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{Ut=∇⋅(∇U−U∇V)in RN×(0,T),0=ΔV+Uin RN×(0,T). We say that a solution (U,V)(U,V) blows up at t=Tt=T if lim supt→T|U(t)|∞=∞lim supt→T|U(t)|∞=∞ with L∞L∞-norm |⋅|∞|⋅|∞ in RNRN. When a solution (U,V)(U,V) blows up at t=Tt=T, the blowup is called of type I if lim supt→T(T−t)|U(t)|∞<∞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=2N=2 or N⩾11N⩾11. When 3⩽N⩽93⩽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⩽93⩽N⩽9.