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 ut=Δu+|u|p−1u which blows up at some time T>0, where u:RN×[0,T)→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 C2. 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