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Sharp threshold of global existence for the Klein-Gordon equations with critical nonlinearity

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Abstract

In this paper, we present a cross-constrained variational method to study the Cauchy problem of the nonlinear Klein-Gordon equations with critical nonlinearity in two space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish a sharp threshold of global existence and blowup of it. Furthermore, we answer the question: How small are the initial data if the solution exists globally.

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References

  1. Ball, J.M. Finite time blow-up in nonlinear problems, In: M.G. Grandall(Ed.). Nonlinear Evolution Equations, Academic Press, New York, 1978

    Google Scholar 

  2. Berestycki, H., Cazenave, T. Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéairees. C. R. Acad. Sci. Paris, 293: 489–492 (1981)

    MATH  MathSciNet  Google Scholar 

  3. Cazenave, T. An introduction to nonlinear Schrödinger equations. Textos de Metodos Mathematicos, Vol. 22, Rio de Janeiro, 1989

  4. Ginibre, J., Velo, G. The global Cauchy problem for the nonlinear Klein-Gordon equation. Math. Z., 189: 487–505 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Glassey, R.T. Blow-up theorems for nonlinear wave equations. Math. Z., 132: 183–302 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  6. Levine, H.A. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = −Au − F(u). Transactions of the American Mathematical Society, 92: 1–21 (1974)

    Article  Google Scholar 

  7. Pagne, L.E., Sattinger, D.H. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., 22(3–4): 273–303 (1975)

    MathSciNet  Google Scholar 

  8. Pecher, H. L p-Abschätzungen and Klassiche Lösungen für nichtlineare wellengeichungen, I. Math. Z., 50: 159–183 (1976)

    Article  MathSciNet  Google Scholar 

  9. Strauss, W.A. Nonlinear wave equations. C.B.M.S., No. 73, Amer. Math. Soc., Providence, RI, 1989

  10. Wang, B.X. On existence and scattering for critical and subcritical nonlinear Klein-Gordon equations in H s. Nonlinear Anal TMA, 31(5/6): 573–587 (1998)

    MATH  Google Scholar 

  11. Zhang, J. Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Analysis TMA, 48: 191–207 (2002)

    Article  MATH  Google Scholar 

  12. Zhang, J. On the standing wave in coupled non-linear Klein-Gordon equations. Math. Meth. Appl. Sci., 26: 11–25 (2003)

    Article  MATH  Google Scholar 

  13. Zhang, J. Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential. Comm. PDE, 30: 1429–1443 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, 610068, China

    Zai-hui Gan & Jian Zhang

  2. Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing, 100088, China

    Zai-hui Gan & Bo-ling Guo

Authors
  1. Zai-hui Gan
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  2. Bo-ling Guo
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  3. Jian Zhang
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Corresponding author

Correspondence to Zai-hui Gan.

Additional information

Supported by the National Natural Science Foundation of China (No. 10771151, 10801102, 10726034) and Sichuan Youth Sciences and Technology Foundation (07ZQ026-009) and China Postdoctoral Science Foundation Funded Project.

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Gan, Zh., Guo, Bl. & Zhang, J. Sharp threshold of global existence for the Klein-Gordon equations with critical nonlinearity. Acta Math. Appl. Sin. Engl. Ser. 25, 273–282 (2009). https://doi.org/10.1007/s10255-007-7085-7

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  • DOI: https://doi.org/10.1007/s10255-007-7085-7

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