Abstract :
LetT*(λ, φ) denote the life span of the positive, bounded solutionu(x, t) to the equationut=Δu+a(u)upinRdandu(x, 0)=λφ(x), where 0 a(x) Cα(Rd), 0 φ(x) Cb(Rd),p>1, andλ>0 is a parameter. Depending ona,φ,p, andd, it is possible thatT*(λ, φ)=∞, forλ>0 sufficiently small, or thatT*(λ, φ)<∞, for allλ>0, in which case limλ→0 T*(λ, φ)=∞. It is always true that limλ→∞ T*(λ, φ)=0. In this paper we investigate the asymptotic behavior ofT*(λ, φ) asλ→0 in the case thatT*(λ, φ)<∞, for allλ>0, and asλ→∞ in all cases. The asymptotic order depends heavily ona,φ,p, anddin the case thatλ→0, whereas in the case thatλ→∞, it depends only on whether there exists anx0witha(x0),φ(x0)≠0, or whether the supports ofaandφare separated by a positive distance.
