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Study of the First Boundary Value Problem for a Fourth Order Parabolic Equation in a Nonregular Domain of

Yıl 2015, Cilt: 5 Sayı: 1, 145 - 157, 01.06.2015

Öz

This paper is concerned with the extension of solvability results obtained for a fourth order parabolic equation, set in a nonregular domain of R 3 obtained in [1], to the case where the domain is cylindrical, not with respect to the time variable, but with respect to N space variables, N > 1. More precisely, we determine optimal conditions on the shape of the boundary of a N + 1 -dimensional domain, N > 1, under which the solution is regular

Kaynakça

  • Kheloufi, A., On a fourth order parabolic equation in a nonregular domain ofR3, Mediterr. J. Math., doi: 10.1007/s00009-014-0429-7
  • Sadallah, B. K., (1983), Etude d’un probl`eme 2m-parabolique dans des domaines plan non rectangu- laires. Boll. Un. Mat. Ital., 2-B (5), pp. 51-112.
  • Baderko, E. A., (1992), On the solution of boundary value problems for linear parabolic equations of arbitrary order in noncylindrical domains by the method of boundary integral equations, Ph D Thesis, Moscow.
  • Cherepova, M. F., (2006), On the solvability of boundary value problems for a higher order parabolic equation with growing coefficients, Doklady Mathematics, 74 (3), pp. 819-820.
  • Cherepova, M. F., (2013), Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted H¨older spaces, Differential Equations, 1 (49), pp. 79-87.
  • Galaktionov, V. A., (2009), On regularity of boundary point for higer-order parabolic equations: Towards Petrovskii-type criterion by blow-up approach, Nonlinear Differential Equations and Appli- cations, 16, pp. 597-655.
  • Baderko, E. A., (1976), On the solution of the boundary value problems for parabolic equations of high order in domains with curvilinear lateral boundaries, Diff. Urav., 12 (2), pp. 1781-1792.
  • Mikhailov, V. P., (1963), The Dirichlet problem for a parabolic equation I, Mat. Sb. (N.S.), 61 (103), pp. 40-64.
  • Savar´e, G., (1997), Parabolic problems with mixed variable lateral conditions: an abstract approach, J. Math. Pures Appl., 76, pp. 321-351.
  • Hofmann S. and Lewis J. L., The Lpregularity problems for the heat equation in noncylindrical domains, Journal of Functional Analysis, 220, pp. 1-54.
  • Labbas, R., Medeghri, A. and Sadallah, B. K., (2002), On a parabolic equation in a triangular domain. Applied Mathematics and Computation, 2002, (130), pp. 511-523
  • Labbas, R., Medeghri, A. and Sadallah, B. K., (2005), An Lpapproach for the study of degenerate parabolic equation, Elec. J. Diff. Equs., 2005 (36), pp. 1-20.
  • Kheloufi, A. and Sadallah, B. K., (2010), Parabolic equations with Robin type boundary conditions in a non-rectangular domain, Elec. J. Diff. Equs., 2010 (25), pp. 1-14.
  • Kheloufi, A., Labbas, R. and Sadallah, B. K., (2010), On the resolution of a parabolic equation in a non-regular domain ofR3, Differential Equations and Applications, 2 (2), pp. 251-263.
  • Kheloufi, A., (2012), Resolutions of parabolic equations in non-symmetric conical domains, Elec. J. Diff. Equs., 2012 (116), pp. 1-14.
  • Kheloufi, A., (2013), Existence and uniqueness results for parabolic equations with Robin type bound- ary conditions in a non-regular domain ofR, Applied Mathematics and Computation, 220, pp. 756- 769.
  • Kheloufi, A. and Sadallah, B. K., (2014), Study of the heat equation in a symmetric conical type domain ofRN +1, Mathematical Methods in the Applied Sciences, 37, pp. 1807-1818.
  • Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural’tseva, N. N., (1968), Linear and Quasi-Linear Equations of Parabolic Type, A.M.S., providence, Rhode Island.
  • Besov, V., (1967), The Continuation of Function in Lpand Wp. Proc. Steklov Inst. Math.,89, pp. 5-17.
  • Lions, J. L. and Magenes, E., (1968), Probl`emes aux Limites Non Homog`enes et Applications, V ol.1, 2, Dunod, Paris.
Yıl 2015, Cilt: 5 Sayı: 1, 145 - 157, 01.06.2015

