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

Alkhutov, Yu.; Denisov, V.

doi:10.1090/S0077-1554-2014-00233-8

Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero
HTML articles powered by AMS MathViewer

by Yu. A. Alkhutov and V. N. Denisov
Translated by: V. E. Nazaikinskii

Trans. Moscow Math. Soc. 2014, 233-258

DOI: https://doi.org/10.1090/S0077-1554-2014-00233-8

Published electronically: November 5, 2014

PDF

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

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
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
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, DOI 10.1070/RM2005v060n04ABEH003675
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, DOI 10.1070/IM2011v075n01ABEH002527
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
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 440200, DOI 10.1017/S0308210500019478
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, DOI 10.1134/S1061920813040122
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
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
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
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, DOI 10.1070/SM1992v071n02ABEH002130
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
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.
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
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, DOI 10.1007/s10958-011-0442-3
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.
D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
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
E. M. Landis, Uravneniya vtorogo poryadka èllipticheskogo i parabolicheskogo tipov, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0320507
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 399659, DOI 10.1007/BF02415021
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