On non-existence of type II blow-up for a supercritical nonlinear heat equation Hiroshi Matano Graduate School of Mathematical Sciences, University of Tokyo Frank Merle Institut Universitaire de France and Universit´ e de Cergy-Pontoise Preprint version Abstract In this paper we study blow-up of radially symmetric solutions of the nonlinear heat equation u t =Δu + |u| p-1 u either on R N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, p>p s := N+2 N-2 . We prove that if p s <p<p * then blow-up is always of Type I, where p * is a certain (explicitly given) positive number. More precisely, the rate of blow-up in the L ∞ norm is always the same as that for the corresponding ODE dv/dt = |v| p-1 v. As it is known that ‘Type II’ blow-up (or equivalently ‘fast blow-up’) can occur if p>p * , the above range of exponent p is optimal. We will also derive various fundamental estimates for blow-up that hold for any p>p s and regardless of the type of blow-up. Among other things we classify local profiles of type I and type II blow-ups in the rescaled coordinates. We then establish useful estimates for the so-called ‘incomplete’ blow-up, which reveal that incomplete blow-up solutions belong to nice function spaces even after the blow-up time. 1 Introduction We consider in this paper the nonlinear heat equation { u t =Δu + |u| p−1 u (x ∈ Ω,t> 0) u(x, 0) = u 0 (x) (x ∈ Ω), (1.1) where either Ω = R N or Ω = B R := {x ∈ R N   |x| <R}. In the latter case, we impose the Dirichlet boundary condition u(x,t)=0 (x ∈ ∂ Ω,t> 0). (1.2) The exponent p is supercritical or critical in the Sobolev sense, that is, p>p s := N +2 N − 2 or p = p s , 1 Communications on Pureand Applied Mathematics, Vol. LVII, 1494–1541 (2004)