A sufficient condition for type I blowup in a parabolic–elliptic system

This paper is concerned with blowup of positive solutions to a Cauchy problem for a parabolic–elliptic system We say that a solution (U,V) blows up at t=T if lim supt→TU(t)∞=∞ with L∞-norm ∞ in RN. When a solution (U,V) blows up at t=T, the blowup is called of type I if lim supt→T(T−t)U(t)∞<∞ and of type II otherwise. It was shown in Herrero and Velázquez (1996) [12] and Mizoguchi and Senba (2007) [20] that there exist radial type II blowup solutions for N=2 or N 11. When 3 N 9, type II blowup solutions were given by formal analysis in Brenner et al. (1999) [2] and Herrero et al. (1997) [10]. We give a sufficient condition for a solution to exhibit type I blowup in the case of 3 N 9.