Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

Request Permissions   Purchase Content 
 

 

Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero


Authors: Yu. A. Alkhutov and V. N. Denisov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 75 (2014), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2014, 233-258
MSC (2010): Primary 35K10
DOI: https://doi.org/10.1090/S0077-1554-2014-00233-8
Published electronically: November 5, 2014
MathSciNet review: 3308611
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the first boundary value problem in a cylindrical domain for a uniformly parabolic second-order equation in nondivergence form. The solution satisfies the homogeneous Dirichlet condition on the lateral surface of the cylinder, and the initial function is bounded. We show that if the coefficients of the equation satisfy the local and global Dini conditions, then a necessary and sufficient condition for the stabilization of the solution to zero coincides with a similar condition for the heat equation.


References [Enhancements On Off] (What's this?)

  • 1. A. M. Il′in, A. S. Kalašnikov, and O. A. Oleĭnik, Second-order linear equations of parabolic type, Uspehi Mat. Nauk 17 (1962), no. 3 (105), 3–146 (Russian). MR 0138888
  • 2. A. K. Gushchin, Dependence of the behavior, for large time values, of a solution of a parabolic equation on the data of the problem, Problems in the qualitative theory of differential equations (Russian) (Irkutsk, 1986) “Nauka” Sibirsk. Otdel., Novosibirsk, 1988, pp. 72–82, 282 (Russian). MR 991152
  • 3. V. N. Denisov, On the behavior of solutions of parabolic equations for large time values, Uspekhi Mat. Nauk 60 (2005), no. 4(364), 145–212 (Russian, with Russian summary); English transl., Russian Math. Surveys 60 (2005), no. 4, 721–790. MR 2190927, https://doi.org/10.1070/RM2005v060n04ABEH003675
  • 4. V. F. Gilimshina and F. Kh. Mukminov, On the decay of a solution of a nonuniformly elliptic equation, Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 1, 53–70 (Russian, with Russian summary); English transl., Izv. Math. 75 (2011), no. 1, 53–71. MR 2815995, https://doi.org/10.1070/IM2011v075n01ABEH002527
  • 5. V. V. Žikov, The stabilization of the solutions of parabolic equations, Mat. Sb. (N.S.) 104(146) (1977), no. 4, 597–616, 663 (Russian). MR 0473524
  • 6. S. Kamin, On stabilisation of solutions of the Cauchy problem for parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 1, 43–53. MR 0440200, https://doi.org/10.1017/S0308210500019478
  • 7. M. D. Surnachev and V. V. Zhikov, Stabilization of solutions to nonlinear parabolic equations of the 𝑝-Laplace type, Russ. J. Math. Phys. 20 (2013), no. 4, 523–541. MR 3144431, https://doi.org/10.1134/S1061920813040122
  • 8. Ju. N. Čeremnyh, The behavior of solutions of boundary value problems for second-order parabolic equations when 𝑡 increases indefinitely, Mat. Sb. (N.S.) 75 (117) (1968), 241–254 (Russian). MR 0223715
  • 9. A. M. Il′in, A sufficient condition for stabilization of the solution of a parabolic equation, Mat. Zametki 37 (1985), no. 6, 851–856, 941 (Russian). MR 802428
  • 10. A. K. Gushchin, A criterion for uniform stabilization of solutions of the second mixed problem for a parabolic equation, Dokl. Akad. Nauk SSSR 264 (1982), no. 5, 1041–1045 (Russian). MR 672010
    A. K. Gushchin, Uniform stabilization of solutions of the second mixed problem for a parabolic equation, Mat. Sb. (N.S.) 119(161) (1982), no. 4, 451–508, 590 (Russian). MR 682495
  • 11. F. Kh. Mukminov, Uniform stabilization of solutions of the first mixed problem for a parabolic equation, Mat. Sb. 181 (1990), no. 11, 1486–1509 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 2, 331–353. MR 1090912
  • 12. V. N. Denisov, Necessary and sufficient conditions for the stabilization of the solution of the Dirichlet problem for the heat equation, Dokl. Akad. Nauk 407 (2006), no. 2, 163–166 (Russian). MR 2348310
  • 13. V. N. Denisov, Necessary and sufficient conditions for the stabilization of the solution to the first boundary value problem for a parabolic equation, Tr. Semin. im. I. G. Petrovskogo 29 (2013), 248-280.
  • 14. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • 15. V. N. Denisov, Conditions for stabilization of solutions to the first boundary value problem for parabolic equations, J. Math. Sci. (N.Y.) 176 (2011), no. 6, 870–890. Problems in mathematical analysis. No. 58. MR 2838981, https://doi.org/10.1007/s10958-011-0442-3
  • 16. Yu. A. Alkhutov and V. N. Denisov, Necessary and sufficient condition for the stabilization of the solution to the initial-boundary value problem for second-order nondivergence parabolic equations, Dokl. Akad. Nauk 451 (2013), no. 1, 7-10; English transl., Dokl. Math. 88 (2013), no. 1, 381-384.
  • 17. D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694. MR 0435594
  • 18. S. D. Èĭdel′man, Parabolic equations, Current problems in mathematics. Fundamental directions, Vol. 63 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 201–313 (Russian). MR 1036530
  • 19. E. M. Landis, \cyr Uravneniya vtorogo poryadka èllipticheskogo i parabolicheskogo tipov., Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0320507
  • 20. Ermanno Lanconelli, Sul problema di Dirichlet per equazioni paraboliche del secondo ordine a coefficienti discontinui, Ann. Mat. Pura Appl. (4) 106 (1975), 11–38. MR 0399659, https://doi.org/10.1007/BF02415021
  • 21. M. D. Ivanovič, On the nature of continuity of solutions of linear parabolic equations of the second order, Vestnik Moskov. Univ. Ser. I Mat. Meh. 21 (1966), no. 4, 31–41 (Russian, with English summary). MR 0204836

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 35K10

Retrieve articles in all journals with MSC (2010): 35K10


Additional Information

Yu. A. Alkhutov
Affiliation: A. G. and N. G. Stoletov Vladimir State University, Vladimir, Russia
Email: yurij-alkhutov@yandex.ru

V. N. Denisov
Affiliation: Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia
Email: vdenisov2008@yandex.ru

DOI: https://doi.org/10.1090/S0077-1554-2014-00233-8
Keywords: Parabolic equation, initial--boundary value problem, Dini condition, stabilization, Wiener capacity, heat capacity
Published electronically: November 5, 2014
Additional Notes: This work was supported by RBFR grant no. 12-01-00058-a, grant no. NSh-3685.2014.1 for support of leading scientific schools, and by a government assignment from the Ministry of Education and Science of the Russian Federation (assignment no. 2014/13, project code 3037). The results in Sections \ref{sec:4} and \ref{sec:5} were obtained with support of the Russian Scientific Foundation (project no. 14-11-00398).
Article copyright: © Copyright 2014 American Mathematical Society