Stability Of The Essential Spectrum For Singular Transport Equations With Periodic Boundary Conditions

In this paper, the well posedness of the Cauchy problem associated to transport equations with singular cross-sections (i.e. unbounded collisions frequencies and unbound-ed collision operators) on Lp spaces with periodic boundary conditions was 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 was proved on Lp(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. And consequently, the stability of the essential spectrum is derived.