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Classical Solutions of a Multidimensional Hyperbolic Differential–Difference Equation with Shifts of Various Directions in the Potentials

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Abstract

We study the existence of smooth solutions of a multidimensional hyperbolic equation containing the sum of differential operators and shift operators along arbitrary spatial coordinate directions. For this equation, we construct a three-parameter family of solutions. It is proved that the resulting solutions are classical under the condition that the real part of the symbol of the differential–difference operator in the equation is positive. Classes of equations for which this condition holds are given.

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Acknowledgments

The author wishes to express deep gratitude to A. B. Muravnik for posing the problem and valuable advice and also to A. L. Skubachevskii for constant attention to the research.

Funding

This 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. Lomonosov Moscow State University, Moscow, 119991, Russia

    N. V. Zaitseva

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    N. V. Zaitseva

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  1. N. V. Zaitseva
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Correspondence to N. V. Zaitseva.

Additional information

Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 810–819 https://doi.org/10.4213/mzm13524.

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Zaitseva, N.V. Classical Solutions of a Multidimensional Hyperbolic Differential–Difference Equation with Shifts of Various Directions in the Potentials. Math Notes 112, 872–880 (2022). https://doi.org/10.1134/S0001434622110219

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  • DOI: https://doi.org/10.1134/S0001434622110219

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