Well-posedness of elliptic-parabolic differential problem with integral condition
DOI:
https://doi.org/10.31489/2025m4/125-133Keywords:
elliptic-parabolic equation, nonlocal boundary value problem, integral condition, Hölder spaces, well-posedness, coercivity inequalities, stability, mixed-type differential equationsAbstract
In this paper, we study a class of nonlocal boundary value problems for elliptic-parabolic equations subject to integral-type conditions. Such problems naturally emerge in various physical and engineering contexts, including diffusion processes in composite materials and systems with memory or nonlocal interactions. The model considered involves a mixed-type equation in which the elliptic and parabolic components are coupled through nonlocal boundary terms, while the boundary conditions incorporate integral constraints that generalize the traditional Dirichlet and Neumann formulations. To investigate the solvability of this problem, we employ analytical methods based on the theory of parabolic and elliptic operators in weighted Ho¨lder spaces, which are particularly suitable for handling boundary singularities and ensuring regularity of solutions. We establish the existence, uniqueness, and continuous dependence of solutions on the input data, thereby proving the well-posedness of the problem. Furthermore, we derive coercivity inequalities for solutions of the associated mixed nonlocal boundary problems, which guarantee their stability and provide essential tools for studying related inverse and control problems. The findings extend several classical results and offer a unified approach to the analysis of nonlocal elliptic-parabolic models.
References
Krein, S.G. (1966). Lineinye differentsialnye uravneniia v banakhovom prostranstve [Linear Differential Equations in a Banach Space]. Moscow: Nauka [in Russian].
Skubachevskii, A.L. (1997). Elliptic Functional Differential Equations and Applications (Operator Theory: Advances and Applications, Vol.91). Basel: Birkh¨auser Verlag. https://doi.org/10.1007/978-3-0348-9033-5
Skubachevskii, A.L. (2016). Boundary value problems for elliptic functional differential equations and their applications. Russian Mathematical Surveys, 71(5), 801–906. https://doi.org/10.4213/rm9739
Salakhitdinov, M.S. (1974). Uravneniia smeshanno-sostavnogo tipa [Equations of mixed-composite type]. Tashkent: Fan [in Russian].
Dzhuraev, T.D. (1979). Kraevye zadachi dlia uravnenii smeshannogo i smeshanno-sostavnogo tipov [Boundary value problems for equations of mixed and mixed-composite types]. Tashkent: Fan [in Russian].
Kal’menov, T.Sh., & Sadybekov, M.A. (2017). On a Frankl-type problem for a mixed parabolichyperbolic equation. Siberian Mathematical Journal, 58(2), 227–231. https://doi.org/10.1134/S0037446617020057
Nakhushev, A.M. (1995). Uravneniia matematicheskoi biologii [Equations of mathematical biology]. Moscow: Vysshaшa Shkola [in Russian].
Ashyralyyev, C. (2017). Numerical solution to Bitsadze–Samarskii type elliptic overdetermined multipoint nonlocal boundary value problem. Boundary Value Problems, 2017, Article 74, 1–22. https://doi.org/10.1186/s13661-017-0804-y
Sapagovas, M., Stikoniene, O., Ciupaila, R. & Joksiene, Z. (2016). Convergence of iterative methods for elliptic equations with integral boundary conditions. Electronic Journal of Differential Equations, 2016(118), 1–14.
Ashyralyev, A., & Hamad, A. (2025). On the well-posedness of elliptic equations with nonlocal boundary conditions. TWMS Journal of Pure and Applied Mathematics, 16(1), 59–70. https://doi.org/10.30546/2219-1259.16.1.2025.59
Kozhanov, A.I. (2023). Nelokalnye zadachi s obobshchennym usloviem Samarskogo–Ionkina dlia nekotorykh klassov nestatsionarnykh differentsialnykh uravnenii [Nonlocal problems with generalized Samarskii–Ionkin condition for some classes of nonstationary differential equations]. Doklady RAN Matematika, informatika, protsessy upravleniia — RAS Reports Mathematics, Computer Science, Control Processes, 509(1), 50–53 [in Russian]. https://doi.org/10.31857/S2686954323700091
Sadybekov, M.A., & Dukenbayeva, A.A. (2022). On boundary value problems of the Samarskii–Ionkin type for the Laplace operator in a ball. Complex Variables and elliptic equations, 67(2), 369–383. https://doi.org/10.1080/17476933.2020.1828377
Ashyralyyeva, M., & Ashyraliyev, M. (2024). A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations. Bulletin of the Karaganda University. Mathematics Series, 3(115), 46–54. https://doi.org/10.31489/2024m3/46-54
Ashyralyev, A., & Al-Hammouri, A. (2023). The spacewise dependent source identification problem for the elliptic-telegraph differential equation with involution and Robin condition. e-Journal of Analysis and Applied Mathematics, 2023, 64–70. https://doi.org/10.62780/ejaam/2023-006
Ashyralyyeva, M., & Ashyraliyev, M. (2024). Stable difference schemes for hyperbolic-parabolic equations with unknown parameter. Boletin de la Sociedad Matematica Mexicana, 30(1), Article 14. https://doi.org/10.1007/s40590-023-00585-1
Ashyralyev, A., & Sarsenbi, A. (2015). Well-posedness of an elliptic equation with involution. Electronic Journal of Differential Equations, 2015(284), 1–8.
Ashyralyev, A., & Ashyralyyev, C. (2024). The second-order accuracy difference schemes for integral-type time-nonlocal parabolic problems. Journal of Mathematical Sciences, 283, 195– 210. https://doi.org/10.1007/s10958-024-07249-z
Ashyralyev, A., Al-Hammouri, A., & Ashyralyyev, C. (2021). On the absolute stable difference scheme for the space-wise dependent source identification problem for elliptic-telegraph equation. Numerical Methods for Partial Differential Equations, 37(2), 962–986. https://doi.org/10.1002/num.22561
Shakhmurov, V., & Musaev, H. (2017). Maximal regular convolution-differential equations in weighted Besov spaces. Applied and Computational Mathematics, 16(2), 190–200.
Ashyralyev, A., Gercek, O., & Zusi, E. (2018). A note on the second order of accuracy difference scheme for elliptic-parabolic equations in Holder spaces. Bulletin of the Karaganda University. Mathematics Series, 3(91), 108–116. https://doi.org/10.31489/2018m3/108-116
Gercek, O., & Ashyralyev, A. (2012). On a difference scheme of the second order of accuracy for elliptic-parabolic equations. Boundary Value Problems, 2012, Article 80, 1–15. https://doi.org/10.1186/1687-2770-2012-80
Sobolevskii, P.E. (1977). Teoriia polugrupp i ustoichivost raznostnykh skhem (Teoriia operatorov v funktsionalnykh prostranstvakh) [The theory of semigroups and the stability of difference schemes (Theory of operators in function spaces). Novosibirsk: Nauka [in Russian].
Gercek, O., & Zusi, E. (2014). Well-posedness of difference scheme for elliptic-parabolic equations in Holder spaces without a weight. AIP Conference Proceedings, 1611, 84–88. https://doi.org/10.1063/1.4893809
Sobolevskii, P.E. (1975). Difference Methods for the Approximate Solution of Differential Equations. Voronezh: Voronezh State University Press.
Ashyralyev, A., Agirseven, D., & Muazu, S.B. (2024). A note on the delay nonlinear parabolic differential equations. Filomat, 38(16), 5761–5778. https://doi.org/10.2298/FIL2416761A
