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Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant

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Abstract

For the telegraph equation with a nonlinear potential in a curvilinear quadrant, we consider a mixed problem with the Cauchy conditions on a spatial half-line and the Dirichlet condition on a noncharacteristic curve. The solution of the problem is constructed by the method of characteristics in an implicit analytical form as a solution of integral equations. We study the solvability of these equations depending on the initial data and their smoothness. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. A mild solution is constructed in the case of insufficiently smooth data of the problem.

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Notes

  1. https://mathoverflow.net/questions/320300/what-mild-solution-means-and-howto- find-it.

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Funding

The work was supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284.

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Authors and Affiliations

  1. Belarusian State University, Minsk, 220030, Belarus

    V. I. Korzyuk & J. V. Rudzko

  2. Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 220072, Belarus

    V. I. Korzyuk

Authors
  1. V. I. Korzyuk
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  2. J. V. Rudzko
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Correspondence to V. I. Korzyuk or J. V. Rudzko.

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Translated by V. Potapchouck

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Korzyuk, V.I., Rudzko, J.V. Classical Solution of the First Mixed Problem for the Telegraph Equation with a Nonlinear Potential in a Curvilinear Quadrant. Diff Equat 59, 1075–1089 (2023). https://doi.org/10.1134/S0012266123080062

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  • DOI: https://doi.org/10.1134/S0012266123080062

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