Nonexplosion of a class of semilinear equations via branching particle representations

摘要

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 P-k, k = 2, 3,.... The corresponding branching process is related to the semilinear partial differential equation partial derivative u/partial derivative t = Au(t, x) + Sigma(infinity)(k=2) p(k)(x)u(k)(t, x) for x is an element of R-d, where A is the infinitesimal generator of a multiplicative semigroup and the p(k)s, k = 2, 3, ..., are nonnegative functions such that Sigma(k) p(k) =1. 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.