Сингулярные решения задачи (3 + 1)-D Проттера для волнового уравнения
Аннотация
Изучаются некоторые краевые задачи для неоднородного волнового уравнения с тремя пространственными и одной временной переменными. Эти задачи можно рассматривать как четырехмерный аналог плоской задачи Дарбу. В отличие от плоской задачи четырехмерная задача оказывается некорректной, поскольку однородная сопряженная к ней задача имеет бесконечно много классических решений. Таким образом, в рамках классической теории изучаемая задача не является фредгольмовой. С другой стороны, известно, что для гладкой правой части существует однозначно определенное обобщенное решение, которое может иметь сильную особенность в граничной точке. Эта особенность будет изолирована в вершине характеристического светового конуса и не будет распространяться вдоль конуса. В работе доказывается общий результат о существовании, существование сингулярних решений и для них устанавливаются априорные оценки. Прилагается обширная библиография.
Ключевые слова
волновое уравнение,
краевая задача,
обобщенное решение,
сингулярное решение,
распространение сингулярности,
специальные функции
При поддержке: The research of N. Popivanov and T. Popov was partially supported by the Bulgarian NSF Under Grant DCVP 02/1/2009 “Centre of Excellence on Supercomputer Applications” and by Sofia University Grants 94/2014 and 142/2015.
Об авторах
N. Popivanov
University of Sofia
Болгария
Болгария
Nedyu Popivanov
Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
T. Popov
University of Sofia
Болгария
Болгария
Todor Popov
Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
R. Scherer
Karlsruhe Institute of Technology (KIT)
Германия
Германия
Rudolf Scherer
Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany
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Рецензия
Для цитирования:
Popivanov N., Popov T., Scherer R. Сингулярные решения задачи (3 + 1)-D Проттера для волнового уравнения. Математические заметки СВФУ. 2015;22(1):69-77.
For citation:
Popivanov N., Popov T., Scherer R. SINGULAR SOLUTIONS OF THE (3+1)–D PROTTER PROBLEM FOR THE WAVE EQUATION. Mathematical notes of NEFU. 2015;22(1):69-77. (In Russ.)
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