Ukrainian Mathematical Journal

, Volume 58, Issue 9, pp 1347–1368 | Cite as

Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

  • S. P. Lavrenyuk
  • N. P. Protsakh


We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established.


Cauchy Problem Diffusion Equation Planck Equation Mixed Problem Kolmogorov Equation 
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  1. 1.
    A. N. Kolmogorov, “Zufällige Bewegungen (zur Theorie der Brownschen Bewegung),” Ann. Math., 35, 116–117 (1934).CrossRefMathSciNetGoogle Scholar
  2. 2.
    W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin (1975).MATHGoogle Scholar
  3. 3.
    S. Polidoro, “On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance,” Nonlin. Anal., 47, 491–502 (2001).CrossRefMathSciNetGoogle Scholar
  4. 4.
    S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser (2004).Google Scholar
  5. 5.
    E. Lanconelli, A. Pascucci, and S. Polidoro, “Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance,” in: Nonlinear Problems in Mathematical Physics and Related Topics. II. In Honor of Prof. O. A. Ladyzhenskaya, Vol. 2, Kluwer, New York (2002), pp. 243–265.Google Scholar
  6. 6.
    V. S. Dron’ and S. D. Ivasyshen, “On the correct solvability of the Cauchy problem for degenerate parabolic equations of Kolmogorov type,” Ukr. Mat. Visn., No. 1, 61–68 (2004).Google Scholar
  7. 7.
    O. H. Voznyak and S. D. Ivasyshen, “Fundamental solutions of the Cauchy problem for one class of degenerate parabolic equations and their applications,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 11–16 (1996).Google Scholar
  8. 8.
    S. D. Éidel’man and A. P. Malitskaya, “On fundamental solutions and stabilization of a solution of the Cauchy problem for one class of degenerate parabolic equations,” Differents. Uravn., 11, No. 7, 1316–1331 (1975).Google Scholar
  9. 9.
    S. G. Pyatkov, “Solvability of boundary-value problems for an ultraparabolic equation,” in: Nonclassical Equations and Equations of Mixed Type [in Russian], Novosibirsk (1990), pp. 182–197.Google Scholar
  10. 10.
    Sh. Amirov, “Mixed problem for an ultraparabolic equation in a bounded domain,” in: Well-Posed Boundary-Value Problems for Nonclassical Equations of Mathematical Physics [in Russian], Novosibirsk (1984), pp. 173–179.Google Scholar
  11. 11.
    J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires [Russian translation], Mir, Moscow (1972).Google Scholar
  12. 12.
    H. M. Barabash, S. P. Lavrenyuk, and N. P. Protsakh, “Mixed problem for a semilinear ultraparabolic equation,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 4, 27–34 (2002).MathSciNetGoogle Scholar
  13. 13.
    N. P. Protsakh, “Mixed problem for a nonlinear ultraparabolic equation,” Nauk. Visn. Cherniv. Univ., Ser. Mat., Issue 134, 97–103 (2002).Google Scholar
  14. 14.
    F. Lascialfari and D. Morbidelli, “A boundary-value problem for a class of quasilinear ultraparabolic equations,” Commun. Part. Different. Equat., 23, Nos. 5, 6, 847–868 (1998).MathSciNetGoogle Scholar
  15. 15.
    H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen [Russian translation], Mir, Moscow (1978).Google Scholar
  16. 16.
    K. Yosida, Functional Analysis [Russian translation], Mir, Moscow (1967).MATHGoogle Scholar
  17. 17.
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. P. Lavrenyuk
    • 1
  • N. P. Protsakh
    • 2
  1. 1.Lviv National UniversityLviv
  2. 2.Institute for Applied Problems in Mechanics and MathematicsUkrainian Academy of SciencesLviv

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