Solvability of a boundary-value problem for degenerate equations

  • T. Gadjiev Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku
  • M. Kerimova Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku
  • G. Gasanova Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku
Keywords: solvability, weighted Sobolev space, elliptic-parabolic equations, degenerated

Abstract

UDC 517.9

We consider a boundary-value problem for degenerate equations with discontinuous coefficients and establish the unique strong solvability (almost everywhere) of this problem in the corresponding weighted Sobolev space.



References

Fichera, Gaetano. On a unified theory of boundary value problems for elliptic-parabolic equations of second order. 1960 Boundary problems in differential equations pp. 97--120 Univ. of Wisconsin Press, Madison http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mat&paperid=288&option_lang=eng

Alt, Hans Wilhelm; Luckhaus, Stephan. Quasilinear elliptic-parabolic differential equations. Math. Z. 183, no. 3, 311--341 (1983). https://doi.org/10.1007/BF01176474 DOI: https://doi.org/10.1007/BF01176474

Benilan, Philippe; Wittbold, Petra. On mild and weak solutions of elliptic-parabolic problems. Adv. Differential Equations. 1, no. 6, 1053--1073 (1996).

Gadjiev, T. S.; Gasimova, E. R. On smoothness of solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations. ; translated from Ukrain. Mat. Zh. 60, no. 6, 723--736 (2008) Ukrainian Math. J. 60, no. 6, 831--847 (2008), https://doi.org/10.1007/s11253-008-0095-7 DOI: https://doi.org/10.1007/s11253-008-0095-7

Gajewski, H.; Skrypnik, I. V. To the uniqueness problem for nonlinear elliptic equations. Nonlinear Anal. 52, no. 1, 291--304 (2003). https://doi.org/10.1016/S0362-546X(02)00112-8 DOI: https://doi.org/10.1016/S0362-546X(02)00112-8

Gajewski, H.; Skrypnik, I. V. On the uniqueness of solutions for nonlinear elliptic-parabolic equations. J. Evol. Equ. 3, no. 2, 247--281 (2003). https://doi.org/10.1007/978-3-0348-7924-8_14 DOI: https://doi.org/10.1007/978-3-0348-7924-8_14

Chanillo, Sagun; Wheeden, Richard L. Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Amer. J. Math. 107, no. 5, 1191--1226 (1985). https://doi.org/10.2307/2374351 DOI: https://doi.org/10.2307/2374351

Gadjiev, Tahir S.; Kerimova, Mehriban N. Coercive estimate for degenerate elliptic parabolic equations. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azera. 41, no. 1, 123--134 (2015). http://proc.imm.az/volumes/41-1/41-01-13.pdf

Bokalo, M. M.; Domansʹka, G. P. A mixed problem for linear elliptic-parabolic-pseudoparabolic equations. (Ukrainian) Mat. Stud. 40, no. 2, 193--197 (2013).

Mamedov, Ilkham T. The first boundary value problem for second-order elliptic-parabolic equations with discontinuous coefficients. (Russian) ; translated from Sovrem. Mat. Fundam. Napravl. 39 (2011), 102--129 J. Math. Sci. (N.Y.) 190 (2013), no. 1, 104--134 https://doi.org/10.1007/s10958-013-1248-2 DOI: https://doi.org/10.1007/s10958-013-1248-2

Ladyženskaja, O. A.; Solonnikov, V. A.; Uralʹceva, N. N. Линейные и квазилинейные уравнения параболического типа. (Russian) [[Linear and quasi-linear equations of parabolic type]] Izdat. ``Nauka'', Moscow 1967 736 pp. https://www.amazon.com/Linear-Quasi-linear-Equations-Parabolic-Type/dp/0821815733

Published
28.03.2020
How to Cite
Gadjiev, T., M. Kerimova, and G. Gasanova. “Solvability of a Boundary-Value Problem for Degenerate Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 4, Mar. 2020, pp. 435-51, doi:10.37863/umzh.v72i4.6000.
Section
Research articles