kinetic equations with positive collision operators

kinetic equations with positive collision operators Abstract We consider " forward-backward " parabolic equations in the abstract form Jdψ/dx+Lψ = 0, 0 < x < τ ≤ ∞, where J and L are operators in a Hilbert space H such that J = J * = J −1 , L = L * ≥ 0, and ker L = 0. The following theorem is proved: if the operator B = JL is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation µ ∂ψ ∂x (x, µ) = b(µ) ∂ 2 ψ ∂µ 2 (x, µ), 0 < x < τ , µ ∈ R, as well as to other parabolic equations of the " forward-backward " type. The abstract kinetic equation T dψ/dx = −Aψ(x) + f (x), where T = T * is injective and A satisfies a certain positivity assumption, is considered also.