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Mat. Sb. (N.S.), 1986, Volume 129(171), Number 2, Pages 186–200 (Mi msb1815)  

This article is cited in 13 scientific papers (total in 13 papers)

On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation

A. V. Lezhnev

Abstract: The author studies the behavior, for large time values $t$, of a nonnegative solution of the second mixed problem for a uniformly parabolic equation
$$ \frac{\partial u(x,t)}{\partial t}=\sum_{i,j=1}^n\frac\partial{\partial x_i}(a_{ij}(x,t)\frac{\partial u(x,t)}{\partial x_j}) $$
in a cylindrical domain $\Omega\times\{t>0\}$, where $\Omega$ is an unbounded domain in $\mathbf R^n$. It is shown that for a certain class of unbounded domains $\Omega$, the behavior of the solution of the problem as $t\to\infty$ is determined by the behavior, for large values of the parameter $R$, of the means of the initial function over the sets $\{x\in\Omega:|x-\xi|<R\}$, $\xi\in\Omega$, $R>0$.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 57:1, 195–209

Bibliographic databases:

UDC: 517.9
MSC: 35K20, 35B40
Received: 24.04.1985

Citation: A. V. Lezhnev, “On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a parabolic equation”, Mat. Sb. (N.S.), 129(171):2 (1986), 186–200; Math. USSR-Sb., 57:1 (1987), 195–209

Citation in format AMSBIB
\by A.~V.~Lezhnev
\paper On the behavior, for large time values, of nonnegative solutions of the second mixed problem for a~parabolic equation
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 129(171)
\issue 2
\pages 186--200
\jour Math. USSR-Sb.
\yr 1987
\vol 57
\issue 1
\pages 195--209

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    This publication is cited in the following articles:
    1. A. K. Gushchin, V. P. Mikhailov, “On uniform quasiasymptotics of solutions of the second mixed problem for a hyperbolic equation”, Math. USSR-Sb., 59:2 (1988), 409–427  mathnet  crossref  mathscinet  zmath
    2. Lezhnev A., “Bounds of Green-Function and Solutions of a 2nd Mixed Problem for a Parabolic Equation”, Differ. Equ., 25:4 (1989), 478–486  mathnet  mathscinet  zmath  isi
    3. Valikov K., “Generalization of the Concept of Uniform Stabilization and Uniform Proximity of Solutions of a Cauchy-Problem”, Differ. Equ., 26:2 (1990), 213–220  mathnet  mathscinet  zmath  isi
    4. Tedeev A., “Estimation of the Stabilization Rate for T -] Infinity of a Solution of a 2nd Mixed Problem for a 2nd-Order Quasi-Linear Parabolic Equation”, Differ. Equ., 27:10 (1991), 1274–1283  mathnet  mathscinet  zmath  isi
    5. Andreucci D. Tedeev A., “Optimal Bounds and Blow Up Phenomena for Parabolic Problems in Narrowing Domains”, Proc. R. Soc. Edinb. Sect. A-Math., 128:Part 6 (1998), 1163–1180  crossref  mathscinet  zmath  isi
    6. 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
    7. 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
    8. L. M. Kozhevnikova, “Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation”, Sb. Math., 196:7 (2005), 999–1032  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. N. A. Kul'sarina, V. F. Gilimshina, “Exact estimate for the rate of decrease of a solution to a parabolic equation of the 2nd kind for $t\to\infty$”, Russian Math. (Iz. VUZ), 51:4 (2007), 32–41  mathnet  crossref  mathscinet  zmath
    10. Gilimshina V.F., “On the decay of a solution of a nonuniformly parabolic equation”, Differential Equations, 46:2 (2010), 239–254  crossref  isi  elib
    11. A. V. Lezhnëv, “Ob odnoi teoreme vlozheniya dlya funktsii s summiruemym gradientom”, Trudy sedmoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem (3–6 iyunya 2010 g.). Chast 3, Differentsialnye uravneniya i kraevye zadachi, Matem. modelirovanie i kraev. zadachi, Samarskii gosudarstvennyi tekhnicheskii universitet, Samara, 2010, 149–152  mathnet
    12. V. F. Gilimshina, F. Kh. Mukminov, “Ob ubyvanii resheniya vyrozhdayuschegosya lineinogo parabolicheskogo uravneniya”, Ufimsk. matem. zhurn., 3:4 (2011), 43–56  mathnet  zmath
    13. V. F. Vil'danova, “On decay of solution to linear parabolic equation with double degeneracy”, Ufa Math. J., 8:1 (2016), 35–50  mathnet  crossref  elib
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