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A Cauchy-Kovalevsky Theorem for Nonlinear and Nonlocal Equations

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Analysis and Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 127))

Abstract

For a generalized Camassa-Holm equation it is shown that the solution to the Cauchy problem with analytic initial data is analytic in both variables, locally in time and globally in space. Furthemore, an estimate for the analytic lifespan is provided. To prove these results, the equation is written as a nonlocal autonomous differential equation on a scale of Banach spaces and then a version of the abstract Cauchy-Kovalevsky theorem is applied, which is derived by the power series method in these spaces. Similar abstract versions of the nonlinear Cauchy-Kovalevsky theorem have been proved by Ovsyannikov, Treves, Baouendi and Goulaouic, Nirenberg, and Nishida.

In memory of M. Salah Baouendi

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Acknowledgments

This work was partially supported by a grant from the Simons Foundation (#246116 to Alex Himonas). The third author was partially supported by CNPq and Fapesp.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de São Carlos, Sao Carlos, SP, 13565-905, Brazil

    Rafael F. Barostichi & Gerson Petronilho

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    A. Alexandrou Himonas

Authors
  1. Rafael F. Barostichi
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  2. A. Alexandrou Himonas
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  3. Gerson Petronilho
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Corresponding author

Correspondence to A. Alexandrou Himonas .

Editor information

Editors and Affiliations

  1. Département de Mathématiques, Université de Sfax, Sfax, Tunisia

    Ali Baklouti

  2. LAMAV, ISTV2, Université de Valenciennes, Valenciennes cedex 9, France

    Aziz El Kacimi

  3. Department of Mathematics and Statistics, The American University of Sharjah, Sharjah, Utd.Arab.Emir.

    Sadok Kallel

  4. Texas A&M University at Qatar, Doha, Qatar

    Nordine Mir

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Barostichi, R.F., Himonas, A.A., Petronilho, G. (2015). A Cauchy-Kovalevsky Theorem for Nonlinear and Nonlocal Equations. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir, N. (eds) Analysis and Geometry. Springer Proceedings in Mathematics & Statistics, vol 127. Springer, Cham. https://doi.org/10.1007/978-3-319-17443-3_5

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