Cauchy Type Problem for Some Quasilinear Equations with Riemann – Liouville Derivatives and a Sectorial Operator
DOI:
https://doi.org/10.52575/2687-0959-2024-56-4-261-272Keywords:
Riemann — Liouville Derivative, Cauchy Type Problem, Quasilinear Equation, Contraction Mapping Theorem, Local Solvability, Global SolvabilityAbstract
We studies the issues of solvability of the Cauchy type problem for quasi-linear equations solved with respect to the highest fractional Riemann – Liouville derivative, the operator in the linear part at an unknown function in the equation is assumed to be sectorial. In this case, the nonlinear operator depends on low-order fractional derivatives with an arbitrary fractional part. Theorems on the local and global existence of a unique solution are obtained under the condition of local Lipschitz continuity and Lipschitz continuity of a nonlinear operator, respectively, in the case of its continuity in the norm of the graph of the sectorial operator. The Cauchy type problem for a quasi-linear equation is reduced to an integro-differential equation in a specially selected functional space. To prove the existence of a unique solution, Banach Theorem on the fixed point of a compressive map in a complete metric space is used. The abstract result obtained is applied for the study of the existence and uniqueness of a solution of a class of initial boundary value problems for nonlinear partial differential equations with polynomials from a self-adjoint elliptic operator in spatial variables and with fractional derivatives in time.
Downloads
References
Самко С.Г., Килбас А.А., Маричев О.И. Интегралы и производные дробного порядка и некоторые их приложения. Минск: Наука и техника; 1987. 688 с.
Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Boston, Heidelberg: Elsevier Science Publishing; 2006. 541 p.
Нахушев А.М. Дробное исчисление и его применение. М.: Физматлит; 2003. 272 c.
Prüss J. Evolutionary integral equations and applications. Basel: Springer; 1993. 366 p. https://doi.org/10.1007/978-3-0348-8570-6
Bazhlekova E.G. Subordination principle for fractional evolution equations. Fractional Calculus & Applied Analysis. 2000;3(3):213–230.
Bazhlekova E.G. Subordination in a class of generalized time-fractional diffusion-wave equations. Fractional Calculus & Applied Analysis. 2018;21(4):869–900. https://doi.org/10.1515/fca-2018-0048
Глушак А.В. О задаче типа Коши для абстрактного дифференциального уравнения с дробной производной. Вестн. Воронеж. гос. ун-та. Сер. Физика, математика. 2001;2:74–77. http://www.vestnik.vsu.ru/content/physmath/2001/02/toc_ru.asp
Глушак А.В., Манаенкова Т.А. Прямая и обратная задачи для абстрактного дифференциального уравнения, содержащего дробные производные Адамара. Дифференц. уравнения. 2011;(47)9:1294–1304. https://www.elibrary.ru/item.asp?id=16766509
Bajlekova E.G. Fractional Evolution Equations in Banach Spaces. PhD thesis. Eindhoven: Eindhoven University of Technology; 2001. 107 p.
Fedorov V.E., Romanova E.A. Inhomogeneous fractional evolutionary equation in the sectorial. Journal of Mathematical Sciences. 2020;250(5):819–829. https://doi.org/10.1007/s10958-020-05047-x
Fedorov V.E., Avilovich A.S., Borel L.V. Initial Problems for Semilinear Degenerate Evolution Equations of Fractional Order in the Sectorial Case. Nonlinear Analysis and Boundary Value Problems. NABVP 2018, Santiago de Compostela, Spain, September 4–7. Ed. by I.Area, A.Cabada, J.A.Cid etc. Springer Proceedings in Mathematics and Statistics. 2019;292:41–62. https://doi.org/10.1007/978-3-030-26987-6_4
Fedorov V.E., Zakharova T.A. Nonlocal solvability of quasilinear degenerate equations with Gerasimov – Caputo derivatives. Lobachevskii Journal of Mathematics. 2023;44(2):595–606. https://doi.org/10.1134/S1995080223020178
Fedorov V.E., Kostic M., Zakharova T.A. Quasilinear fractional order equations and fractional powers of sectorial operators. Fractal and Fractional. 2023;7(5)385. https://doi.org/10.3390/fractalfract7050385
Федоров В.Е., Авилович А.С. Задача типа Коши для вырожденного уравнения с производной Римана — Лиувилля в секториальном случае. Сиб. мат. журн. 2019;60(2):461–477. https://doi.org/10.33048/smzh.2019.60.216
Авилович А.С., Гордиевских Д.М., Федоров В.Е. Вопросы однозначной разрешимости и приближённой управляемости для линейных уравнений дробного порядка с гёльдеровой правой частью. Челябинский физико-математический журнал. 2020;(5)1:5–21. https://doi.org/10.24411/2500-0101-2020-15101
Fedorov V.E., Avilovich A.S. Semilinear fractional-order evolution equations of Sobolev type in the sectorial case. Complex Variables and Elliptic Equations. 2021;66(6–7):1108–1121. https://doi.org/10.1080/17476933.2020.1833870
Fedorov V.E., Turov M.M. Sectorial tuples of operators and quasilinear fractional equations with multi-term linear part. Lobachevskii Journal of Mathematics. 2022;(43)6:1502–1512. https://doi.org/10.1134/S1995080222090074
Федоров В.Е., Туров М.М. Дефект задачи типа Коши для линейных уравнений с несколькими производными Римана — Лиувилля. Сиб. мат. журн. 2021;62(5):1143–1162. https://doi.org/10.33048/smzh.2021.62.514
Туров М.М. Квазилинейные уравнения с несколькими производными Римана — Лиувилля произвольных порядков. Челябинский физико-математический журнал. 2022;(7)4:434–446. https://doi.org/10.47475/2500-0101-2022-17404
Fedorov V.E., Turov M.M.. Multi-term equations with Riemann — Liouville derivatives and Hölder type function spaces. Boletin de la Sociedad Matematica Mexicana. 2023;29:42. https://doi.org/10.1007/s40590-023-00509-z
Трибель Х. Теория интерполяции. Функциональные пространства. Дифференциальные операторы. М.: Мир; 1980. 664 c.
Хэссард Б., Казаринов Н., Вэн И. Теория и приложения бифуркации рождения цикла. М.: Мир; 1985. 280 c.
Abstract views: 0
##submission.share##
Published
How to Cite
Issue
Section
Copyright (c) 2024 Applied Mathematics & Physics
This work is licensed under a Creative Commons Attribution 4.0 International License.
