The Fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain

Consider the equation u(t) = Delta u - Vu + au(p) in R(n) x (0, T): u(x, 0) = phi (x) not greater than or equal to 0 in R(n). (0,1) where p > 1. n >= 2, T is an element of (0, infinity vertical bar, V(x) similar to omega/vertical bar x vertical bar(2) as vertical bar x vertical bar -> infinity, for some omega not equal 0, and a(x) is on the order vertical bar x vertical bar(m) as vertical bar x vertical bar -> infinity, for some Keywords: M E (-infinity, infinity). A solution to the above equation is called global if Critical exponent T = infinity. Under some additional technical conditions, we calculate Blow-up a critical exponent p* such that global solutions exist for p > p*, Global solution while for 1 < p <= p*, all solutions blow up in finite time. We Fujita exponent also show that when V equivalent to 0, the blow-up/global solution dichotomy Exterior domain for (0.1) coincides with that for the corresponding problem in an exterior domain with the Dirichlet boundary condition, including the case in which p is equal to the critical exponent. (C) 2008 Elsevier Inc. All rights reserved.