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

  • Á. FigulaEmail author
  • M. Z. Menteshashvili
Chapter
Part of the UNIPA Springer Series book series (USS)

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.

Notes

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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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, University of DebrecenDebrecenHungary
  2. 2.Muskhelishvili Institute of Computational Mathematics of the Georgian Technical UniversityTbilisiGeorgia
  3. 3.Sokhumi State UniversityTbilisiGeorgia

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