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Mat. Sb. (N.S.), 1982, Volume 119(161), Number 4(12), Pages 451–508 (Mi msb2896)  

This article is cited in 55 scientific papers (total in 55 papers)

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

A. K. Gushchin


Abstract: A number of properties of the Green function of the second mixed problem for a parabolic equation of second order on $(t>0)\times\Omega$ ($\Omega$ an arbitrary domain in $\mathbf R_n$) are established. By means of these results a criterion is proved for the uniform stabilization of a solution: the existence of a uniform limit of the spherical mean of the initial function (extended by zero outside $\Omega$) is necessary and sufficient for the uniform stabilization of a solution of the problem considered, with a bounded initial function under a certain condition on the unbounded domain $\Omega$.
The basic properties of the Green function are obtained on the basis of an estimate of the solution of the problem with a compactly supported initial function in terms of the norm of the initial function in $L_1(\Omega)$.
Bibliography: 53 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 47:2, 439–498

Bibliographic databases:

Document Type: Article
UDC: 517.945.9
MSC: Primary 35K20, 35B40; Secondary 35C10
Received: 22.04.1982

Citation: A. K. Gushchin, “On the uniform stabilization of solutions of the second mixed problem for a parabolic equation”, Mat. Sb. (N.S.), 119(161):4(12) (1982), 451–508; Math. USSR-Sb., 47:2 (1984), 439–498

Citation in format AMSBIB
\Bibitem{Gus82}
\by A.~K.~Gushchin
\paper On the uniform stabilization of solutions of the second mixed problem for a parabolic equation
\jour Mat. Sb. (N.S.)
\yr 1982
\vol 119(161)
\issue 4(12)
\pages 451--508
\mathnet{http://mi.mathnet.ru/msb2896}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=682495}
\zmath{https://zbmath.org/?q=an:0554.35055}
\transl
\jour Math. USSR-Sb.
\yr 1984
\vol 47
\issue 2
\pages 439--498
\crossref{https://doi.org/10.1070/SM1984v047n02ABEH002654}


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    This publication is cited in the following articles:
    1. Landis E., “The Dependence of Classes of Uniqueness of the Solution to 2nd Initial Boundary-Value-Problems for the Heat-Conductivity Equation in an Unbounded Domain on the Geometry of the Domain”, 275, no. 4, 1984, 790–793  mathscinet  zmath  isi
    2. A. K. Gushchin, V. P. Mikhailov, Yu. A. Mikhailov, “On uniform stabilization of the solution of the second mixed problem for a second order parabolic equation”, Math. USSR-Sb., 56:1 (1987), 141–162  mathnet  crossref  mathscinet  zmath
    3. Gushchin A. Mikhailov V. Mikhailov I., “Estimates of Green-Function and the Criterion of the Uniform Stabilization of the 2nd Mixed Problem Solution for a 2nd-Order Parabolic Equation”, 282, no. 1, 1985, 15–18  mathscinet  zmath  isi
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    5. Yu. A. Mikhailov, “On uniform quasistabilization of solutions of the second mixed problem for a second-order hyperbolic equation”, Math. USSR-Sb., 57:1 (1987), 243–262  mathnet  crossref  mathscinet  zmath
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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