NONEXPLOSION OF A CLASS OF SEMILINEAR EQUATIONS VIA BRANCHING PARTICLE REPRESENTATIONS.

We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3,…. The corresponding branching process is related to the semilinear partial differential equation ∂u/∂t = Au(t, x) + ∑_k=2^∞ Pk (x)u^k (t, x) for x ∈ ℝ^d, where A is the infinitesimal generator of a multiplicative semigroup and the p_ks, k = 2,3,…, are nonnegative functions such that ∑_k P_k = 1. We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others. [ABSTRACT FROM AUTHOR]