Advertisement

One multidimensional version of the Darboux first problem for one class of semilinear second order hyperbolic systems

  • S. KharibegashviliEmail author
  • B. Midodashvili
Article
  • 78 Downloads

Abstract

One multidimensional version of the Darboux first problem for one class of semilinear second order hyperbolic systems is investigated. The questions on local and global solvability and nonexistence of a global solution of this problem are considered.

Mathematics Subject Classification

35L51 35L71 

Keywords

Multidimensional version of the Darboux first problem Semilinear second order hyperbolic systems Local and global solvability Non-existence of a global solution 

References

  1. 1.
    Bitsadze A.V.: Some Classes of Partial Differential Equations. Nauka (Russian), Moscow (1981)zbMATHGoogle Scholar
  2. 2.
    Jörgens K.: Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen. Math. Z. 77, 295–308 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Segal L.E.: The global Cauchy problem for a relativistic scalar field with power interaction. Bull. Soc. Math. France 91, 129–135 (1963)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Levine H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −Au + F(u). Trans. Am. Math. Soc. 192, 1–21 (1974)zbMATHGoogle Scholar
  5. 5.
    John F.: Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscripta Math. 28(1–3), 235–268 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Kato T.: Blow-up of solutions of some nonlinear hyperbolic equations. Commun. Pure Appl. Math. 33(4), 501–505 (1980)zbMATHCrossRefGoogle Scholar
  7. 7.
    Sideris T.C.: Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differ. Equ. 52(3), 378–406 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Georgiev V., Lindblad H., Sogge C.D.: Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math.(6), 1291–1319 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Mitidieri, È., Pohozhaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. (Russian) Trudy Mat. Inst. Steklova, 234 (2001), 1–384; English transl.: Proc. Steklov Inst. Math. 234(3), 1–362 (2001)Google Scholar
  10. 10.
    Ikehata R., Tanizawa K.: Global existence of solutions for semilinear damped wave equations in R n with noncompactly supported initial data. Nonlinear Anal. 61, 1189–1208 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Zhou Y.: Global existence and nonexistence for a nonlinear wave equation with damping and source terms. Math. Nachr. 278(11), 1341–1358 (2008)CrossRefGoogle Scholar
  12. 12.
    Bitsadze A.V.: Mixed type equations on three-dimensional domains. (Russian) Dokl. Akad. Nauk SSSR 143(5), 1017–1019 (1962)Google Scholar
  13. 13.
    Vragov V.N.: The Goursat and Darboux problems for a certain class of hyperbolic equations. (Russian) Differential’nye Uravneniya 8(1), 7–16 (1972)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Vragov, V.N.: Boundary value problems for nonclassical equations of mathematical physics. Novosibirsk State University (Russian), Novosibirsk (1983)Google Scholar
  15. 15.
    Nakhushev A.M.: A multidimensional analogue of the Darboux problem for hyperbolic equations. (Russian) Dokl. Akad. Nauk SSSR 194, 31–34 (1970)Google Scholar
  16. 16.
    Nakhushev A.M.: Equations of mathematical biology. Vysshaya Shkola (Russian), Moscow (1995)zbMATHGoogle Scholar
  17. 17.
    Kalmenov, T.Sh.: On multidimensional regular boundary value problems for the wave equation. (Russian) Izv. Akad. Nauk Kazakh. SSR. Ser. Fiz.-Mat., no. 3, 18–25 (1982)Google Scholar
  18. 18.
    Kharibegashvili S.: On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations. Nonlinear Anal. 68(4), 912–924 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bogveradze G., Kharibegashvili S.: On the global and local solution of the multidimensional Darboux problem for some nonlinear wave equations. Georgian Math. J. 14(1), 65–80 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kharibegashvili S., Midodashvili B.: On the solvability of one boundary value problem for some semilinear wave equations with source terms. Nonlinear Differ. Equ. Appl. 18, 117–138 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Ladyzhenskaya O.A.: Boundary Value Problems of Mathematical Physics. Nauka (Russian), Moscow (1973)Google Scholar
  22. 22.
    Krasnosel’skiǐ M.A., Zabreǐko P.P., Pustyl’nik E.I., Sobolevskiǐ P.E.: Integral Operators in Spaces of Summable Functions. Nauka (Russian), Moscow (1966)Google Scholar
  23. 23.
    Kufner, A., Futchik, S.: Nonlinear differential equations. (Translated into Russian) Nauka, Moscow (1988); English original: Studies in Applied Mechanics, vol. 2. Elsevier Scientific Publishing Co., Amsterdam (1980)Google Scholar
  24. 24.
    Kharibegashvili S.: Some multidimensional problems for hyperbolic partial differential equations and systems. Mem. Differ. Equ. Math. Phys. 37, 1–136 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Henry, D.: Geometrical theory of semi-linear parabolic equations. (Translated into Russian) Mir, Moscow (1985); English original: Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)Google Scholar
  26. 26.
    Hörmander, L.: Linear partial differential operators. (Translated into Russian) Mir, Moscow (1965); English original: die Grundlehren der mathematischen Wissenschaften, Bd. 116. Academic Press Inc., Publishers, New York; Springer, Berlin (1963)Google Scholar
  27. 27.
    Trenogin V.A.: Functional Analysis. Nauka (Russian), Moscow (1993)zbMATHGoogle Scholar
  28. 28.
    Vulikh B.Z.: Concise course of the theory of functions of a real variable. Nauka (Russian), Moscow (1973)Google Scholar
  29. 29.
    Fichtengolz G.M.: Course of differential and integral calculus. I. Nauka (Russian), Moscow (1969)Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.A. Razmadze Mathematical InstituteI. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.Department of MathematicsGeorgian Technical UniversityTbilisiGeorgia
  3. 3.Faculty of Exact and Natural SciencesI. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  4. 4.Faculty of Education, Exact and Natural SciencesGori Teaching UniversityGoriGeorgia

Personalised recommendations