Positivity and monotonicity properties of transport equations with spatially dependent cross sections

Abstract

We investigate the transport equation

with suitable boundary conditions through an equivalent integral equation. Assuming the incoming fluxes, the internal source term f(x,μ), the cross section c(x) and the parameter ξ to be nonnegative, we prove the existence of a unique dominant eigenvalue ξ=ξ₀(τ) for which the homogeneous problem has a positive solution (critical case), the existence of a unique positive solution for ξ < ξ₀(τ) (non-critical case), and the absence of positive solutions for ξ > ξ₀(τ) (supercritical case). We show ξ₀(τ) to decrease continuously from ∞ to some ξ₀(∞)>0 whenever ξ increases from 0 to ∞ (monotonicity). The results are obtained by studying an operator that leaves invariant the cone of nonnegative functions in L_∞(0,τ).