Классическое решение задачи Коши для полулинейного волнового уравнения с потенциалом Дирака
https://doi.org/10.29235/1561-8323-2025-69-1-7-12
Аннотация
Для одномерного полулинейного волнового уравнения со свободным членом, являющимся значением решения в одной заданной точке (потенциал Дирака), рассматривается задача Коши в верхней полуплоскости. Решение строится методом характеристик в неявном аналитическом виде как решение некоторых интегральных уравнений. Проводится исследование разрешимости этих уравнений, а также зависимости от начальных данных и гладкости их решений. Для рассматриваемой задачи доказывается единственность решения и устанавливаются условия, при выполнении которых существует ее классическое решение.
Об авторах
В. И. Корзюк
Институт математики Национальной академии наук Беларуси; Белорусский государственный университет
Беларусь
Беларусь
Корзюк Виктор Иванович – академик, д-р физ.-мат. наук, профессор
ул. Сурганова, 11, 220072, Минск
Я. В. Рудько
Институт математики Национальной академии наук Беларуси
Беларусь
Беларусь
Рудько Ян Вячеславович – магистр (математика и компьютерные науки), аспирант
ул. Сурганова, 11, 220072, Минск
Список литературы
1. Nakhushev A. M. Loaded equations and their applications. Differential Equations, 1983, vol. 19, no. 1, pp. 74–81.
2. Sabitov K. B. Initial-boundary problem for parabolic-hyperbolic equation with loaded summands. Russian Mathematics, 2015, vol. 59, no. 6, pp. 23–33. https://doi.org/10.3103/s1066369x15060055
3. Sabitova Yu. K. Dirichlet problem for Lavrent’ev–Bitsadze equation with loaded summands. Russian Mathematics, 2018, vol. 62, no. 9, pp. 35–51. https://doi.org/10.3103/s1066369x18090050
4. Baranovskaya S. N., Yurchuk N. I. Cauchy problem for the Euler–Poisson–Darboux Equation with a Dirac potential concentrated at finitely many given points. Differential Equations, 2020, vol. 56, no. 1, pp. 93–97. https://doi.org/10.1134/s0012266120010103
5. Korzyuk V. I., Rudzko J. V. Classical solution of the first mixed problem for the telegraph equation with a nonlinear potential. Differential Equations, 2022, vol. 58, no. 2, pp. 175–186. https://doi.org/10.1134/s0012266122020045
6. Korzyuk V. I., Rudzko J. V. Classical solution of the initial-value problem for a one-dimensional quasilinear wave equation. Doklady National’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2023, vol. 67, no. 1, pp. 14–19. https://doi.org/10.29235/1561-8323-2023-67-1-14-19
7. Korzyuk V. I., Rudzko J. V. Classical and mild solution of the first mixed problem for the telegraph equation with a nonlinear potential. Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 43, pp. 48–63. https://doi.org/10.26516/1997-7670.2023.43.48
8. Moiseev E. I., Yurchuk N. I. Classical and generalized solutions of problems for the telegraph equation with a Dirac potential. Differential Equations, 2015, vol. 51, no. 10, pp. 1330–1337. https://doi.org/10.1134/s0012266115100080
9. Baranovskaya S. N., Novikov E. N., Yurchuk N. I. Directional derivative problem for the telegraph equation with a Dirac potential. Differential Equations, 2018, vol. 54, no. 9, pp. 1147–1155. https://doi.org/10.1134/s0012266118090033
10. Baranovskaya S. N., Yurchuk N. I. Cauchy problem and the second mixed problem for parabolic equations with the Dirac potential. Differential Equations, 2015, vol. 51, no. 6, pp. 819–821. https://doi.org/10.1134/s0012266115060130
11. Baranovskaya S. N., Yurchuk N. I. Cauchy problem and the second mixed problem for parabolic equations with a Dirac potential concentrated at finitely many given points. Differential Equations, 2019, vol. 55, no. 3, pp. 348–352. https://doi.org/10.1134/s001226611903008x
12. Attaev A. Kh. To the question of solvability of the Cauchy problem for one loaded hyperbolic equation of the second order. News of the Kabardino-Balkarian Scientific Center of the RAS, 2018, no. 6, pp. 5–9 (in Russian).
13. Attaev A. Kh. On some problems for loaded partial differential equation of the first order. Vestnik KRAUNC. FizikoMatematicheskie Nauki = Bulletin KRASEC, Physical and Mathematical Sciences, 2016, no. 4-1(16), pp. 9–14 (in Russian).
14. Khubiev K. U. Cauchy problem for one loaded wave equation. Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk = Adyghe International Scientific Journal, 2020, vol. 20, no. 4, pp. 9–14 (in Russian). https://doi.org/10.47928/1726-9946-2020-20-4-9-14
15. Korzyuk V. I. Equations of Mathematical Physics. Moscow, 2021. 480 p. (in Russian).
16. Courant R., Hilbert D. Methods of Mathematical Physics: Partial Differential Equations. Singapore, 1962.
17. Trenogin V. A. Global invertibility of nonlinear operators and the method of continuation with respect to a parameter. Doklady Mathematics, 1996, vol. 54, no. 2, pp. 730–732.
18. Qin Y. Integral and Discrete Inequalities and Their Applications. Volume I: Linear Inequalities. Cham, 2016. https://doi.org/10.1007/978-3-319-33301-4
19. Korzyuk V. I., Rudzko J. V. On the absence and non-uniqueness of classical solutions of mixed problems for the telegraph equation with a nonlinear potential. Available at: https://arxiv.org/abs/2303.17483 (accessed 18 February 2024).
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