A compactness result for perturbed semigroups and application to a transport model

Abstract

In this paper we are concerned with the compactness properties of remainder terms of the Dyson-Phillips expansion of perturbed semigroups on general Banach spaces. More specifically, we derive conditions which ensure the compactness of the remainder term R n (t) for some integer n. Our result applies directly to discuss the time asymptotic behaviour (for large times) of the solution of a one-dimensional transport equation with reentry boundary conditions on L 1 -spaces without regularity conditions on the initial data.

Spectral Distribution of Transport Operator Arising in Growing Cell Populations

“…Lemma 2 (see [9]). Let B be the generator of a strongly continuous semigroup ( ) on a Banach space , and denote the bounded linear operators in .…”

“…It is well known that the streaming operator generates a strongly continuous semigroup ( ( ) ≥0 ) (see, e.g., [2][3][4][5]). If the collision operator is bounded, then the classical perturbation theory (see, [6][7][8]) shows that the transport operator generates also a strongly continuous semigroup ( ( ) ≥0 ) (see, e.g., [9]) given by the Dyson-Phillips expansion:…”