danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2023-67-1-14-19dan-1106Research ArticleМАТЕМАТИКАMATHEMATICSКлассическое решение задачи Коши для одномерного квазилинейного волнового уравненияClassical solution of the initial-value problem for a one-dimensional quasilinear wave equationКорзюкВ. 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 andMathematics), Professor

11, Surganov Str., 220072, Minsk

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

Рудько Ян Вячеславович – аспирант

ул. Сурганова, 11, 220072, Минск 

Rudzko Jan V. – Postgraduate Student

11, Surganov Str., 220072, Minsk

janycz@yahoo.comИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus2023030320236711419Copyright © 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/1106

Для одномерного слабо квазилинейного волнового уравнения, заданного в верхней полуплоскости, рассматривается задача Коши. Решение строится в неявном аналитическом виде как решение некоторого интегродифференциального уравнения. Проводится исследование разрешимости этого уравнения, а также гладкости его решения. Для рассматриваемой задачи доказывается единственность решения и устанавливаются условия, при выполнении которых существует ее классическое решение. При недостаточной гладкости начальных данных строится слабое решение.

For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integro-differential equation. The solvability of this equation, as well the smoothness of its solution, is studied. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established. When given data is not enough smooth a mild solution is constructed.

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