Öz

Kaynakça

  • Kheloufi, A., On a fourth order parabolic equation in a nonregular domain ofR3, Mediterr. J. Math., doi: 10.1007/s00009-014-0429-7
  • Sadallah, B. K., (1983), Etude d’un probl`eme 2m-parabolique dans des domaines plan non rectangu- laires. Boll. Un. Mat. Ital., 2-B (5), pp. 51-112.
  • Baderko, E. A., (1992), On the solution of boundary value problems for linear parabolic equations of arbitrary order in noncylindrical domains by the method of boundary integral equations, Ph D Thesis, Moscow.
  • Cherepova, M. F., (2006), On the solvability of boundary value problems for a higher order parabolic equation with growing coefficients, Doklady Mathematics, 74 (3), pp. 819-820.
  • Cherepova, M. F., (2013), Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted H¨older spaces, Differential Equations, 1 (49), pp. 79-87.
  • Galaktionov, V. A., (2009), On regularity of boundary point for higer-order parabolic equations: Towards Petrovskii-type criterion by blow-up approach, Nonlinear Differential Equations and Appli- cations, 16, pp. 597-655.
  • Baderko, E. A., (1976), On the solution of the boundary value problems for parabolic equations of high order in domains with curvilinear lateral boundaries, Diff. Urav., 12 (2), pp. 1781-1792.
  • Mikhailov, V. P., (1963), The Dirichlet problem for a parabolic equation I, Mat. Sb. (N.S.), 61 (103), pp. 40-64.
  • Savar´e, G., (1997), Parabolic problems with mixed variable lateral conditions: an abstract approach, J. Math. Pures Appl., 76, pp. 321-351.
  • Hofmann S. and Lewis J. L., The Lpregularity problems for the heat equation in noncylindrical domains, Journal of Functional Analysis, 220, pp. 1-54.
  • Labbas, R., Medeghri, A. and Sadallah, B. K., (2002), On a parabolic equation in a triangular domain. Applied Mathematics and Computation, 2002, (130), pp. 511-523
  • Labbas, R., Medeghri, A. and Sadallah, B. K., (2005), An Lpapproach for the study of degenerate parabolic equation, Elec. J. Diff. Equs., 2005 (36), pp. 1-20.
  • Kheloufi, A. and Sadallah, B. K., (2010), Parabolic equations with Robin type boundary conditions in a non-rectangular domain, Elec. J. Diff. Equs., 2010 (25), pp. 1-14.
  • Kheloufi, A., Labbas, R. and Sadallah, B. K., (2010), On the resolution of a parabolic equation in a non-regular domain ofR3, Differential Equations and Applications, 2 (2), pp. 251-263.
  • Kheloufi, A., (2012), Resolutions of parabolic equations in non-symmetric conical domains, Elec. J. Diff. Equs., 2012 (116), pp. 1-14.
  • Kheloufi, A., (2013), Existence and uniqueness results for parabolic equations with Robin type bound- ary conditions in a non-regular domain ofR, Applied Mathematics and Computation, 220, pp. 756- 769.
  • Kheloufi, A. and Sadallah, B. K., (2014), Study of the heat equation in a symmetric conical type domain ofRN +1, Mathematical Methods in the Applied Sciences, 37, pp. 1807-1818.
  • Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural’tseva, N. N., (1968), Linear and Quasi-Linear Equations of Parabolic Type, A.M.S., providence, Rhode Island.
  • Besov, V., (1967), The Continuation of Function in Lpand Wp. Proc. Steklov Inst. Math.,89, pp. 5-17.
  • Lions, J. L. and Magenes, E., (1968), Probl`emes aux Limites Non Homog`enes et Applications, V ol.1, 2, Dunod, Paris.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

A. Kheloufi Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 5 Sayı: 1

Kaynak Göster