Advertisement

Differential Equations

, Volume 51, Issue 10, pp 1369–1386 | Cite as

Time-periodic problem for a weakly nonlinear telegraph equation with directional derivative in the boundary condition

  • S. S. KharibegashviliEmail author
  • O. M. Dzhokhadze
Partial Differential Equations
  • 37 Downloads

Abstract

We study a time-periodic problem for the wave equation with a power-law nonlinearity and with a directional derivative in the boundary condition. We study the existence, uniqueness, and absence of solutions of the problem.

Keywords

Periodic Solution Nonlocal Condition Nonlinear Hyperbolic Equation Unique Classical Solution Quasilinear Wave Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Rabinowitz, P., Periodic Solutions of Nonlinear Hyperbolic Partial Differential Equations, Comm. Pure Appl. Math., 1967, vol. 20, pp. 145–205.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Paris: Dunod, 1969. Translated under the title Nekotorye metody resheniya nelineinykh kraevykh zadach, Moscow: Mir, 1972.Google Scholar
  3. 3.
    Brezis, H. and Nirenberg, L., Characterizations of the Ranges of Some Nonlinear Operators and Applications to Boundary Value Problems, Ann. Sc. Norm. Super Pisa Cl. Sci., 1978, vol. 5, no. 2, pp. 225–325.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Nirenberg, L., Variational and Topological Methods in Nonlinear Problems, Bull. Amer. Math. Soc. (N. S.), 1981, vol. 4, no. 3, pp. 267–302.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Vejvoda, O., Herrmann, L., and Lovicar, V., Partial Differential Equations: Time-Periodic Solutions, Maryland; Bockville, 1981.Google Scholar
  6. 6.
    Brezis, H., Periodic Solutions of Nonlinear Vibrating String and Duality Principles, Bull. Amer. Math. Soc. (N. S.), 1983, vol. 8, no. 3, pp. 409–426.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Rabinowitz, P., Large Amplitude Time Periodic Solutions of a Semilinear Wave Equations, Comm. Pure Appl. Math., 1984, vol. 37, pp. 189–206.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feireisl, E., On the Existence of Periodic Solutions of a Semilinear Wave Equation with a Superlinear Forcing Term, Chechosl. Math. J., 1988, vol. 38, no. 1, pp. 78–87.MathSciNetGoogle Scholar
  9. 9.
    Plotnikov, P.I., Existence of a Countable Set of Periodic Solutions of a Problem on Forced Oscillations for a Weakly Nonlinear Wave Equation, Mat. Sb., 1988, vol. 136 (178), no. 4 (8), pp. 546–560.Google Scholar
  10. 10.
    Mustonen, V. and Pohozaev, S.I., On the Nonexistence of Periodic Radial Solutions for Semilinear Wave Equations in Unbounded Domain, Differential Integral Equations, 1998, vol. 11, no. 1, pp. 133–145.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kiguradze, T., On Periodic in the Plane Solutions of Nonlinear Hyperbolic Equations, Nonlinear Anal., 2000, vol. 39, no. 2, pp. 173–185.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kiguradze, T., On Bounded and Time-Periodic Solutions of Nonlinear Wave Equations, J. Math. Anal. Appl., 2001, vol. 259, no. 1, pp. 253–276.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mitidieri, E. and Pokhozhaev, S.I., A Priori Estimates and the Absence of Solutions of Nonlinear Partial Differential Equations and Inequalities, Tr. Mat. Inst. Steklova, 2001, vol. 234.Google Scholar
  14. 14.
    Rudakov, I.A., Periodic Solutions of a Quasilinear Wave Equation with Variable Coefficients, Mat. Sb., 2007, vol. 198, no. 7, pp. 91–108.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kondrat’ev, V.A. and Rudakov, I.A., Periodic Solutions of a QuasilinearWave Equation, Mat. Zametki, 2009, vol. 85, no. 1, pp. 37–53.MathSciNetGoogle Scholar
  16. 16.
    Pava, J.A., Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Amer. Math. Soc. Math. Surv. Monogr., 2009, vol. 156.CrossRefGoogle Scholar
  17. 17.
    Ladyzhenskaya, O.A. and Ural’tseva, N.N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa (Linear and Quasilinear Equations of Elliptic Type), Moscow: Nauka, 1973.zbMATHGoogle Scholar
  18. 18.
    Bitsadze, A.V., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1982.Google Scholar
  19. 19.
    Kharibegashvili, S. and Midodashvili, B., Solvability of Nonlocal Problems for Semilinear One-Dimensional Wave Equations, Electron. J. Differ. Equ., 2012, no. 28, pp. 1–16.MathSciNetGoogle Scholar
  20. 20.
    Kharibegashvili, S.S. and Dzhokhadze, O.M., Second Darboux Problem for the Wave Equation with a Power-Law Nonlinearity, Differ. Uravn., 2013, vol. 49, no. 12, pp. 1623–1640.MathSciNetGoogle Scholar
  21. 21.
    Gilbarg, D. and Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlag, 1983. Translated under the title Ellipticheskie differentsial’nye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Moscow: Nauka, 1989.zbMATHCrossRefGoogle Scholar
  22. 22.
    Narasimhan, R., Analysis on Real and Complex Manifolds, Amsterdam: North-Holland Publ., 1968. Translated under the title Analiz na deistvitel’nykh i kompleksnykh mnogoobraziyakh, Moscow, 1971.zbMATHGoogle Scholar
  23. 23.
    Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1993.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Andrea Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.I. Javakhishvili Tbilisi State UniversityTbilisiGeorgia

Personalised recommendations