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Asymptotic Behavior of the Solution to the Klein–Gordon–Zakharov Model in Dimension Two

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Abstract

Consider the Klein–Gordon–Zakharov equations in \({\mathbb {R}}^{1+2}\), and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The Klein–Gordon–Zakharov equations can be regarded as a coupled semilinear wave and Klein–Gordon system with quadratic nonlinearities which do not satisfy the null conditions, and the fact that wave components and Klein–Gordon components do not decay sufficiently fast makes it harder to conduct the analysis. In order to conquer the difficulties, we will rely on the hyperboloidal foliation method and a minor variance of the ghost weight method. As a side result of the analysis, we are also able to show the small data global existence result for a class of quasilinear wave and Klein–Gordon system violating the null conditions.

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Notes

  1. Originally E takes values in \({\mathbb {R}}^2\), but more general cases of taking values in \({\mathbb {C}}^{N_0}\) with \(N_0 = 1, 2, \cdots \) can also be treated.

References

  1. Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions I. Invent. Math. 145(3), 597–618 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  2. Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions II. Am. J. Math. 123(6), 1071–1101 (2001)

    Article  MathSciNet  Google Scholar 

  3. Bachelot, A.: Problème de Cauchy global pour des systèmes de Dirac–Klein–Gordon. Ann. Inst. Henri Poincaré 48, 387–422 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39(2), 267–282 (1986)

    Article  MathSciNet  Google Scholar 

  5. Dendy, R.O.: Plasma Dynamics. Oxford University Press, Oxford (1990)

    Google Scholar 

  6. Dong, S.: Stability of a wave and Klein–Gordon system with mixed coupling, preprint arXiv:1912.05578

  7. Dong, S.: Global solution to the wave and Klein–Gordon system under null condition in dimension two, preprint arXiv:2005.04767

  8. Dong, S., Wyatt, Z.: Stability of a coupled wave-Klein–Gordon system with quadratic nonlinearities. J. Differ. Equ. 269(9), 7470–7497 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  9. Dong, S., LeFloch, P., Wyatt, Z.: Global evolution of the \(U(1)\) Higgs Boson: nonlinear stability and uniform energy bounds, preprint arXiv:1902.02685, to appear in Annales Henri Poincare

  10. Fang, A., Wang, Q., Yang, S.: Global solution for massive Maxwell–Klein–Gordon equations with large Maxwell field, preprint arXiv:1902.08927

  11. Georgiev, V.: Global solution of the system of wave and Klein–Gordon equations. Math. Z. 203, 683–698 (1990)

    Article  MathSciNet  Google Scholar 

  12. Guo, B., Yuan, G.: Global smooth solution for the Klein–Gordon–Zakharov equations. J. Math. Phys. 36, 4119 (1995)

    Article  MathSciNet  Google Scholar 

  13. Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin (1997)

    MATH  Google Scholar 

  14. Ionescu, A.D., Pausader, B.: On the global regularity for a wave-Klein–Gordon coupled system. Acta Math. Sin. 35, 933–986 (2019)

    Article  MathSciNet  Google Scholar 

  15. Ionescu, A.D., Pausader, B.: The Einstein–Klein–Gordon coupled system: global stability of the Minkowski solution, preprint arXiv:1911.10652

  16. Katayama, S.: Global existence for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions. Math. Z. 270, 487–513 (2012)

    Article  MathSciNet  Google Scholar 

  17. Katayama, S.: Global existence for systems of nonlinear wave and Klein–Gordon equations with compactly supported initial data. Commun. Pure Appl. Anal. 17, 1479–1497 (2018)

    Article  MathSciNet  Google Scholar 

  18. Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four spacetime dimensions. Commun. Pure Appl. Math. 38, 631–641 (1985)

    Article  Google Scholar 

  19. Klainerman, S.: The null condition and global existence to nonlinear wave equations. Nonlinear systems of partial differential equations in applied mathematics, part 1 (Santa Fe, N.M., 1984), lectures in applied mathematics, vol 23. American Mathematical Society, Providence, RI, pp. 293–326 (1986)

  20. Klainerman, S., Wang, Q., Yang, S.: Global solution for massive Maxwell–Klein–Gordon equations. Commun. Pure Appl. Math. 73, 63–109 (2020)

    Article  MathSciNet  Google Scholar 

  21. LeFloch, P.G., Ma, Y.: The Hyperboloidal Foliation Method. World Scientific Press, Singapore (2014)

    MATH  Google Scholar 

  22. LeFloch, P.G., Ma, Y.: The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein–Gordon model. Commun. Math. Phys. 346, 603–665 (2016)

