SciELO - Scientific Electronic Library Online

 
vol.41 issue3Two-parameter generalization of bihyperbolic Jacobsthal numbersLyapunov stability and weak attraction for control systems author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Share


Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.41 no.3 Antofagasta June 2022

http://dx.doi.org/10.22199/issn.0717-6279-3993 

Artículos

On the resolution of the heat equation in unbounded non-regular domains of R³

Tahir Boudjeriou1 

Arezki Khelouf2 

1Laboratoire de Mathématiques Appliquées, Département des Mathématiques, Faculté des Sciences Exactes, Université de Bejaia, 06000 Bejaia, Algérie. e-mail: re.tahar@yahoo.com

2Department of Technology, Faculty of Technology, Lab. of Applied Mathematics, Bejaia University, 06000 Bejaia, Algeria. e-mail: arezkinet2000@yahoo.fr

Abstract

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.

Keywords: heat equation; unbounded non-regular domains; Dirichlet-Neumann condition; anisotropic Sobolev spaces

Texto completo disponible sólo en PDF

Full text available only in PDF format.

References

[1] 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. doi: 10.1007/s10958-007-0110-9 [ Links ]

[2] 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. doi: 10.1134/s0037446615060038 [ Links ]

[3] 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. doi: 10.1007/bf02310297 [ Links ]

[4] O. V. Besov, “Continuation of functions from Lp ¹ and Wp ¹”, Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 89, pp. 5-17, 1967. [Online]. Available: https://bit.ly/3wUgKOLLinks ]

[5] 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. doi: 10.3934/dcdsb.2001.1.319 [ Links ]

[6] 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. doi: 10.1111/j.1467-9590.2011.00545.x [ Links ]

[7] 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. doi: 10.1007/s00013-013-0555-7 [ Links ]

[8] 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. doi: 10.1016/0020-7225(81)90056-2 [ Links ]

[9] S. Hofmann and J. L. Lewis, “The L Neumann problem for the heat equation in non-cylindrical domains”, Journal of Functional Analysis, vol. 220, no. 1, pp. 1-54, 2005. doi: 10.1016/j.jfa.2004.10.016 [ Links ]

[10] A. Kheloufi, R. Labbas, and B.-K. Sadallah, “On the resolution of a parabolic equation in a nonregular domain of R³”, Differential Equations and Applications, no. 2, pp. 251-263, 2010. doi: 10.7153/dea-02-17 [ Links ]

[11] 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. doi: 10.1016/j.amc.2013.07.027 [ Links ]

[12] 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. doi: 10.1002/mma.2936 [ Links ]

[13] 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 [ Links ]

[14] 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. doi: 10.1007/s00009-015-0616-1 [ Links ]

[15] 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. doi: 10.1007/s10492-007-0021-6 [ Links ]

[16] 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. doi: 10.1007/bf01062085 [ Links ]

[17] 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. doi: 10.1016/s1631-073x(02)02592-x [ Links ]

[18] R. Labbas , A. Medeghri , B.-K. Sadallah, “An L-approach for the study of degenerate parabolic equations”, Electronic Journal of Differential Equations, vol. 2005, no. 36, pp. 1-20, 2005. [Online]. Available: https://bit.ly/3LRlquaLinks ]

[19] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Providence, R.I.: American Mathematical Society, 1968. [ Links ]

[20] C. Lederman, J. L. Vazquez, and N. Wolanski, "A mixed semilinear parabolic problem from combustion theory", Electronic Journal of Differential Equations , vol. 06, pp. 203-214, 2001. [Online]. Available: https://bit.ly/3NFLPwsLinks ]

[21] J. L. Lions, E. Magenes, Problèmes aux Limites Non Homogènes et Applications, vols. 1- 2. Paris: Dunod, 1968. [ Links ]

[22] 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. doi: 10.1007/s00030-013-0227-0 [ Links ]

[23] B.-K. Sadallah, “Etude d’un problème 2m-parabolique dans des domaines plan non rectangulaires”, The Bollettino dell'Unione Matematica Italiana B, vol. 2, no. 1, pp. 51-112, 1983. [ Links ]

[24] 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. doi: 10.1016/s0764-4442(99)80110-4 [ Links ]

[25] 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. doi: 10.1017/s1446788708000268 [ Links ]

[26] B. K. Sadallah, “A remark on a parabolic problem in a sectorial domain”, Applied Mathematics E-Notes, vol. 8, pp. 263-270, 2008. [ Links ]

[27] 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. doi: 10.1016/s0021-7824(97)89955-2 [ Links ]

Received: February 28, 2020; Accepted: December 30, 2021

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License