SPECTRAL PROPERTIES OF TRANSPORT EQUATIONS FOR SLAB GEOMETRY IN L¹WITH REENTRY BOUNDARY CONDITIONS

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

“…Since (H) is satisfied, Theorem 3.8 in [12] implies that, for Re λ > −λ * , the operator [K (λ − T ) −1 ] 2 K is compact on L 1 (G). Now the result follows from the fact that, for m 3, the operator (λ Proof.…”

“…Now the result follows from the fact that, for m 3, the operator (λ Proof. If (H) is satisfied, then, according to Lemma 3.1 in [12], there exist constants Ξ 0 > 0 andτ such that K (λ − T ) −1 K Ξ 0 ω + λ * + iτ δ−1 ln ω + λ * + iτ uniformly on {λ = β + iτ : β ω, |τ | τ }. Further, for every λ with Re λ > −λ * we have |Im λ| (λ − T ) −1 K (λ − T ) −1 2m = 0 uniformly on R ω which ends the proof.…”

“…2 Proof of Theorem 3.1. Since the semigroup (S(t)) t 0 is positive and the spectral bound of T is equal to −λ * (see [12] or [13]), it follows from [21] that ω(S) = −λ * . Hence the essential type ω e (S) of (S(t)) t 0 satisfies ω e (S) −λ * and therefore r e S(t) = e ω e (S)t e −λ * t for all t 0.…”

“…There are much works in this direction motivated by various problems arising in mathematical physics, bio-mathematics and, in particular, the time dependent neutron transport equation (see, for example, [1,4,5,8,9,11,10,12,[14][15][16]18,20] and the references therein). For transport equations, the compactness of some order remainder term of the Dyson Phillips expansion in L p -spaces, 1 p < +∞, was established only for no-reentry boundary conditions (i.e.…”