matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосоваmatmess-416Research ArticleМАТЕМАТИКАСингулярные решения задачи (3 + 1)-D Проттера для волнового уравненияSINGULAR SOLUTIONS OF THE (3+1)–D PROTTER PROBLEM FOR THE WAVE EQUATIONPopivanovN.PopivanovN.

Nedyu Popivanov

Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria 

Nedyu Popivanov

Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria 

nedyu@fmi.uni-sofia.bgPopovT.PopovT.

Todor Popov

Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria 

Todor Popov

Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria 

topopover@fmi.uni-sofia.bgSchererR.SchererR.

Rudolf Scherer

Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany

Rudolf Scherer

Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany

rudolf.scherer@kit.eduUniversity of SofiaБолгарияUniversity of SofiaBulgariaKarlsruhe Institute of Technology (KIT)ГерманияKarlsruhe Institute of Technology (KIT)Germany2015070720262216977Copyright © Popivanov N., Popov T., Scherer R., 20262026Popivanov N., Popov T., Scherer R.Popivanov N., Popov T., Scherer R.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/416

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.

We study some boundary value problems for the nonhomogeneous wave equation with three space and one time variables. The problems could be viewed as R4 analogs of Darboux problems in R2. In contrast to the planar Darboux problem the fourdimensional version is ill-posed, since its homogeneous adjoint problem has infinitely many classical solutions. Thus, in the framework of the classical solvability the problem is not Fredholm. Alternatively, it is known that for smooth right-hand side functions, there is a uniquely determined generalized solution that may have strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. In this article we give a general existence result and find a priori estimates for singular solutions. A lengthy reference list is appended.

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