Research Of The Stabiliyty Of Transport Equation’s Solution With Abstract Boundary Conditions

In this paper we research transport equations with anisotropic continuous energy inhomogeneous in slab geometry for abstract boundary conditions by the methods of modern analysis, such as functional analysis,the theory of operator and the theory of semigroup, and we have got a series of new results about spectral analysis of the transport operator A and the asymptotic behaviors of solutions.The main results are showed below:Firstly, when operator H is power compact positive, we have got the following conclusions: where B is the streaming operator, K is the collision operator;2. The spectrum of transport operator A consists only of finite isolated eigenvalues with finite algebraic multiplicity.Secondly, when H is general operator, we also have got the following conclusions:1. Let K is regular in Xp (1≤p<∞), then, for all r∈[0,1), have2. The power of the operator (λ-B)-1K is compact on Banach space Xp (1<p<∞) and weakly compact on Banch space X1;3. For abstract boundary condition H, the stability of abstract Cauchy problem solution’s, i.e(i)(?)ε>0,3M>0,such that(ii) If furthermore operator K is positive, p∈(1,+∞),(?)M’>0, such that Where Co semigroup V(t) is generated by transport operator A, the eigenvalues {λ1,λ2…λn, λn+1…} are ordered in such a way that the real part decreases,Pi and Dj denote, respectively, the spectral projection and the nilpotent operator associared with λj(1≤i≤n); s(AH) is the spectral bound of AH. P denotes the projection operator corresponding to{λ∈σ(AH) such that Re λ=s(AH)}.