Skip to main content
SpringerLink
Log in

The global existence of small self-interacting scalar field propagating in the contracting universe

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

We present a condition on the self-interaction term that guaranties the existence of the global in time solution of the Cauchy problem for the semilinear Klein–Gordon equation in the Friedmann–Lamaître–Robertson–Walker model of the contracting universe. For the Klein–Gordon equation with the Higgs potential we give a lower estimate for the lifespan of solution.

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

Fig. 1

Similar content being viewed by others

References

  1. Baskin, D.: Strichartz estimates on asymptotically de Sitter spaces. Ann. Henri Poincaré 14(2), 221–252 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Brenner, P.: On the existence of global smooth solutions of certain semilinear hyperbolic equations. Math. Z. 167(2), 99–135 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Brenner, P., Kumlin, P.: On wave equations with supercritical nonlinearities. Arch. Math. (Basel) 74(2), 129–147 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bros, J., Epstein, H., Moschella, U.: Scalar tachyons in the de Sitter universe. Lett. Math. Phys. 93(2), 203–211 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Ebert, M.R., Reissig, M.: Methods for partial differential equations. Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models. Birkhäuser/Springer, Cham (2018)

    MATH  Google Scholar 

  6. Ebert, M.R., do Nascimento, W.N.: A classification for wave models with time-dependent mass and speed of propagation. Adv. Differ. Equ. 23(11–12), 847–888 (2018)

    MATH  Google Scholar 

  7. Epstein, H., Moschella, U.: de Sitter tachyons and related topics. Commun. Math. Phys. 336(1), 381–430 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Galstian, A.: \(L_p-L_q\) decay estimates for the wave equations with exponentially growing speed of propagation. Appl. Anal. 82(3), 197–214 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Galstian, A.: \(L_p-L_q\) -decay estimates for the Klein–Gordon equation in the anti-de Sitter space-time. Rend. Istit. Mat. Univ. Trieste 42(suppl.), 27–50 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Galstian, A., Yagdjian, K.: Global solutions for semilinear Klein–Gordon equations in FLRW spacetimes. Nonlinear Anal. 113, 339–356 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Galstian, A., Yagdjian, K.: Global in time existence of the self-interacting scalar field in De Sitter spacetimes. Nonlinear Anal. Real World Appl. 34, 110–139 (2017)

    MathSciNet  MATH  Google Scholar 

  12. Hintz, P., Vasy, A.: Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes. Anal. PDE 8(8), 1807–1890 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Hintz, P.: Global analysis of quasilinear wave equations on asymptotically de Sitter spaces. Ann. Inst. Fourier (Grenoble) 66(4), 1285–1408 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Hirosawa, F., Wirth, J.: Generalised energy conservation law for wave equations with variable propagation speed. J. Math. Anal. Appl. 358(1), 56–74 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Hirosawa, F., do Nascimento, W.N.: Energy estimates for the Cauchy problem of Klein–Gordon-type equations with non-effective and very fast oscillating time-dependent potential. Ann. Mat. Pura Appl. (4) 197(3), 817–841 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Møller, C.: The Theory of Relativity. Clarendon Press, Oxford (1952)

    MATH  Google Scholar 

  17. Nakamura, M.: The Cauchy problem for semi-linear Klein–Gordon equations in de Sitter spacetime. J. Math. Anal. Appl. 410(1), 445–454 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Pecher, H.: \(L^p\)-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I. Math. Z. 150, 159–183 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Reissig, M., Yagdjian, K.: \(L^p -L^q\) decay estimates for the solutions of strictly hyperbolic equations of second order with increasing in time coefficients. Math. Nachr. 214, 71–104 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Wirth, J.: Solution representations for a wave equation with weak dissipation. Math. Methods Appl. Sci. 27(1), 101–124 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Wirth, J.: Wave equations with time-dependent dissipation. II. Effective dissipation. J. Differ. Equ. 232(1), 74–103 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Yagdjian, K.: Global existence for the n-dimensional semilinear Tricomi-type equations. Commun. Partial Differ. Equ. 31(4–6), 907–944 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Yagdjian, K.: The Cauchy Problem for Hyperbolic Operators. Multiple Characteristics. Micro-local Approach. Akademie Verlag, Berlin (1997)

    MATH  Google Scholar 

  24. Yagdjian, K., Galstian, A.: Fundamental solutions for the Klein–Gordon equation in de Sitter spacetime. Commun. Math. Phys. 285, 293–344 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Yagdjian, K.: The semilinear Klein–Gordon equation in de Sitter spacetime. Discrete Contin. Dyn. Syst. Ser. S 2(3), 679–696 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Yagdjian, K.: Global existence of the scalar field in de Sitter spacetime. J. Math. Anal. Appl. 396(1), 323–344 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Yagdjian, K.: Integral transform approach to solving Klein–Gordon equation with variable coefficients. Math. Nachr. 288(17–18), 2129–2152 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Yagdjian, K.: Global existence of the self-interacting scalar field in the de Sitter universe. J. Math. Phys. 60(5), 051503 (2019). 29 pp

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas RGV, 1201 W. University Drive, Edinburg, TX, 78539, USA

    Anahit Galstian & Karen Yagdjian

Authors
  1. Anahit Galstian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karen Yagdjian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karen Yagdjian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galstian, A., Yagdjian, K. The global existence of small self-interacting scalar field propagating in the contracting universe. Nonlinear Differ. Equ. Appl. 27, 28 (2020). https://doi.org/10.1007/s00030-020-00632-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-020-00632-0

Keywords

Mathematics Subject Classification

Search

Navigation