We consider the fundamental solution E(t, x, s; s0) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t + c(s, gradx) + γ)E(t, x, s; s0) = γν ∫ ((s, s′))E(t, x, s′ s0)ds′ + Ωδ, (t) δ(x)δ(s-s0) that is assumed to be valid for any (t, x) ∈ Rn+1; moreover, for t < 0 the condition E(t, x, s; s 0) = 0 holds. By using the Fourier-Laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variable s. For 0 < ν ≤ 1, the uniqueness and existence of the solution of the original problem are proved for any fixed s in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.) = 1), the solution of the integral equation is given in explicit form. In the approximation of "small mean-free paths, " various weak limits of the solution are obtained with the help of a Tauberian-type theorem for distributions. ©2000 Kluwer Academic/Plenum Publishers.
