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

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 Rn(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 L¹-spaces without regularity conditions on the initial data.