On the resolution of the heat equation in unbounded non-regular domains of R³
DOI:
https://doi.org/10.22199/issn.0717-6279-3993Keywords:
heat equation, unbounded non-regular domains, Dirichlet-Neumann condition, anisotropic Sobolev spacesAbstract
We will prove well posedness and regularity results for the bidimensional heat equation, subject to mixed Dirichlet-Neumann type boundary conditions on the parabolic boundary of an unbounded (in one space variable direction) time-dependent domain. Our results are proved in anisotropic Hilbertian Sobolev spaces by using the domain decomposition method. This work complements the results obtained in [13] in the one-space variable case.
References
Y. A. Alkhutov, “LP-estimates of solutions of the Dirichlet problem for the heat equation in a ball”, Journal of Mathematical Sciences, vol. 142, no. 3, pp. 2021–2032, 2007. https://doi.org/10.1007/s10958-007-0110-9
M. M. Amangalieva, M. T. Dzhenaliev, M. T. Kosmakova, and M. I. Ramazanov, “On one homogeneous problem for the heat equation in an infinite angular domain”, Siberian Mathematical Journal, vol. 56, no. 6, pp. 982–995, 2015. https://doi.org/10.1134/s0037446615060038
V. N. Aref'ev and L. A. Bagirov, “Solutions of the heat equation in domains with singularities”, Mathematical Notes, vol. 64, no. 2, pp. 139–153, 1998. https://doi.org/10.1007/bf02310297
O. V. Besov, “Continuation of functions from Lpl and Wpl” (in Russian), Trudy Mat. Inst. Steklov., vol. 89, pp. 5-17, 1967. [On line]. Available: https://bit.ly/3wUgKOL
M. Chipot and A. Rougirel, “On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions”, Discrete and Continuous Dynamical Systems – Series B, vol. 1, no. 3, pp. 319–338, 2001. https://doi.org/10.3934/dcdsb.2001.1.319
A. S. Fokas and B. Pelloni, “Generalized dirichlet-to-neumann map in time-dependent domains”, Studies in Applied Mathematics, vol. 129, no. 1, pp. 51–90, 2012. https://doi.org/10.1111/j.1467-9590.2011.00545.x
S. Guesmia, “Large time and space size behaviour of the heat equation in non-cylindrical domains”, Archiv der Mathematik, vol. 101, no. 3, pp. 293–299, 2013. https://doi.org/10.1007/s00013-013-0555-7
S. C. Gupta, “Two-dimensional heat conduction with phase change in a semi-infinite mould”, International Journal of Engineering Science, vol. 19, no. 1, pp. 137–146, 1981. https://doi.org/10.1016/0020-7225(81)90056-2
S. Hofmann and J. L. Lewis, “The Lp Neumann problem for the heat equation in non-cylindrical domains”, Journal of Functional Analysis, vol. 220, no. 1, pp. 1–54, 2005. https://doi.org/10.1016/j.jfa.2004.10.016
A. Kheloufi, R. Labbas, and B.-K. Sadallah, “On the resolution of a parabolic equation in a nonregular domain of ℝ^3”, Differential Equations and Applications, no. 2, pp. 251–263, 2010. https://doi.org/10.7153/dea-02-17
A. Kheloufi, “Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of ℝ3”, Applied Mathematics and Computation, vol. 220, pp. 756–769, 2013. https://doi.org/10.1016/j.amc.2013.07.027
A. Kheloufi and B.-K. Sadallah, “Study of the heat equation in a symmetric conical type domain of ℝN+1”, Mathematical Methods in the Applied Sciences, vol. 37, no. 12, pp. 1807–1818, 2013. https://doi.org/10.1002/mma.2936
A. Kheloufi, B. K. Sadallah, “Study of a parabolic equation with mixed Dirichlet-Neumann type boundary conditions in unbounded noncylindrical domains”, Journal of Advanced Research in Applied Mathematics, vol. 7, no. 4, pp. 62—77, 2015
A. Kheloufi, “On parabolic equations with mixed Dirichlet–Robin type boundary conditions in a non-rectangular domain”, Mediterranean Journal of Mathematics, vol. 13, no. 4, pp. 1787–1805, 2015. https://doi.org/10.1007/s00009-015-0616-1
K. Kuliev and L.-E. Persson, “An extension of Rothe’s method to non-cylindrical domains”, Applications of Mathematics, vol. 52, no. 5, pp. 365–389, 2007. https://doi.org/10.1007/s10492-007-0021-6
V. V. Kurta and A. E. Shishkov, “Uniqueness classes of solutions of boundary problems for nondivergent second order parabolic equations in noncylindrical domains”, Ukrainian Mathematical Journal, vol. 42, no. 7, pp. 819–825, 1990. https://doi.org/10.1007/bf01062085
R. Labbas, A. Medeghri, and B.-K. Sadallah, “Sur une équation parabolique dans un domaine non cylindrique”, Comptes Rendus Mathematique, vol. 335, no. 12, pp. 1017–1022, 2002. https://doi.org/10.1016/s1631-073x(02)02592-x
R. Labbas, A. Medeghri, B.-K. Sadallah, “An Lp-approach for the study of degenerate parabolic equations”, Electron. J. Differential Equations, vol. 2005, no. 36, pp. 1-20, 2005. [On line]. Available: https://bit.ly/3LRlqua
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Providence, R.I.: American Mathematical Society, 1968.
Lederman, Claudia, Vazquez, Juan Luis, and Wolanski, Noemi. "A mixed semilinear parabolic problem from combustion theory", Electronic Journal of Differential Equations, vol. 06, pp. 203-214, 2001. [On line]. Available: https://bit.ly/3NFLPws
J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogènes et Applications, vols. 1, 2, Travaux et Recherches Mathématiques Series, vol. 17, no. 18. Paris: Dunod, 1968.
F. Paronetto, “An existence result for evolution equations in non-cylindrical domains”, Nonlinear Differential Equations and Applications, vol. 20, no. 6, pp. 1723–1740, 2013. https://doi.org/10.1007/s00030-013-0227-0
B.-K. Sadallah, “Etude d’un problème 2m-parabolique dans des domaines plan non rectangulaires”, The Bollettino dell'Unione Matematica Italiana B (6), vol. 2, no. 1, pp. 51-112, 1983.
B.-K. Sadallah, “Existence de la solution de l'équation de la chaleur dans un disque”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 327, no. 9, pp. 813–816, 1998. https://doi.org/10.1016/s0764-4442(99)80110-4
B.-K. Sadallah, “Regularity of a parabolic equation solution in a nonsmooth and unbounded domain,” Journal of the Australian Mathematical Society, vol. 84, no. 2, pp. 265–276, 2008. https://doi.org/10.1017/s1446788708000268
B. K. Sadallah, “A remark on a parabolic problem in a sectorial domain”, Applied Mathematics E-Notes, vol. 8, pp. 263-270, 2008.
G. Savaré, “Parabolic problems with mixed variable lateral conditions: An abstract approach”, Journal de Mathématiques Pures et Appliquées, vol. 76, no. 4, pp. 321–351, 1997. https://doi.org/10.1016/s0021-7824(97)89955-2
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