    Article  ADS  Google Scholar 

  23. LeFloch, P.G., Ma, Y.: The global nonlinear stability of Minkowski space. Einstein equations, f(R)-modified gravity, and Klein–Gordon fields, preprint arXiv:1712.10045

  24. Ma, Y.: Global solutions of quasilinear wave-Klein–Gordon system in two-space dimensions: technical tools. J. Hyperbolic Differ. Equ. 14(4), 591–625 (2017)

    Article  MathSciNet  Google Scholar 

  25. Ma, Y.: Global solutions of quasilinear wave-Klein–Gordon system in two-space dimension: completion of the proof. J. Hyperbolic Differ. Equ. 14(4), 627–670 (2017)

    Article  MathSciNet  Google Scholar 

  26. Ma, Y.: Global solutions of non-linear wave-Klein–Gordon system in two space dimension: semi-linear interactions, preprint, arXiv:1712.05315

  27. Ma, Y.: Global solutions of nonlinear wave-Klein–Gordon system in two spatial dimensions: weak coupling case, preprint, arXiv:1907.03516

  28. Masmoudi, N., Nakanishi, K.: From the Klein–Gordon–Zakharov systems to the nonlinear Schrodinger equation. J. Hyperbolic Differ. Equ. 2(4), 975–1008 (2005)

    Article  MathSciNet  Google Scholar 

  29. Ozawa, T., Tsutaya, K., Tsutsumi, Y.: Normal form and global solutions for the Klein–Gordon–Zakharov equations. Ann. IHP 12(4), 459–503 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Ozawa, T., Tsutaya, K., Tsutsumi, Y.: Well-posedness in energy space for the Cauchy problem of the Klein–Gordon–Zakharov equations with different propagation speeds in three space dimensions. Math. Ann. 313, 127–140 (1999)

    Article  MathSciNet  Google Scholar 

  31. Psarelli, M.: Asymptotic behavior of the solutions of Maxwell–Klein–Gordon field equations in 4-dimensional Minkowski space. Commun. Partial Differ. Equ. 24, 223–272 (1999)

    Article  MathSciNet  Google Scholar 

  32. Psarelli, M.: Time decay of Maxwell–Klein–Gordon equations in 4-dimensional Minkowski space. Commun. Partial Differ. Equ 24, 273–282 (1999)

    Article  MathSciNet  Google Scholar 

  33. Shatah, J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Commun. Pure Appl. Math. 38, 685–696 (1985)

    Article  MathSciNet  Google Scholar 

  34. Shi, Q., Wang, S.: Klein–Gordon–Zakharov system in energy space: blow-up profile and subsonic limit. Math. Methods Appl. Sci. 42(9), 3211–3221 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  35. Sogge, C.D.: Lectures on Nonlinear Wave Equations. International Press, Boston (2008)

    MATH  Google Scholar 

  36. Stingo, A.: Global existence of small amplitude solutions for a model quadratic quasi-linear coupled wave-Klein–Gordon system in two space dimension, with mildly decaying Cauchy data, preprint, arXiv:1810.10235

  37. Tsutaya, K.: Global existence of small amplitude solutions for the Klein–Gordon–Zakharov equations. Nonlinear Anal. Theory Methods Appl. 27(12), 1373–1380 (1996)

    Article  MathSciNet  Google Scholar 

  38. Tsutsumi, Y.: Global solutions for the Dirac–Proca equations with small initial data in \(3+ 1\) spacetime dimensions. J. Math. Anal. Appl. 278, 485–499 (2003)

    Article  MathSciNet  Google Scholar 

  39. Wang, Q.: An intrinsic hyperboloid approach for Einstein Klein–Gordon equations. J. Differ. Geom. 115, 27–109 (2020)

    Article  MathSciNet  Google Scholar 

  40. Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972)

    ADS  Google Scholar 

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Acknowledgements

The author is very grateful to the anonymous referees for their patience with the manuscript. Their suggestions and comments help greatly improve the presentation and precision of the paper.

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  1. School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, China

    Shijie Dong

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  1. Shijie Dong
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Correspondence to Shijie Dong.

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Communicated by A. Ionescu

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Dong, S. Asymptotic Behavior of the Solution to the Klein–Gordon–Zakharov Model in Dimension Two. Commun. Math. Phys. 384, 587–607 (2021). https://doi.org/10.1007/s00220-021-04003-3

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