Asymptotics of Transport Equations for Spherical Geometry in L 2 with Reflecting Boundary Conditions

The time dependent transport equation in a sphere with reflecting boundary conditions is discussed in the setting of L 2. Some aspects of the spectral properties of the strongly continuous semigroup T(t) generated by the corresponding transport operator A are studied, and it is shown that the spectrum of T(t) outside the disk {λ: |λ| ≤ exp(−λ*t)} (where λ* is the essential infimum of the total collision frequency σ (r, v), or λ* = ess inf r lim v →0+ σ (r, v)) consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of σ(T(t))∩{λ : |λ| > exp(−λ*t)} can only appear on the circle {λ : |λ| = exp(−λ*t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained.