$ are linked through a new mechanism of algebraic cancellations that we explain in detail