Study of a 2m-th order parabolic equation in a non-regular type of prism of ℝN+1

Arezki Kheloufi

doi:10.1515/gmj-2016-0005

Published by De Gruyter March 5, 2016

Study of a 2m-th order parabolic equation in a non-regular type of prism of ℝ^N+1

Arezki Kheloufi

From the journal Georgian Mathematical Journal

https://doi.org/10.1515/gmj-2016-0005

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Abstract

This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2m (with m∈ℕ*)

∂tu+(-1)m∑k=1N∂xk2mu=f,

defined on a type of the prism

Q=(t,x1)∈ℝ2:0<t<T,φ1(t)<x1<φ2(t)×∏i=1N-1]0,bi[

of ℝ^N+1, which is a nonstandard domain since its base shrinks at t = 0 (φ1(0)=φ2(0)). The equation is associated with Cauchy–Dirichlet boundary conditions and the right-hand side term f is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functions φ1 and φ2 under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in [19] and of the case m = 1 studied in [11].

Keywords: High-order parabolic equations; non-regular domains; anisotropic Sobolev spaces

MSC: 35K05; 35K55

The author wants to thank the anonymous referee for a careful reading of the manuscript and for his/her helpful suggestions.

References

1 E. A. Baderko, The solvability of boundary value problems for higher order parabolic equations in domains with curvilinear lateral boundaries (in Russian), Differ. Uravn. 12 (1976), 10, 1781–1792. Search in Google Scholar

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, Ph.D. thesis, Moscow, 1992. Search in Google Scholar

3 O. V. Besov, Continuation of functions from L_p^l and W_p^l (in Russian), Trudy Mat. Inst. Steklov. 89 (1967), 5–17. Search in Google Scholar

4 M. F. Cherepova, On the solvability of boundary value problems for a higher order parabolic equation with growing coefficients (in Russian), Dokl. Akad. Nauk 411 (2006), 2, 171–172; translation in Dokl. Math. 74 (2006), no. 3, 819–820. 10.1134/S1064562406060093Search in Google Scholar

5 M. F. Cherepova, Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces (in Russian), Differ. Uravn. 49 (2013), 1, 79–87; translation in Differ. Equ. 49 (2013), no. 1, 79–87. 10.1134/S0012266113010084Search in Google Scholar

6 V. A. Galaktionov, On regularity of a boundary point for higher-order parabolic equations: Towards Petrovskii-type criterion by blow-up approach, NoDEA Nonlinear Differential Equations Appl. 16 (2009), 5, 597–655. 10.1007/s00030-009-0025-xSearch in Google Scholar

7 S. Hofmann and J. L. Lewis, The L_p Neumann problem for the heat equation in non-cylindrical domains, J. Funct. Anal. 220 (2005), 1, 1–54. 10.1016/j.jfa.2004.10.016Search in Google Scholar

8 A. Kheloufi, R. Labbas and B. K. Sadallah, On the resolution of a parabolic equation in a nonregular domain of ℝ³, Differ. Equ. Appl. 2 (2010), 2, 251–263. 10.7153/dea-02-17Search in Google Scholar

9 A. Kheloufi, Resolutions of parabolic equations in non-symmetric conical domains, Electron. J. Differential Equations 2012 (2012), Article ID 116. Search in Google Scholar

10 A. Kheloufi, Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of ℝ³, Appl. Math. Comput. 220 (2013), 756–769. Search in Google Scholar

11 A. Kheloufi, Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝ^N+1, Georgian Math. J. 21 (2014), 2, 199–209. 10.1515/gmj-2014-0019Search in Google Scholar

12 A. Kheloufi and B. K. Sadallah, Study of the heat equation in a symmetric conical type domain of ℝ^N+1, Math. Methods Appl. Sci. 37 (2014), 12, 1807–1818. 10.1002/mma.2936Search in Google Scholar

13 R. Labbas, A. Medeghri and B. K. Sadallah, On a parabolic equation in a triangular domain, Appl. Math. Comput. 130 (2002), 2–3, 511–523. 10.1016/S0096-3003(01)00113-8Search in Google Scholar

14 R. Labbas, A. Medeghri and B. K. Sadallah, An L_p-approach for the study of degenerate parabolic equations, Electron. J. Differential Equations 2005 (2005), Article ID 36. Search in Google Scholar

15 O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar

16 J.-L. Lions and E. Magenes, Problémes aux limites non homogènes et applications, Vols. 1–2, Travaux et Recherches Mathématiques 17–18, Dunod, Paris, 1968. Search in Google Scholar

17 V. P. Mikhailov, The Dirichlet problem for a parabolic equation. I (in Russian), Mat. Sb. 61 (1963), 40–64. Search in Google Scholar

18 V. P. Mikhailov, The Dirichlet problem for a parabolic equation. II (in Russian), Mat. Sb. 62 (1963), 140–159. Search in Google Scholar

19 B. K. Sadallah, Étude d'un problème 2m-parabolique dans des domaines plans non rectangulaires, Boll. Unione Mat. Ital. B (6) 2 (1983), 1, 51–112. Search in Google Scholar

20 G. Savaré, Parabolic problems with mixed variable lateral conditions: An abstract approach, J. Math. Pures Appl. (9) 76 (1997), 4, 321–351. 10.1016/S0021-7824(97)89955-2Search in Google Scholar

Received: 2014-4-13

Revised: 2014-11-13

Accepted: 2014-11-28

Published Online: 2016-3-5

Published in Print: 2016-6-1

Study of a 2m-th order parabolic equation in a non-regular type of prism of ℝ^N+1

Abstract

References

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