Time asymptotic description of the solution to an abstract Cauchy problem and application to transport equation (Q636773)

The main result regarding the time asymptotic behavior of solutions associated to a standard abstract Cauchy problem in Banach spaces, without restrictions on initial data, is proved by considering some slightly different hypotheses than the ones used in [\textit{M. Mokhtar-Kharroubi}, ``Time asymptotic behavior and compactness in neutron transport theory'', Eur. J. Mech., B, Fluid. 11, No. 1, 39--68 (1992)]. The proof uses techniques developed in [\textit{D. Song}, ``On the spectrum of neutron transport equations with reflecting boundary conditions'', Ph.D.\ Thesis, Blacksburg, Virginia (2000, Zbl 1301.47061)]. There is a quite extensive presentation of generalities on the transport equation, perturbation operators, compactness and estimation of resolvent. The asymptotic behavior of solutions in \(L^{p}\) spaces is investigated, too.