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

In this paper, we study the initial-boundary value problem of a semilinear parabolic equation with localized reaction, (P)   ut = Δu + up + uq (x∗, t ), (x, t ) ∈ B ×(0,T ), u(x, t ) = 0, (x, t ) ∈ ∂B ×(0,T ), u(x, 0) = u0(x), x ∈ B, where B is a unit ball in RN, 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.  2003 Elsevier Science (USA). All rights reserved