Total versus single point blow-up of solutions of a semilinear parabolic equation with localized reaction

Abstract In this paper, we study the initial-boundary value problem of a semilinear parabolic equation with localized reaction, ( P ) u t =Δu+u p +u q (x ∗ ,t), (x,t)∈B×(0,T), u(x,t)=0, (x,t)∈∂B×(0,T), u(x,0)=u 0 (x), x∈B, where B is a unit ball in R N , x ∗ ∈B , and p,q>0. For the case x ∗ =0 , we completely classify blow-up solutions of (P) into total blow-up cases and single point blow-up cases according to the values p and q. Moreover, we give the blow-up rates of solutions near the blow-up time. For the other case x ∗ ≠0 , we show total blow-up and single point blow-up of solutions of (P) in some cases depending on p and q.