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Spectral Theory in Inner Product Spaces and Applications pp 175–195Cite as

Abstract Kinetic Equations with Positive Collision Operators

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 188))

Abstract

We consider “forward-backward” parabolic equations in the abstract form Jdψ/dx+Lψ = 0, 0 < x ≤∞, 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 \( \mu \frac{{\partial \psi }} {{\partial x}}(x,\mu ) = b(\mu )\frac{{\partial ^2 \psi }} {{\partial \mu ^2 }}(x,\mu ) \) 0 < x < τ, μ ∈ ℝ, as well as to other parabolic equations of the “forwardbackward” type. The abstract kinetic equation Tdψ/dx = −Aψ(x)+f(x), where T = T* is injective and A satisfies a certain positivity assumption, is also considered. The method is based on the Krein space theory.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary 2500 University Drive NW, Calgary, T2N 1N4, Alberta, Canada

    I. M. Karabash

  2. Department of PDE, Institute of Applied Mathematics and Mechanics, R. Luxemburg str. 74, Donetsk, 83114, Ukraine

    I. M. Karabash

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  1. I. M. Karabash
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Editors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, MA 6-4, 10623, Berlin, Germany

    Jussi Behrndt  & Karl-Heinz Förster  & 

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8, 1040, Wien, Austria

    Heinz Langer

  3. Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, 98684, Ilmenau, Germany

    Carsten Trunk

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Karabash, I.M. (2008). Abstract Kinetic Equations with Positive Collision Operators. In: Behrndt, J., Förster, KH., Langer, H., Trunk, C. (eds) Spectral Theory in Inner Product Spaces and Applications. Operator Theory: Advances and Applications, vol 188. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8911-6_9

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