On the uniqueness of L-1-continuation after blowup

摘要

This paper is concerned with the uniqueness of L-1-continuation beyond blowup for a Cauchy problem of a semilinear heat equation [GRAPHICS] with p > 1, 0 < (T) over tilde <= infinity and u(o) is an element of L-infinity(R-N). Here we say that u is an L-1-solution of (P) if u is an element of C([0, (T) over tilde); L-loc(1)(R-N)) with u is an element of L-loc(p)(R-N x (0, (T) over tilde)) satisfies (P) in the distributional sense. In the case of pS < p < pJL, a counter example for the uniqueness of radial L-1-solution of (P) after blowup was given in [M. Fila, N. Mizoguchi, Multiple continuation beyond blow-up, Differential Integral Equations 20 (2007) 671-680], where pS and PJL are the exponent of Sobolev and of Joseph and Lundgren, respectively. We give a sufficient condition for the uniqueness of L-1-continuation beyond blowup for p > PJL in the radial case. If for an L-1-solution u of (P) there exists a sequence {u(n)} I of classical solutions of (P) such that u(0,n) -> u(0) in L-infinity(R-N) as n -> infinity for the sequence {u(0,n)) of initial data and that u(n)(t) -> u(t)in L-loc(p) (R-N) as n -> infinity for t is an element of (0, (T) over tilde), then u is called a limit L-1-solution. Based on the sufficient condition, we prove the uniqueness of limit L-1-solution with radial symmetry after blowup for p > pJL. (c) 2008 Published by Elsevier Inc.