Premium
On Nonexistence of type II blowup for a supercritical nonlinear heat equation
Author(s) -
Matano Hiroshi,
Merle Frank
Publication year - 2004
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20044
Subject(s) - mathematics , sobolev space , exponent , type (biology) , supercritical fluid , nonlinear system , mathematical analysis , norm (philosophy) , dirichlet boundary condition , range (aeronautics) , pure mathematics , boundary (topology) , physics , thermodynamics , ecology , philosophy , linguistics , quantum mechanics , political science , law , biology , materials science , composite material
In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation u t = Δ u + | 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,$$p > p_{s} := {N + 2 \over N - 2} .$$ We prove that if p s < p < p * , then blowup is always of type I, where p * is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L ∞ norm is always the same as that for the corresponding ODE dv / dt = | v | p −1 v . Because it is known that “type II” blowup (or, equivalently, “fast blowup”) can occur if p > p * , the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > p s and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so‐called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc.