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Mat. Sb., 1990, Volume 181, Number 11, Pages 1486–1509 (Mi msb1241)  

This article is cited in 14 scientific papers (total in 14 papers)

On uniform stabilization of solutions of the first mixed problem for a parabolic equation

F. Kh. Mukminov

Abstract: The first mixed problem with a homogeneous boundary condition is considered for a linear parabolic equation of second order. It is assumed that the unbounded domain $\Omega$ satisfies the following condition: there exists a positive constant $\theta$ such that for any point $x$ of the boundary $\partial\Omega$
$$ \operatorname{mes}(\{y\colon|x-y|<r\}\setminus\Omega)\geqslant\theta r^n, \quad r>0. $$
For a certain class of initial functions $\varphi$, which includes all bounded functions, the following condition is a necessary and sufficient condition for uniform stabilization of the solution to zero: $\displaystyle r^{-n}\int_{|x-y|<r}\varphi (y) dy\to0$ as $r\to\infty$ uniformly with respect to all $x$ in $\Omega$ such that $\operatorname{dist}(x,\partial\Omega)\geqslant r+1$.
The proof of the stabilization condition is based on an estimate of the Green function that takes account of its decay near the boundary.

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English version:
Mathematics of the USSR-Sbornik, 1992, 71:2, 331–353

Bibliographic databases:

UDC: 517.9
MSC: 35K20, 35B40
Received: 10.05.1990

Citation: F. Kh. Mukminov, “On uniform stabilization of solutions of the first mixed problem for a parabolic equation”, Mat. Sb., 181:11 (1990), 1486–1509; Math. USSR-Sb., 71:2 (1992), 331–353

Citation in format AMSBIB
\by F.~Kh.~Mukminov
\paper On uniform stabilization of solutions of the first mixed problem for a~parabolic equation
\jour Mat. Sb.
\yr 1990
\vol 181
\issue 11
\pages 1486--1509
\jour Math. USSR-Sb.
\yr 1992
\vol 71
\issue 2
\pages 331--353

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    This publication is cited in the following articles:
    1. Mukminov F., “On Uniform Stabilization of Solution to the Mixed Problem for the Navier–Stokes Equations in Exterior Domain”, Dokl. Akad. Nauk, 332:1 (1993), 24–25  mathnet  mathscinet  zmath  isi
    2. F. Kh. Mukminov, “On uniform stabilization of solutions of the exterior problem for the Navier–Stokes equations”, Russian Acad. Sci. Sb. Math., 81:2 (1995), 297–320  mathnet  crossref  mathscinet  zmath  isi
    3. L. M. Kozhevnikova, F. Kh. Mukminov, “Estimates of the stabilization rate as $t\to\infty$ of solutions of the first mixed problem for a quasilinear system of second-order parabolic equations”, Sb. Math., 191:2 (2000), 235–273  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. L. M. Kozhevnikova, “On uniqueness classes of solutions of the first mixed problem for a quasi-linear second-order parabolic system in an unbounded domain”, Izv. Math., 65:3 (2001), 469–484  mathnet  crossref  crossref  mathscinet  zmath
    5. V. N. Denisov, “On the behaviour of solutions of parabolic equations for large values of time”, Russian Math. Surveys, 60:4 (2005), 721–790  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. L. M. Kozhevnikova, “Uniqueness classes for solutions in unbounded domains of the first mixed problem for the equation $u_t=Au$ with quasi-elliptic operator $A$”, Sb. Math., 198:1 (2007), 55–96  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. V. N. Denisov, “Conditions for stabilization of solutions to the first boundary value problem for parabolic equations”, J Math Sci, 2011  crossref
    8. L. M. Kozhevnikova, “Examples of the Nonuniqueness of Solutions of the Mixed Problem for the Heat Equation in Unbounded Domains”, Math. Notes, 91:1 (2012), 58–64  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. V. F. Vil'danova, F. Kh. Mukminov, “Anisotropic uniqueness classes for a degenerate parabolic equation”, Sb. Math., 204:11 (2013), 1584–1597  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Alkhutov Yu.A., Denisov V.N., “Necessary and Sufficient Condition for the Stabilization of the Solution to the Initial-Boundary Value Problem for Second-Order Nondivergence Parabolic Equations”, Dokl. Math., 88:1 (2013), 381–384  crossref  isi
    11. V. N. Denisov, “Necessary and sufficient conditions of stabilization of solutions of the first boundary-value problem for a parabolic equation”, J. Math. Sci. (N. Y.), 197:3 (2014), 303–324  mathnet  crossref  elib
    12. Daniele Andreucci, A.F.. Tedeev, “The Cauchy–Dirichlet Problem for the Porous Media Equation in Cone-Like Domains”, SIAM J. Math. Anal, 46:2 (2014), 1427  crossref
    13. Yu. A. Alkhutov, V. N. Denisov, “Necessary and sufficient condition for the stabilization of the solution of a mixed problem for nondivergence parabolic equations to zero”, Trans. Moscow Math. Soc., 75 (2014), 233–258  mathnet  crossref  elib
    14. V. F. Vil'danova, F. Kh. Mukminov, “Täcklind uniqueness classes for heat equation on noncompact Riemannian manifolds”, Ufa Math. J., 7:2 (2015), 55–63  mathnet  crossref  isi  elib
  • Математический сборник - 1989–1990 Sbornik: Mathematics (from 1967)
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