ILL-POSED INITIAL–BOUNDARY VALUE PROBLEM FOR A SYSTEM OF MIXED-TYPE EQUATIONS WITH TWO DEGENERATE PLANES
Keywords:
System of mixed-type equations, two degenerate planes, ill-posed initial–boundary value problem, correctness set, uniqueness of solution, conditional stability, a priori estimates, boundary value problem.Abstract
This paper investigates an ill-posed initial–boundary value problem, in the sense of Hadamard, for a system of mixed-type equations with two degenerate planes. An appropriate correctness set is introduced for the considered problem, and sufficient conditions ensuring the uniqueness of the solution within this set are established. Furthermore, a conditional stability estimate for the solution is derived by means of appropriate a priori estimates. The obtained results contribute to the theory of ill-posed problems for systems of mixed-type equations with two degenerate planes and provide a theoretical foundation for the further investigation of boundary value problems.
References
Tikhonov, A. N., Arsenin, V. Y. Solutions of Ill-Posed Problems. Winston & Sons, Washington, D.C., 1977.
Lavrentiev, M. M., Romanov, V. G., Shishatskii, S. P. Ill-Posed Problems of Mathematical Physics and Analysis. American Mathematical Society, Providence, RI, 1986.
Ivanov, V. K., Vasin, V. V., Tanana, V. P. Theory of Linear Ill-Posed Problems and Its Applications. VSP, Utrecht, 2002.
Kozhanov A. I. Well-posedness and ill-posedness of boundary-value problems for one class of fourth-order differential equations of Sobolev type // Itogi Nauki i Tekhniki. Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory. 2021. Vol. 198. P. 68–75
Pyatkov, S. G. Some Classes of Ill-Posed Problems. VSP, Utrecht, 2012.
Pyatkov, S. G. Operator Theory. Nonclassical Problems. VSP, Utrecht, 2002.
Tersenov, S. A. Parabolic Equations with Changing Time Direction. Novosibirsk: Nauka, 1985. (rus tilida)
Fayazov K. S., Khudaybergenov Ya. K. An ill-posed boundary value problem for a system of mixed-type equations with two degenerate planes, Siberian Electronic Mathematical Reports, Vol. 17, pp. 647–660 (2020). (in Russian)
Kerefov, A. A. Mixed-Type Equations and Boundary Value Problems. Vladikavkaz, 2000.
Egorov, I. E. "Boundary value problems for parabolic equations with degeneration." Differential Equations, 35 (1999), 1123–1131.
Khazhiev I. O. Conditional stability of the boundary value problem for a high-order system of mixed-type equations, Scientific Bulletin of SamSU, No. 1, pp. 60–67 (2022). (in Russian)
Fayazov, K.S., Khudayberganov, Y.K. Nonlocal boundary value problem for a nonhomogeneous parabolic type equation with two degenerate planes, Uzbek Mathematical Journal Open source preview, 2024, 68(3), Pp 53–65.
Fayazov, K.S., Khudayberganov, Y.K., Pyatkov S.G. Journal of Mathematical Sciences United StatesOpen source preview, 2023, 274(2), Pp 201–214.
Пятков С.Г., Неклассические дифференциальные уравнения в частных производных // Новосибирск, (1988), 77 с.
Пятков С.Г., Абашеева Н.Л., Разрешимость краевых задач для операторно-
дифференциальных уравнений смешанного типа. Вырожденный случай, Сиб. матем. журн., 43(3) (2002), c. 678–693.
Крейн М.Г., Гохберг И. Ц. Введение в теорию линейных
несамосопряженных операторов в гильбертовом пространстве // М.: Наука (1965). 448 c.
Лаврентьев М. М., Савельев Л. Я., Теория операторов и некорректные
задачи // 2-е изд. Новосибирск: Изд-во Ин-та математики. (2010), 941 с.
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- 2026-08-07 (2)
- 2026-08-01 (1)