Finite time extinction of super-Brownian motions with deterministic catalyst

In this paper we consider a super-Brownian motion X with branching mechanism k(x)z(alpha), where k(x) > 0 is a bounded Holder continuous function on R-d and inf(xis an element ofRd) k(x) = 0. We prove that if k(x) greater than or equal to parallel toxparallel to(-l) (0 less than or equal to 1 < infinity) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) > exp(-lparallel toxparallel to)(0 less than or equal to 1 < infinity) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d > 3: it is O(parallel toxparallel to(-(alpha+1))) in one dimension, O(parallel toxparallel to(-2)(log parallel toxparallel to)(-(alpha+1))) in two dimensions, and O(parallel toxparallel to(-2)) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2Deltau = k(x)u(alpha).Mathematics, AppliedMathematicsSCI(E)0ARTICLE4538-5514