Stability of the Essential Spectrum for Singular Transport Semigroups with Periodic Boundary Conditions

Abstract

In this article, the well posedness of the Cauchy problem associated to transport equations with singular cross-sections (i.e., unbounded collision frequencies and unbounded collision operators) on L^p spaces with periodic boundary conditions is discussed, and some compactness (or weak compactness) of the first order remainder term of the Dyson-Phillips expansion for a large class of singular collision operators is proved on L^p(1 < p < ∞) (or L¹) spaces. Thus, this allows us to evaluate the essential type of the transport semigroup from which the asymptotic behavior and the well posedness of the solution is derived. Consequently, the stability of the essential spectrum is concluded.

Keywords:

• singular transport equation
• C₀-semigroup
• essential spectrum
• Dyson-Phillips expansion