Extinction of Superdiffusions and Semilinear Partial Differential Equations

A superdiffusion is a measure-valued branching process associated with a nonlinear operatorLu−ψ(u) whereLis a second order elliptic differential operator andψis a function from Rd×R+to R+. In the caseL1⩽0 (the so-called subcritical case), the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1. Most results on connections between the superdiffusion and differential equations involvingLu−ψ(u) were obtained for the subcritical case. In the present paper, we assume only thatL1 is a bounded function. In this more general setting, the probability of extinction can be smaller than 1, and we show that his happens if and only if there exists a strictly positive solution of the equationLu−ψ(u)=0 in Rd. We also establish a relation between the probability of extinction in a domainDand strictly positive solutions of equationLu−ψ(u)=0 inDthat are equal to 0 on the boundary ofD. We call such solutions special. For the equation (Δ+c) u=uα, whereΔis the Beltrami–Laplace operator on a complete Riemannian manifoldE, strictly positive solutions inEwere studied in connection with a geometrical problem: Which two functions represent scalar curvatures of two Riemannian metrics related by a conformal mapping (see [1] and references there)?