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Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential

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Abstract

For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.

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Funding

The work was carried out with financial support from 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

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

    V. I. Korzyuk & J. V. Rudzko

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  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 Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential. Diff Equat 59, 1216–1234 (2023). https://doi.org/10.1134/S0012266123090070

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  • DOI: https://doi.org/10.1134/S0012266123090070

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