Skip to main content
SpringerLink
Log in

Properties of solutions of a mixed problem for a nonlinear ultraparabolic equation

  • Published:
Ukrainian Mathematical Journal Aims and scope

Mixed problems for a nonlinear ultraparabolic equation are considered in domains bounded and unbounded with respect to the space variables. Conditions for the existence and uniqueness of solutions of these problems are established and some estimates for these solutions are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. N. Kolmogorov, “Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung),” Ann. Math., 35, 116–117 (1934).

    Article  MathSciNet  Google Scholar 

  2. G. Citty, A. Pascucci, and S. Polidoro, “On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance,” Different. Integral Equat., 14, No. 6, 701–738 (2001).

    Google Scholar 

  3. 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 Honour of Prof. O. A. Ladyzhenskaya, Kluwer, New York (2002), pp. 243–265.

  4. V. S. Dron’ and S. D. Ivasyshen, “Properties of fundamental solutions and theorems on uniqueness of solutions of the Cauchy problem for one class of ultraparabolic equations,” Ukr. Mat. Zh., 50, No. 11, 1482–1496 (1998).

    Google Scholar 

  5. S. D. Ivasishen and S. D. Éidel’man, “On fundamental solutions of the Cauchy problem for degenerate parabolic equations of Kolmogorov type with \( 2\vec b \) -parabolic part on the basis of the main group of variables,” Differents. Uravn., 34, No. 11, 1536–1545 (1998).

    Google Scholar 

  6. S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type (Operator Theory: Advances and Applications), Birkhäuser, Basel (2004).

    Google Scholar 

  7. S. P. Lavrenyuk and M. O. Oliskevych, “Mixed problem for a semilinear ultraparabolic equation in an unbounded domain,” Ukr. Mat. Zh., 59, No. 12, 1661–1673 (2007).

    Article  MATH  Google Scholar 

  8. S. P. Lavrenyuk and N. P. Protsakh, “Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia,” Ukr. Mat. Zh., 58, No. 9, 1192–1210 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  9. S. P. Lavrenyuk and N. P. Protsakh, “Mixed problem for an ultraparabolic equation in an unbounded domain,” Ukr. Mat. Zh., 54, No. 8, 1053–1066 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  10. N. P. Protsakh, “Mixed problem for a nonlinear ultraparabolic equation,” Nauk. Visn. Chern. Univ., Ser. Mat., Issue 134, 97–103 (2002).

  11. S. A. Tersenov, “On the main boundary-value problems for one ultraparabolic equation,” Sib. Mat. Zh., 42, No. 6, 1413–1430 (2001).

    MATH  MathSciNet  Google Scholar 

  12. F. Lascialfari and D. Morbidelli, “A boundary-value problem for a class of quasilinear ultraparabolic equations,” Commun. Part. Different Equat., 23, No. 5, 6, 847–868 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Lavrenyuk and N. Protsakh, “Boundary-value problem for a nonlinear ultraparabolic equation in domain unbounded with respect to time variable,” Tatra Mt. Math. Publ., 38, 131–146 (2007).

    MATH  MathSciNet  Google Scholar 

  14. M. Bramanti, M. C. Cerutti, and M. Manfredini, “ L p estimates for some ultraparabolic operators with discontinuous coefficients,” J. Math. Anal. Appl., 200, 332–354 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. E. Kogoj and E. Lanconelli, “An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations,” Mediter. J. Math., 1, 51–80 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  16. M. E. Schonbek and E. Süli, “Decay of the total variation and Hardy norms of solutions to parabolic conservation laws,” Nonlin. Anal., 45, 515–528 (2001).

    Article  MATH  Google Scholar 

  17. F. Bernis, “Qualitative properties for some nonlinear higher order degenerate parabolic equations,” Houston J. Math., 14, 319–352 (1988).

    MATH  MathSciNet  Google Scholar 

  18. P. D. Lax and R. S. Phillips, “ Local boundary conditions for dissipative symmetric linear differential operators,” Commun. Pure Appl. Math., 13, 427–455 (1960).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ukrainian National Forestry University, Lviv, Ukraine

    N. P. Protsakh

  2. Institute of Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    N. P. Protsakh

Authors
  1. N. P. Protsakh
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 6, pp. 795 – 809, June, 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Protsakh, N.P. Properties of solutions of a mixed problem for a nonlinear ultraparabolic equation. Ukr Math J 61, 945–963 (2009). https://doi.org/10.1007/s11253-009-0252-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-009-0252-7

Keywords

Search

Navigation