danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2025-69-1-7-12dan-1230Research ArticleМАТЕМАТИКАMATHEMATICSКлассическое решение задачи Коши для полулинейного волнового уравнения с потенциалом ДиракаClassical solution of the Cauchy problem for a semilinear wave equation with a Dirac potentialКорзюкВ. И.KorzyukV. I.

Корзюк Виктор Иванович – академик, д-р физ.-мат. наук, профессор

ул. Сурганова, 11, 220072, Минск

Korzyuk Viktor I. – Academician, D. Sc. (Physics and Ma thematics), Professor

11, Surganov Str., 220072, Minsk

korzyuk@bsu.byhttps://orcid.org/0000-0002-1482-9106РудькоЯ. В.RudzkoJ. V.

Рудько Ян Вячеславович – магистр (математика и компьютерные науки), аспирант

ул. Сурганова, 11, 220072, Минск

Rudzko Jan V. – Master (Mathematics and Computer Sciences), Postgraduate Student

11, Surganov Str., 220072, Minsk

janycz@yahoo.comИнститут математики Национальной академии наук Беларуси; Белорусский государственный университетInstitute of Mathematics of the National Academy of Sciences of BelarusИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202526022025691712Copyright © Корзюк В.И., Рудько Я.В., 20252025Корзюк В.И., Рудько Я.В.Korzyuk V.I., Rudzko J.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1230

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.

For a one-dimensional semilinear wave equation with a free term that is a solution value at one given point (a Dirac potential), we consider the Cauchy problem in the upper half-plane. We construct the solution using the method of characteristics in implicit analytical form as a solution of some integral equations. The solvability of these equations, as well the smoothness of their solutions, is studied. For the problem in question, we prove the uniqueness of the solution, and establish the conditions under which its classical solution exists. 

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The authors declare that there are no conflicts of interest present.