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

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:

UDC: 517.945.9
MSC: Primary 35K20, 35B40; Secondary 35C10

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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