On the solvability of the Burgers equation with dynamic boundary conditions in a degenerating domain. (English) Zbl 1486.35247
Summary: In this article, the well-posedness of the boundary value problem for the Burgers equation with dynamic boundary conditions is studied in Sobolev spaces with a degenerating domain.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B45 | A priori estimates in context of PDEs |
| 35K58 | Semilinear parabolic equations |
References:
| [1] | Benia, Y.; Sadallah, B. K., Existence of solutions to Burgers equations in domains that can be transformed into rectangles, Electron. J. Differ. Equat., 157, 1-13 (2016) · Zbl 1342.35154 |
| [2] | Benia, Y.; Sadallah, B. K., Existence of solutions to Burgers equations in a non-parabolic domain, Electron. J. Differ. Equat., 20, 1-13 (2018) · Zbl 1378.35261 |
| [3] | Burgers, J. M., The Nonlinear Diffusion Equation. Asymptotic Solutions and Statistical Problems (1974), Dordrecht: D. Reidel, Dordrecht · Zbl 0302.60048 · doi:10.1007/978-94-010-1745-9 |
| [4] | M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Vol. 9 of Mathematics and Its Applications (Soviet Series) (Nauka, Moscow, 1980; Springer, Berlin, 1988). · Zbl 0643.35005 |
| [5] | Solonnikov, V. A.; Fasano, A., One-dimensional parabolic problem arising in the study of some free boundary problems, Zap. Nauch. Sem. LOMI, 269, 322-338 (2000) · Zbl 1031.35153 |
| [6] | Kim, E. I.; Omelchenko, V. T.; Kharin, S. N., Mathematical Models of Thermal Processes in Electrical Contacts (1977) |
| [7] | Mitropolsky, Yu. A.; Berezovsky, A. A.; Plotnitskiy, T. A., Problems with free boundaries for the nonlinear evolution equation in metallurgy, medicine, and ecology, Ukr. Math. J., 44, 59-67 (1992) · Zbl 0786.35154 · doi:10.1007/BF01062627 |
| [8] | Verigin, N. N., On one class of hydromechanical problems for domains with moving boundaries, Dinam. Splosh. Sredy, 46, 23-32 (1980) · Zbl 0499.76081 |
| [9] | Kartashov, E. M., “The problem of heat stroke in a domain with a moving boundary based on new integral relations,” Izv. Akad. Nauk, Energet., 4, 122-137 (1997) · Zbl 0928.74025 |
| [10] | Yuldashev, T. K., Nonlinear optimal control of thermal processes in a nonlinear inverse problem, Lobachevskii J. Math., 41, 124-136 (2020) · Zbl 1450.35295 · doi:10.1134/S1995080220010163 |
| [11] | Solonukha, O. V., The first boundary value problem for quasilinear parabolic differential-difference equations, Lobachevskii J. Math., 42, 1067-1077 (2021) · Zbl 1466.35254 · doi:10.1134/S1995080221050188 |
| [12] | Mamedov, Kh. R., An initial boundary value problem for a mixed type equation in a rectangular domain, Lobachevskii J. Math., 42, 572-578 (2021) · Zbl 1465.35319 · doi:10.1134/S1995080221030136 |
| [13] | Yuldashev, T. K.; Abdullaev, O. Kh., Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic equation with nonlinear terms, Lobachevskii J. Math., 42, 1113-1123 (2021) · Zbl 1466.35269 · doi:10.1134/S1995080221050218 |
| [14] | Yuldashev, T. K.; Islomov, B. I.; Alikulov, E. K., Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains, Lobachevskii J. Math., 41, 926-944 (2020) · Zbl 1450.35187 · doi:10.1134/S1995080220050145 |
| [15] | Amangaliyeva, M. M.; Jenaliyev, M. T.; Kosmakova, M. T.; Ramazanov, M. I., About Dirichlet boundary value problem for the heat equation in the infinite angular domain, Bound. Value Probl., 213, 1-21 (2014) · Zbl 1304.35322 |
| [16] | Amangaliyeva, M. M.; Jenaliyev, M. T.; Kosmakova, M. T.; Ramazanov, M. I., On one homogeneous problem for the heat equation in an infinite angular domain, Sib. Math. J., 56, 982-995 (2015) · Zbl 1334.35070 · doi:10.1134/S0037446615060038 |
| [17] | Amangaliyeva, M. M.; Jenaliyev, M. T.; Ramazanov, M. I.; Iskakov, S. A., On a boundary value problem for the heat equation and a singular integral equation associated with it, Appl. Math. Comput., 399, 126009 (2021) · Zbl 1508.35018 |
| [18] | Lions, J. L.; Magenes, E., Problemes aux limites non homogenes et applications (1968), Paris: Dunod, Paris · Zbl 0165.10801 |
| [19] | Riesz, F.; Bela, Sz., Nagy, Lecons d’Analyse Fonctionelle (1972), Budapest: Akademiai Kiado, Budapest |
| [20] | Adams, R. A.; Fournier, J. J. F., Sobolev Spaces (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1098.46001 |
| [21] | Gantmacher, F. R.; Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (2002), Providence, RI: AMS Chelsea, Providence, RI · Zbl 1002.74002 · doi:10.1090/chel/345 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
