Обратная задача для квазилинейного волнового уравнения с памятью

Обратная задача для квазилинейного волнового уравнения с памятью

Романов В. Г., Бугуева Т. В.
Сибирский журнал индустриальной математики, 28, 1(101), 38-66 (2025)
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УДК 517.956 
DOI: 10.33048/SIBJIM.2025.28.104

Аннотация:

Исследованы прямая и обратная задачи для квазилинейного волнового уравнения $\square u − qu^2 − K ∗ u = 0$, в котором ядро $K(x, t)$ представимо в виде $K(x, t) = p(x)K_{0}(t)$, где $p(x)$ — непрерывная функция. Обратная задача посвящена определению функций $q(x)$ и $p(x)$. В качестве дополнительной информации в обратной задаче задаются следы производной по переменной $x$ двух решений прямой начально краевой задачи, соответствующих различным краевыми условиям, при $x = 0$ на конечном отрезке [$0, T$]. Найдены условия однозначной разрешимости прямой задачи. Для обратной задачи установлена теорема о локальном существовании решения задачи.

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Данная работа выполнена в рамках государственного задания Института математики им. С. Л. Соболева СО РАН (проект FWNF-2022-0009). Других источников финансирования проведения или руководства данным конкретным исследованием не было.

В. Г. Романов
  1. Институт математики им. С. Л. Соболева СО РАН, 
    просп. Акад. Коптюга, 4, г. Новосибирск 630090, Россия

E-mail: romanov@math.nsc.ru

Т. В. Бугуева
  1. Институт математики им. С. Л. Соболева СО РАН, 
    просп. Акад. Коптюга, 4, г. Новосибирск 630090, Россия
  2. Новосибирский государственный университет, 
    ул. Пирогова, 1, г. Новосибирск 630090, Россия

E-mail: bugueva@math.nsc.ru

Статья поступила 22.11.2024 г.
После доработки — 26.02.2025 г.
Принята к публикации 10.03.2025 г.

Abstract:

The forward and inverse problems are investigated for the quasilinear wave equation $\square u − qu^2 − K ∗ u = 0$ where the kernel $K(x, t)$ is represented in the form $K(x, t) = p(x)K_{0}(t)$ with $p(x)$ being a continuous function. The inverse problem is devoted to the determination of the compact functions $q(x)$ and $p(x)$. Traces of the derivative with respect to $x$ of two solutions to the forward initial–boundary value problem related to two arbitrary boundary data are given for $x = 0$ on the finite segment [$0, T$] as an additional information for the solution to the inverse problem. The conditions for the unique solvability of the forward problem.

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