Abstract kinetic equations with positive collision operators

Details Abstract kinetic equations with positive collision operators Abstract kinetic equations with positive collision operators

2007-08-18

arxiv.org/abs/0708.2510v2

Operator Theory: Advances and Applications, 2009, Volume 188, pp. 175-195

Mathematics

Spectral Theory

20 pages, LaTeX2e, version 2, references have been added, changes in the introduction

Scientific paper

10.1007/978-3-7643-8911-6_9

We consider "forward-backward" parabolic equations in the abstract form $Jd \psi / d x + L \psi = 0$, $ 0< x < \tau \leq \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^*=J^{-1}$, $L=L^* \geq 0$, 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 $ \mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu)$, $ 0

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