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Mathematics of the USSR-Sbornik, 1976, Volume 30, Issue 4, Pages 403–440
DOI: https://doi.org/10.1070/SM1976v030n04ABEH002281
(Mi sm2926)
 

This article is cited in 27 scientific papers (total in 27 papers)

Stabilization of the solutions of the second boundary value problem for a second order parabolic equation

A. K. Gushchin
References:
Abstract: The paper is a continuation of work (RZhMat., 1973, 10B301) in which in the case of a “noncontracting” unbounded domain $\Omega$ there is distinguished a geometric characteristic $v(R)=\operatorname{mes}(\Omega\cap\{|x|<R\})$ of the domain $\Omega$ that determines (under the fulfillment of a certain condition of “regularity” of the domain) the rate of stabilization for $t\to\infty$ of the solution in $(t>0)\times\Omega$ of the following second boundary value problem for a parabolic equation:
$$ u_t=\sum_{i,j=1}^n\bigl(a_{i,j}(t,x)u_{x_i}\bigr)_{x_j},\qquad\frac{\partial u}{\partial N}\Bigr|_{x\in\partial\Omega}=0,\quad u|_{t=0}=\varphi(x) $$
in which the initial function $\varphi(x)$ decreases sufficiently rapidly as $|x|\to\infty$. It is proved in the present paper that the same characteristic also determines the rate of stabilization of the solution in a class of “contracting” ($\lim_{R\to\infty}v(R)/R=0$) domains $\Omega$. In this case, as in the case of a “noncontracting” domain, $\|u(t,x)\|_{L_\infty(\Omega)}$ tends to zero as $t\to\infty$ like $1/v(\sqrt{t})$: there exist estimates of the function $\|u(t,x)\|_{L_\infty(\Omega)}$ from above and from below having such an order of decrease.
Bibliography: 11 titles.
Received: 09.03.1976
Bibliographic databases:
Document Type: Article
UDC: 517.945.9
MSC: 35K20, 35B40
Language: English
Original paper language: Russian
Citation: A. K. Gushchin, “Stabilization of the solutions of the second boundary value problem for a second order parabolic equation”, Math. USSR-Sb., 30:4 (1976), 403–440
Citation in format AMSBIB
\Bibitem{Gus76}
\by A.~K.~Gushchin
\paper Stabilization of the solutions of the second boundary value problem for a~second order parabolic equation
\jour Math. USSR-Sb.
\yr 1976
\vol 30
\issue 4
\pages 403--440
\mathnet{http://mi.mathnet.ru//eng/sm2926}
\crossref{https://doi.org/10.1070/SM1976v030n04ABEH002281}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=487024}
\zmath{https://zbmath.org/?q=an:0339.35011}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1976FU24400001}
Linking options:
  • https://www.mathnet.ru/eng/sm2926
  • https://doi.org/10.1070/SM1976v030n04ABEH002281
  • https://www.mathnet.ru/eng/sm/v143/i4/p459
  • This publication is cited in the following 27 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:539
    Russian version PDF:155
    English version PDF:32
    References:80
    First page:3
     
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