Abstract
We study the well-posedness of a third boundary value problem for a multidimensional parabolic equation in the case when the coefficient of the conormal derivative vanishes at some points. We show that under some conditions on the sign of this coefficient there exists nonexistence or nonuniqueness of a solution in the conventional anisotropic Sobolev space. Using the regularization method, we prove existence and uniqueness theorems for the regular solution in suitable weighted spaces.
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References
Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).
Ikeda M., Jleli M., and Samet M., “On the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial Robin boundary conditions,” J. Differ. Equ., vol. 269, no. 1, 563–594 (2020).
Gesztesy F., Mitrea M., and Nichols R., “Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions,” J. Anal. Math., vol. 122, no. 1, 229–287 (2014).
Choi J. and Kim S., “Green’s functions for elliptic and parabolic systems with Robin-type boundary conditions,” J. Funct. Anal., vol. 267, no. 9, 3205–3261 (2014).
Li Y., Liu Y., and Xiao S., “Blow-up phenomena for some nonlinear parabolic problems under Robin boundary conditions,” Math. Comput. Model., vol. 54, no. 11, 3065–3069 (2011).
Ding J., “Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions,” Comput. Math. Appl., vol. 65, no. 11, 1808–1822 (2013).
Vyborny R., “On the properties of the solutions of some boundary value problems for equations of parabolic type,” Dokl. Akad. Nauk SSSR, vol. 117, no. 4, 563–565 (1957).
Kamynin L. I. and Khimchenko B. N., “Analogs of Giraud’s theorem for a second-order parabolic equation,” Sib. Math. J., vol. 14, no. 1, 59–77 (1973).
Lavrentev M. A. and Shabat B. V., Theory of Functions of Complex Variables, Nauka, Moscow (1973) [Russian].
Triebel H., Interpolation Theory. Function Spaces. Differential Operators, Leipzig, Barth (1995).
Rudin W., Principles of Mathematical Analysis, McGraw-Hill, New York etc. (1976).
Ladyzhenskaya O. A., “On the closure of the elliptic operator,” Dokl. Akad. Nauk SSSR, vol. 79, no. 5, 723–726 (1951).
Funding
The work is supported by the Mathematical Center in Akademgorodok under Agreement 075–15–2022–281 on April 5, 2022 with the Ministry of Science and Higher Education of the Russian Federation.
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 870–883. https://doi.org/10.33048/smzh.2022.63.413
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Kozhanov, A.I., Shubin, V.V. Existence and Uniqueness of the Solution to a Degenerate Third Boundary Value Problem for a Multidimensional Parabolic Equation. Sib Math J 63, 723–734 (2022). https://doi.org/10.1134/S0037446622040139
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DOI: https://doi.org/10.1134/S0037446622040139