danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2023-67-3-183-188dan-1127Research ArticleМАТЕМАТИКАMATHEMATICSКлассическое решение задачи Коши для квазилинейного волнового уравнения с разрывными начальными условиямиClassical solution of the Cauchy problem for a quasi-linear wave equation with discontinuous initial conditionsКорзюкВ. И.KorzyukV. I.

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

ул. Сурганова, 11, 220072, Минск, Республика Беларусь

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

11, Surganov Str., 220072, Minsk, Republic of Belarus

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

Рудько Ян Вячеславович – аспирант, мл. науч. сотрудник

ул. Сурганова, 11, 220072, Минск, Республика Беларусь

Rudzko Jan V. – Postgraduate Student, Junior Researсher

11, Surganov Str., 220072, Minsk, Republic of Belarus

janycz@yahoo.comИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202306072023673183188Copyright © Корзюк В.И., Рудько Я.В., 20232023Корзюк 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/1127

Для одномерного слабо квазилинейного волнового уравнения, заданного в верхней полуплоскости, рассматривается задача Коши. Начальные условия имеют разрыв первого рода в одной точке. Решение строится в неявном аналитическом виде как решение некоторых интегро-дифференциальных уравнений. Проводится исследование разрешимости этих уравнений, а также гладкости их решений. Для рассматриваемой задачи доказывается единственность решения и устанавливаются условия, при выполнении которых существует ее классическое решение.

We consider the Cauchy problem for a one-dimensional weakly quasi-linear wave equation given in the upper half-plane. The initial conditions have a first-kind discontinuity at one point. We construct the solution using the method of characteristics in implicit analytical form as a solution of some integro-differential 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.