Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups e t ( T + K e ) t ≥ 0 and e t ( T + K c + K e ) t ≥ 0 , generated by the operators T + K e and T + K c + K e , respectively, have the same essential type. According to Proposition 1, it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that R 2 ( t ) is compact on L 2 ( Ω × V , d x d v ) , and, using an interpolation argument (see Proposition 2), we establish the compactness of R 2 ( t ) on L p ( Ω × V , d x d v ) -spaces for 1 < p < + ∞ . To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem 1, is new for periodic boundary conditions.