Singular one-dimensional transport equations on Lp spaces

We prove the well-posedness of the Cauchy problem governed by a linear mono-energetic singular transport equation (i.e., transport equation with unbounded collision frequency and unbounded collision operator) with specular reflecting and periodic boundary conditions on Lp spaces. The large time behaviour of its solution is also considered. We discuss the compactness properties of the second-order remainder term of the Dyson–Phillips expansion for a large class of singular collision operators. This allows us to evaluate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived.  2003 Elsevier Inc. All rights reserved.