Skip to main content
Log in

Well-posedness of boundary-value problems for multidimensional hyperbolic systems

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

By using the method of characteristics, we investigate the well-posedness of local problems (Cauchy problem and mixed problems) and nonlocal problems (with nonseparable and integral conditions) for some multidimensional almost-linear hyperbolic systems of the first order. We reduce these problems to systems of integro-operator equations and prove theorems on the existence and uniqueness of classical solutions.

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

Access this article

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. I. G. Petrovskii, Lectures on Partial Differential Equations [in Russian], Fizmatlit, Moscow (1961).

    Google Scholar 

  2. V. É. Abolinya and A. D. Myshkis, “On a mixed problem for a linear hyperbolic system on a plane,” Uch. Zap. Latv. Univ., 20, No. 3, 87–104 (1958).

    Google Scholar 

  3. V. É. Abolinya and A. D. Myshkis, “A mixed problem for an almost linear hyperbolic system on a plane,” Mat. Sb., 50, No. 4, 423–442 (1960).

    MathSciNet  Google Scholar 

  4. R. V. Andrusyak, V. M. Kirilich, and A. D. Myshkis, “Local and global solvabilities of the quasilinear hyperbolic Stefan problem on a straight line,” Differents. Uravn., 42, No. 4, 489–503 (2006).

    MathSciNet  Google Scholar 

  5. I. Ya. Kmit’, “A nonlocal problem for a quasilinear hyperbolic system of the first order with two independent variables,” Ukr. Mat. Zh., 45, No. 9, 1307–1311 (1993).

    MathSciNet  Google Scholar 

  6. A. D. Myshkis and A. M. Filimonov, “Continuous solutions of hyperbolic systems of quasilinear equations with two independent variables,” in: Nonlinear Analysis and Nonlinear Differential Equations [in Russian], (2003), pp. 337–351.

  7. A. D. Myshkis and A. M. Filimonov, “Continuous solutions of quasilinear hyperbolic systems with two independent variables,” Differents. Uravn., 17, No. 3, 488–500 (1981).

    MATH  MathSciNet  Google Scholar 

  8. B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002).

    Google Scholar 

  9. A. M. Filimonov, Sufficient Conditions for the Global Solvability of a Mixed Problem for Quasilinear Hyperbolic Systems with Two Independent Variables [in Russian], Dep. in VINITI, No. 6-81, Moscow (1980).

  10. K. O. Friedrichs, “Symmetric positive linear differential equations,” Commun. Pure Appl. Math., 11, 333–418 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Hersh, “Mixed problems in several variables,” J. Math. Mech., 12, 317–334 (1963).

    MATH  MathSciNet  Google Scholar 

  12. R. L. Higdon, “Initial-boundary value problems for hyperbolic systems,” SIAM Rev., 27, 177–217 (1986).

    Article  MathSciNet  Google Scholar 

  13. H. O. Kreiss, “Initial boundary value problems for hyperbolic systems,” Commun. Pure Appl. Math., 23, 277–289 (1970).

    Article  MathSciNet  Google Scholar 

  14. P. 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 

  15. A. Majada and S. Osher, “Initial-boundary value problem for hyperbolic equations with uniformly characteristic boundary,” Commun. Pure Appl. Math., 28, 607–675 (1975).

    Article  Google Scholar 

  16. G. Metivier, “The block structure condition for symmetric hyperbolic systems,” Bull. London Math. Soc., 32, No. 6, 689–702 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Rauch, “Symmetric positive systems with boundary characteristic of constant multiplicity,” Trans. Amer. Math. Soc., 291, 167–187 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Secchi, “The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity,” Different. Integr. Equat., 9, 671–700 (1996).

    MATH  MathSciNet  Google Scholar 

  19. P. Secchi, “Well-posedness of characteristic symmetric hyperbolic systems,” Arch. Ration. Mech. Anal., 134, No. 2, 155–197 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  20. P. Secchi, “Full regularity of solutions to a nonuniformly characteristic boundary value problem for symmetric hyperbolic systems,” Adv. Math. Appl., 10, No. 1, 39–55 (2000).

    MATH  MathSciNet  Google Scholar 

  21. R. Courant, Partial Differential Equations, Wiley, New York (1962).

    MATH  Google Scholar 

  22. I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  23. I. Kmit, “Generalized solutions to singular initial-boundary hyperbolic problems with non-Lipschitz nonlinearities,” Bull. Acad. Serbe Sci. Arts. Classe Sci. Math. Natur., Sci. Math., 133, No. 31, 87–99 (2006).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 192–203, February, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kmit’, I.Y., Ptashnyk, B.I. Well-posedness of boundary-value problems for multidimensional hyperbolic systems. Ukr Math J 60, 221–234 (2008). https://doi.org/10.1007/s11253-008-0054-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-008-0054-3

Keywords

Navigation