Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: ut=Δu+up+uq(x*,t) in B(R) where p,q>0,B(R)={x∈RN:|x|1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution u(x,t)=u(r,t)(r=|x|) of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.