Regularity and time asymptotic behaviour of solutions to transport equations.

We study certain regularity properties of solutions to evolution transport problems which are closely related to spectral properties of transport operators. We consider one-dimensional transport equation for a large class of anisotropic scattering operators and a variety of boundary conditions which cover, in particular, the classical known ones. The analysis uses the geometrical properties of the functional spaces, the results by Gearhart, Slemrod and Wrobel (Theorem 1); and the recent ones by Degond Song. Under adequate assumptions, various descriptions of the large time behaviour of solutions to the associated Cauchy problems are obtained.