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Lie Groups, Differential Equations, and Geometry pp 205–221Cite as

On the Geometry of the Domain of the Solution of Nonlinear Cauchy Problem

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Abstract

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the considered inverse problem has a solution under certain regularity condition. We illustrate the Cauchy and the inverse problems in some interesting examples such that the families of the characteristic curves have either common envelopes or singular points. In these cases the definition domain of the solution of the differential equation contains a gap.

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Acknowledgements

The authors were supported by the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreements no. 317721, no. 318202.

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, P.O. Box 400, 4002, Debrecen, Hungary

    Á. Figula

  2. Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 4, Grigol Peradze Str., 0131, Tbilisi, Georgia

    M. Z. Menteshashvili

  3. Sokhumi State University, 0186, Tbilisi, Georgia

    M. Z. Menteshashvili

Authors
  1. Á. Figula
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  2. M. Z. Menteshashvili
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Correspondence to Á. Figula .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Palermo, Palermo, Italy

    Prof. Giovanni Falcone

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Figula, Á., Menteshashvili, M.Z. (2017). On the Geometry of the Domain of the Solution of Nonlinear Cauchy Problem. In: Falcone, G. (eds) Lie Groups, Differential Equations, and Geometry. UNIPA Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-319-62181-4_9

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