Abstract kinetic equations with positive collision operators

Abstract

We consider "forward-backward" parabolic equations in the abstract form $Jd\psi /dx+L\psi =0$, $0<x<\tau \le \infty$, where $J$ and $L$ are operators in a Hilbert space $H$ such that $J=J^{\ast }=J^{-1}$, $L=L^{\ast }\ge 0$, 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 $\mu \frac{\partial \psi }{\partial x}(x,\mu )=b(\mu )\frac{{\partial }^2\psi }{\partial {\mu }^2}(x,\mu )$, $0<x<\tau$, $\mu \in \mathbb{R}$, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation $Td\psi /dx=-A\psi (x)+f(x)$, where $T=T^{\ast }$ is injective and $A$ satisfies a certain positivity assumption, is considered also.

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

References in corpus (4)

• The similarity problem for $J$-nonnegative Sturm-Liouville operators
• Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x
• Indefinite Sturm-Liouville operators with the singular critical point zero
• Indefinite Sturm-Liouville operators $ (\sgn x) (- \frac{d^2}{dx^2} +q(x))$ with finite-zone potentials

Cited by in corpus (4)

• The similarity problem for $J$-nonnegative Sturm-Liouville operators
• The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
• On the nature of ill-posedness of the forward-backward heat equation
• A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators