Yıl 2020,
Cilt: 49 Sayı: 1, 180 - 194, 06.02.2020
Saida Cherfaoui
Amor Kessab
Arezki Kheloufi
Kaynakça
- [1] E.A. Baderko, The solvability of boundary value problems for higher order parabolic
equations in domains with curvilinear lateral boundaries, Differ. Uravn. 10 (12), 1781–
1792, 1976.
- [2] E.A. Baderko, On the solution of boundary value problems for linear parabolic equations
of arbitrary order in noncylindrical domains by the method of boundary integral
equations, PhD Thesis, Moscow, 1992.
- [3] V. Besov, Continuation of functions from $L_p^l$ and $W_p^l$, Proc. Steklov Inst. Math.
89, 5–17, 1967.
- [4] M.F. Cherepova, On the solvability of boundary value problems for a higher order
parabolic equation with growing coefficients, Dokl. Math. 74 (3), 819–820 2006.
- [5] S. Cherfaoui, A. Kessab, and A. Kheloufi, On 2m-th order parabolic equations with
mixed boundary conditions in non-rectangular domains, Sib. Èlektron. Mat. Izv. 14,
73–91, 2017.
- [6] V.A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations:
towards Petrovskii-type criterion by blow-up approach, Nonlinear Differ. Equ.
Appl. 5 (16), 597–655, 2009.
- [7] A. Grimaldi Piro and F. Ragnedda, Higher-order parabolic operators in domains with
a "nonsmooth" boundary, Rend. Sem. Fac.Sci. Univ. Cagliari 54, 45–62, 1984.
- [8] P. Grisvard and G. Looss, Problèmes aux limites unilatéraux dans des domaines non
réguliers, Jour. Equ. Dériv. Part. 1–26, 1976.
- [9] A. Kheloufi, Resolutions of parabolic equations in non-symmetric conical domains,
Electron. J. Differ. Equ. 2012 (116), 1–14, 2012.
- [10] A. Kheloufi, On a fourth order parabolic equation in a nonregular domain of $\mathbb{R}^3$,
Mediterr. J. Math. 12, 803–820, 2015.
- [11] A. Kheloufi, Study of a 2m-th order parabolic equation in a non-regular type of prism
of $\mathbb{R}^{N+1}$, Georgian Math. J. 23 (2), 227–237, 2016.
- [12] A. Kheloufi, On the Dirichlet problem for the heat equation in non-symmetric conical
domains of $\mathbb{R}^{N+1}$, Palestine J. Math. 6 (1), 287–300, 2017.
- [13] A. Kheloufi and B.K. Sadallah, On the regularity of the heat equation solution in noncylindrical
domains: two approaches, Appl. Math. Comput. 218, 1623–1633, 2011.
- [14] A. Kheloufi and B.K. Sadallah, Study of the heat equation in a symmetric conical type
domain of $\mathbb{R}^{N+1}$, Math. Methods Appl. Sci. 37, 1807–1818, 2014.
- [15] A. Kheloufi and B.K. Sadallah, Resolution of a high-order parabolic equation in conical
time-dependent domains of $\mathbb{R}^{3}$, Arab J. Math. Sci. 22, 165–181, 2016.
- [16] V.A. Kondrat’ev, Boundary problems for parabolic equations in closed regions, Am.
Math. Soc. Providence. R I. 450–504, 1966.
- [17] V.A. Kozlov, Coefficients in the asymptotic solutions of the Cauchy boundary-value
parabolic problems in domains with a conical point, Siberian Math. J. 29, 222–233,
1988.
- [18] R. Labbas and B.K. Sadallah, Smoothness of the solution of a fourth order parabolic
equation in a polygonal domain, Int. J. Appl. Math. 1, 75–90, 1999.
- [19] O.A. Ladyzhenskaya and V.A. Solonnikov and N.N. Ural’tseva, Linear and quasilinear
equations of parabolic type, (A.M.S., Providence, Rhode Island, 1968).
- [20] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, 1,
2, Dunod, Paris, 1968.
- [21] A. Maghnouji, Problèmes aux limites paraboliques dans un domaine non régulier,
C.R.A.S. 316, 331–336, 1993.
- [22] V.P. Mikhailov, The Dirichlet problem for a parabolic equation I, Mat. Sb. (N.S.) 61
(103), 40–64, 1963.
- [23] V.P. Mikhailov, The Dirichlet problem for a parabolic equation II, Mat. Sb. (N.S.) 62
(104), 140–159, 1963.
- [24] B.K. Sadallah, Etude d’un problème 2m-parabolique dans des domaines plan non rectangulaires,
Boll. Un. Mat. Ital. 5 (2-B), 51–112, 1983.
- [25] B.K. Sadallah, Singularities of the solution of a 2m-parabolic problem in a polygonal
domain, Arab J. Math. Sci. 4 (2), 31–41, 1998.
- [26] B.K. Sadallah, Study of a parabolic problem in a conical domain, Math. J. Okayama
Univ. 56, 157–169, 2014.
