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On the Energy Estimate for Klein–Gordon-Type Equations with Time-Dependent Singular Mass

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Book cover Analysis, Probability, Applications, and Computation

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

We consider the energy estimate of the solution to the Cauchy problem of Klein–Gordon-type equation with time-dependent mass M(t), in particular M(t) has a singularity. The main purpose of this chapter is to give sufficient conditions to M(t) for the energy to be asymptotically stable.

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References

  1. C. Böhme, F. Hirosawa, Generalized energy conservation for Klein-Gordon type equations. Osaka J. Math. 49, 297–323 (2012)

    MathSciNet  MATH  Google Scholar 

  2. M. Cicognani, F. Hirosawa, On the Gevrey well-posedness for second order strictly hyperbolic Cauchy problems under the influence of the regularity of the coefficients. Math. Scand. 102, 283–304 (2008)

    Article  MathSciNet  Google Scholar 

  3. D. Del Santo, T. Kinoshita, M. Reissig, Klein-Gordon type equations with a singular time-dependent potential. Rend. Istit. Mat. Univ. Trieste 39, 141–175 (2007)

    MathSciNet  MATH  Google Scholar 

  4. M.R. Ebert, R.A. Kapp, W.N. Nascimento, M. Reissig, Klein-Gordon type wave equation models with non-effective time-dependent potential, in AMADE 2012, vol. 60, Org. by M.V. Dubatovskaya, S.V. Rogosin (Cambridge Scientific Publishers, Cambridge, 2014), pp. 143–161

    Google Scholar 

  5. F. Hirosawa, On the asymptotic behavior of the energy for the wave equations with time depending coefficients. Math. Ann. 339, 819–838 (2007)

    Article  MathSciNet  Google Scholar 

  6. F. Hirosawa, W.N. Nascimento, 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. 197, 817–841 (2018). https://doi.org/10.1007/s10231-017-0705-9

    Article  MathSciNet  Google Scholar 

  7. F. Hirosawa, J. Wirth, C m-theory of damped wave equations with stabilisation. J. Math. Anal. Appl. 343, 1022–1035 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 26400170.

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Authors and Affiliations

  1. Yamaguchi University, Yamaguchi, Japan

    Fumihiko Hirosawa

Authors
  1. Fumihiko Hirosawa
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Corresponding author

Correspondence to Fumihiko Hirosawa .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Karl‐Olof Lindahl

  2. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Torsten Lindström

  3. Department of Mathematics, University of Torino, Torino, Italy

    Luigi G. Rodino

  4. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

  5. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Patrik Wahlberg

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Hirosawa, F. (2019). On the Energy Estimate for Klein–Gordon-Type Equations with Time-Dependent Singular Mass. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_31

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