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

Abstract

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: u t = Δ u + u p + u q ( x * , t ) in B ( R ) where p , q > 0 , B ( R ) = { x ∈ R N : | x | < R } and x * ≠ 0 . If max ( p , q ) > 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. (i) If q < p ( p > 1 ) or p = q > 2 , then single point blow-up occurs whenever solutions blow up. (ii) If 1 < p < q , both phenomena, total blow-up and single point blow-up, occur depending on the initial data. (iii) If p ⩽ 1 < q , total blow-up occurs whenever solutions blow up. (iv) If max ( p , q ) ⩽ 1 , every solution exists globally in time.