Abstract
In the present paper we investigate a boundary-value problem for a forward-backward parabolic equation in a rectangular domain and prove the existence of unique regular solution to this problem. The proof of the uniqueness of the solution is based on the spectral method, and in the proof of existence of solution we use the method of separation of variables. In the introduction we give a survey of related works.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Gevrey, M. “Sur les équation aux Dérivées Partièlles du Type Parabolique,” J. Math. Appl. 4, 105–137 (1914).
Pagani, C. D., Talenti, G. “On a Forward-Backward Parabolic Equation,” Ann. Mat. Pura Appl. 90, No. 4, 1–58 (1971).
Kerefov, A. A. “The Gevrey Problem for a Certain Mixed-Parabolic Equation,” Differ. Uravn. 13, 76–83 (1977).
Egorov, I. E. “The First Mixed Problem for a Parabolic System,” Sib. Math. J. 18, No. 1, 220–224 (1977).
Katyshev, V. V. “On an Equation of Elliptic-Parabolic Type,” in Boundary-Value Problems for Non-Linear Equations (Novosibirsk, 1982), pp. 130–133.
Tersenov, S. A. Parabolic Equations with Varying Time Direction (Nauka, Novosibirsk, 1985) [in Russian].
Akbarova, M. H. “A Nonlocal Problem for a Mixed Parabolic Equation,” Uzbek. Mat. Zh., No. 3–4, 3–13 (1992).
Pagani, C. D. “On the Parabolic Equation and a Related One,” Ann. Mat. Pura ed Appl. 99, No. 4, 333–399 (1974).
Gekkieva, S. Kh. “Analog of the Gevrey Problem for a Mixed-Parabolic Equation with Fractional Caputo Derivative,” in Proceedings of the Second International Russian-Uzbekistan Symposium’ Equations of Mixed Type and Related Problems of Analysis and Information Science’ (Kabarda-Balkar Republic, Elbrus, 2012), p. 82.
Baouendi, M. S., Grisvard P. “Sur une Equation d’Evolution Changeant de Type,” J. Functional Anal., No. 2, 352–369 (1968).
Amanov, D., Kadirkulov, B. D. “A Boundary Value Problem for a Fourth-Order Mixed Parabolic Equation with Fractional Derivatives,” Uzbek. Mat. Zh., No. 4, 11–20 (2009).
Amanov, D. “Boundary Value Problem for a Mixed-Parabolic Fourth Order Degenerate Equation,” Uzbek. Mat. Zh., No. 2, 26–30 (2010).
Amanov, D., Yuldasheva, A. V. “Boundary-Value Problem for Mixed-Parabolic Equation of High Order,” Dokl. AN RUz, No. 6, 10–13 (2008).
Sabitov, K. B. “Tricomi Problem for a Mixed Parabolic-Hyperbolic Equation in a Rectangular Domain,” Math. Notes 86, No. 2, 249–254 (2009).
Sabitov, K. B., Safin, E.M. “Inverse Problem for a Parabolic-Hyperbolic Equation in a Rectangular Domain,” Dokl. Math. 80, No. 3, 856–859 (2009).
Sabitov, K. B., Yunusova, G. R. “Inverse Problem for an Equation of Parabolic-Hyperbolic Type with a Nonlocal Boundary Condition,” Differ. Equ. 48, No. 2, 246–254 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D. Amanov, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 12, pp. 3–8.
About this article
Cite this article
Amanov, D. Boundary-value problem for degenerate parabolic equation of high order with varying time direction. Russ Math. 58, 1–6 (2014). https://doi.org/10.3103/S1066369X14120019
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X14120019