Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation

Abstract

depends on the dimension d and power /, where G is the infinitesimal generator of a linear nonnegative contraction semigroup on the space B(Rd) of bounded measurable functions on Rd and c is a bounded nonnegative measurable function on Rd. This fact was recently proved by Fujita [2] when G is the Laplacian operator. In this paper we will give upper and lower bounds for the solution of (1.1) constructed by a probabilistic method (cf. (3.4) and (4.7)). As a corollary we shall obtain Fujita's result when G is a fractional power -(-A), 0< a <2, of the Laplacian operator. Our method is based on probabilistic arguments relating to the branching Markov processes (cf. Ikeda-Nagasawa-Watanabe [3], Sirao [8] and Nagasawa [7]). The necessary facts of probabilistic arguments in this context will be summarized in ?2, while in ?3 and ?4 we shall give upper and lower bounds of the probabilistic solution of (1.1) and some applications.