GLOBAL EXISTENCE, ASYMPTOTICS AND UNIQUENESS FOR THE REFLECTION KERNEL OF THE ANGULARLY SHIFTED TRANSPORT-EQUATION

Abstract

A nonlinear integrodifferential initial value problem, which is induced from a ''simple transport model,'' is investigated. The underlying equation contains two parameters c and alpha. Here c (c greater than or equal to O) denotes the fraction of the scattering per collision and alpha (O less than or equal to alpha less than or equal to 1) is an angular shift. In this paper, we exploit the relationship between the solution in the half space and that in slab geometry. We are thus able to show that the problem has a unique, positive, uniformly bounded and globally defined solution for O less than or equal to c less than or equal to 1 and O less than or equal to alpha less than or equal to 1. Moreover, it is shown that such global solution converges to the minimal positive solution of the half space problem as the spatial variable approaches infinity (i.e. the slab becomes thicker).