Advertisement

Siberian Mathematical Journal

, Volume 53, Issue 6, pp 1115–1118 | Cite as

Existence and nonuniqueness of solutions to a functional-differential equation

  • A. I. NoarovEmail author
Article

Abstract

We examine the functional-differential equation Δu(x) — div(u(H(x))f (x)) = 0 on a torus which is a generalization of the stationary Fokker-Planck equation. Under sufficiently general assumptions on the vector field f and the map H, we prove the existence of a nontrivial solution. In some cases the subspace of solutions is established to be multidimensional.

Keywords

stationary Fokker-Planck equation deviating argument 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Noarov A. I., “On the solvability of stationary Fokker-Planck equations close to the laplace equation,” Differential Equations, 42, No. 4, 556–566 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Noarov A. I., “Generalized solvability of the stationary Fokker-Planck equation,” Differential Equations, 43, No. 6, 833–839 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Noarov A. I., “Unique solvability of the stationary Fokker-Planck equation in a class of positive functions,” Differential Equations, 45, No. 2, 197–208 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Noarov A. I., “On some diffusion processes with stationary distributions,” Theory Probab. Appl., 54, No. 3, 525–533 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Zeeman E. C., “Stability of dynamical systems,” Nonlinearity, 1, 115–155 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ladyzhenskaya O. A., Mathematical Problems of the Dynamics of an Incompressible Viscous Fluid [in Russian], Nauka, Moscow (1961).Google Scholar
  7. 7.
    Nazarov S. A., “The essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a cusp,” Funct. Anal. Appl., 43, No. 1, 44–54 (2009).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institute of Numerical MathematicsMoscowRussia

Personalised recommendations