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Mat. Sb., 2000, Volume 191, Number 2, Pages 91–131 (Mi msb454)  

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

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

L. M. Kozhevnikova, F. Kh. Mukminov

Sterlitamak State Pedagogical Institute

Abstract: A quasilinear system of parabolic equations with energy inequality is considered in a cylindrical domain $\{t>0\}\times\Omega$. In a broad class of unbounded domains $\Omega$ two geometric characteristics of a domain are identified which determine the rate of convergence to zero as $t\to\infty$ of the $L_2$-norm of a solution. Under additional assumptions on the coefficients of the quasilinear system estimates of the derivatives and uniform estimates of the solution are obtained; they are proved to be best possible in the order of convergence to zero in the case of one semilinear equation.


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English version:
Sbornik: Mathematics, 2000, 191:2, 235–273

Bibliographic databases:

UDC: 517.95
MSC: 35B35, 35K20, 35K55
Received: 19.04.1999

Citation: 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”, Mat. Sb., 191:2 (2000), 91–131; Sb. Math., 191:2 (2000), 235–273

Citation in format AMSBIB
\by L.~M.~Kozhevnikova, F.~Kh.~Mukminov
\paper 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
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 2
\pages 91--131
\jour Sb. Math.
\yr 2000
\vol 191
\issue 2
\pages 235--273

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    This publication is cited in the following articles:
    1. F. Kh. Mukminov, I. M. Bikkulov, “Stabilization of the norm of the solution of a mixed problem in an unbounded domain for parabolic equations of orders 4 and 6”, Sb. Math., 195:3 (2004), 413–440  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. 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
    3. L. M. Kozhevnikova, F. Kh. Mukminov, “Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms”, J. Math. Sci., 150:5 (2008), 2369–2383  mathnet  crossref  mathscinet  zmath  elib
    4. 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
    5. A. R. Gerfanov, F. Kh. Mukminov, “Shirokii klass edinstvennosti resheniya dlya neravnomerno ellipticheskogo uravneniya v neogranichennoi oblasti”, Ufimsk. matem. zhurn., 1:3 (2009), 11–27  mathnet  zmath  elib
    6. L. M. Kozhevnikova, “Stabilization of solutions of pseudo-differential parabolic equations in unbounded domains”, Izv. Math., 74:2 (2010), 325–345  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. R. Kh. Karimov, L. M. Kozhevnikova, “Stabilization of solutions of quasilinear second order parabolic equations in domains with non-compact boundaries”, Sb. Math., 201:9 (2010), 1249–1271  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. V. F. Gilimshina, F. Kh. Mukminov, “Ob ubyvanii resheniya vyrozhdayuschegosya lineinogo parabolicheskogo uravneniya”, Ufimsk. matem. zhurn., 3:4 (2011), 43–56  mathnet  zmath
    9. L. M. Kozhevnikova, A. A. Leontev, “Otsenki resheniya anizotropnogo parabolicheskogo uravneniya s dvoinoi nelineinostyu”, Ufimsk. matem. zhurn., 3:4 (2011), 64–85  mathnet  zmath
    10. L. M. Kozhevnikova, F. Kh. Mukminov, “Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains”, Proc. Steklov Inst. Math., 278 (2012), 106–120  mathnet  crossref  mathscinet  isi
    11. L. M. Kozhevnikova, A. A. Leontev, “Resheniya anizotropnykh parabolicheskikh uravnenii s dvoinoi nelineinostyu v neogranichennykh oblastyakh”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 82–89  mathnet  crossref
    12. L. M. Kozhevnikova, A. A. Leontiev, “Decay of solution of anisotropic doubly nonlinear parabolic equation in unbounded domains”, Ufa Math. J., 5:1 (2013), 63–82  mathnet  crossref  mathscinet  elib
    13. L. M. Kozhevnikova, A. A. Leont'ev, “Solutions to higher-order anisotropic parabolic equations in unbounded domains”, Sb. Math., 205:1 (2014), 7–44  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. 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
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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