Study of 2m-th order parabolic equation in non-symmetric conical domains
Yıl 2020,
Cilt: 49 Sayı: 1, 180 - 194, 06.02.2020
Saida Cherfaoui
Amor Kessab
Arezki Kheloufi
Öz
This article is devoted to the study of a $N$-space dimensional linear high-order parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a non-symmetric conical domain. The analysis is performed in the framework of weighted anisotropic Sobolev spaces by using the domain decomposition method.
Kaynakça
- [1] E.A. Baderko, The solvability of boundary value problems for higher order parabolic
equations in domains with curvilinear lateral boundaries, Differ. Uravn. 10 (12), 1781–
1792, 1976.
- [2] E.A. Baderko, On the solution of boundary value problems for linear parabolic equations
of arbitrary order in noncylindrical domains by the method of boundary integral
equations, PhD Thesis, Moscow, 1992.
- [3] V. Besov, Continuation of functions from $L_p^l$ and $W_p^l$, Proc. Steklov Inst. Math.
89, 5–17, 1967.
- [4] M.F. Cherepova, On the solvability of boundary value problems for a higher order
parabolic equation with growing coefficients, Dokl. Math. 74 (3), 819–820 2006.
- [5] S. Cherfaoui, A. Kessab, and A. Kheloufi, On 2m-th order parabolic equations with
mixed boundary conditions in non-rectangular domains, Sib. Èlektron. Mat. Izv. 14,
73–91, 2017.
- [6] V.A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations:
towards Petrovskii-type criterion by blow-up approach, Nonlinear Differ. Equ.
Appl. 5 (16), 597–655, 2009.
- [7] A. Grimaldi Piro and F. Ragnedda, Higher-order parabolic operators in domains with
a "nonsmooth" boundary, Rend. Sem. Fac.Sci. Univ. Cagliari 54, 45–62, 1984.
- [8] P. Grisvard and G. Looss, Problèmes aux limites unilatéraux dans des domaines non
réguliers, Jour. Equ. Dériv. Part. 1–26, 1976.
- [9] A. Kheloufi, Resolutions of parabolic equations in non-symmetric conical domains,
Electron. J. Differ. Equ. 2012 (116), 1–14, 2012.
- [10] A. Kheloufi, On a fourth order parabolic equation in a nonregular domain of $\mathbb{R}^3$,
Mediterr. J. Math. 12, 803–820, 2015.
- [11] A. Kheloufi, Study of a 2m-th order parabolic equation in a non-regular type of prism
of $\mathbb{R}^{N+1}$, Georgian Math. J. 23 (2), 227–237, 2016.
- [12] A. Kheloufi, On the Dirichlet problem for the heat equation in non-symmetric conical
domains of $\mathbb{R}^{N+1}$, Palestine J. Math. 6 (1), 287–300, 2017.
- [13] A. Kheloufi and B.K. Sadallah, On the regularity of the heat equation solution in noncylindrical
domains: two approaches, Appl. Math. Comput. 218, 1623–1633, 2011.
- [14] A. Kheloufi and B.K. Sadallah, Study of the heat equation in a symmetric conical type
domain of $\mathbb{R}^{N+1}$, Math. Methods Appl. Sci. 37, 1807–1818, 2014.
- [15] A. Kheloufi and B.K. Sadallah, Resolution of a high-order parabolic equation in conical
time-dependent domains of $\mathbb{R}^{3}$, Arab J. Math. Sci. 22, 165–181, 2016.
- [16] V.A. Kondrat’ev, Boundary problems for parabolic equations in closed regions, Am.
Math. Soc. Providence. R I. 450–504, 1966.
- [17] V.A. Kozlov, Coefficients in the asymptotic solutions of the Cauchy boundary-value
parabolic problems in domains with a conical point, Siberian Math. J. 29, 222–233,
1988.
- [18] R. Labbas and B.K. Sadallah, Smoothness of the solution of a fourth order parabolic
equation in a polygonal domain, Int. J. Appl. Math. 1, 75–90, 1999.
- [19] O.A. Ladyzhenskaya and V.A. Solonnikov and N.N. Ural’tseva, Linear and quasilinear
equations of parabolic type, (A.M.S., Providence, Rhode Island, 1968).
- [20] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, 1,
2, Dunod, Paris, 1968.
- [21] A. Maghnouji, Problèmes aux limites paraboliques dans un domaine non régulier,
C.R.A.S. 316, 331–336, 1993.
- [22] V.P. Mikhailov, The Dirichlet problem for a parabolic equation I, Mat. Sb. (N.S.) 61
(103), 40–64, 1963.
- [23] V.P. Mikhailov, The Dirichlet problem for a parabolic equation II, Mat. Sb. (N.S.) 62
(104), 140–159, 1963.
- [24] B.K. Sadallah, Etude d’un problème 2m-parabolique dans des domaines plan non rectangulaires,
Boll. Un. Mat. Ital. 5 (2-B), 51–112, 1983.
- [25] B.K. Sadallah, Singularities of the solution of a 2m-parabolic problem in a polygonal
domain, Arab J. Math. Sci. 4 (2), 31–41, 1998.
- [26] B.K. Sadallah, Study of a parabolic problem in a conical domain, Math. J. Okayama
Univ. 56, 157–169, 2014.