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Mat. Sb., 2005, Volume 196, Number 7, Pages 67–100 (Mi msb1377)  

This article is cited in 15 scientific papers (total in 15 papers)

Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation

L. M. Kozhevnikova

Sterlitamak State Pedagogical Institute

Abstract: In a cylindrical domain $D=(0,\infty)\times\Omega$, where $\Omega$ is an unbounded subdomain of $\mathbb R_{n+1}$, one considers the first mixed problem for a higher order equation
\begin{gather*} u_t+Lu=0,
Lu\equiv\sum_{i=q}^k(-1)^iD_x^i(a_i(x,{\mathbf y})D_x^iu)+ \sum_{i=l}^m \sum_{|\alpha|=|\beta|=i}(-1)^i D_{\mathbf y}^\alpha(b_{\alpha\beta}(x,{\mathbf y})D_{\mathbf y}^\beta u),
q\leqslant k,\quad l\leqslant m,\quad q,k,l,m\in\mathbb N,\quad x\in\mathbb R,\quad \mathbf y\in\mathbb R_n, \end{gather*}
with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an upper estimate of the $L_2$-norm $\|u(t)\|$ of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains $\{(x,{\mathbf y})\in\mathbb R_{n+1}:|y_1|<x^a\}$, $0<a<q/l$, for the operator $L$ with symbol satisfying a certain condition this estimate takes the following form:
$$ \|u(t)\|\leqslant M\exp(-\kappa_2t^b)\|\varphi\|,\qquad b=\frac{q-{la}}{q-{la}+2laq} . $$
The estimate is shown to be sharp in a broad class of unbounded domains for $q=k=l=m=1$, that is, for second-order parabolic equations.


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English version:
Sbornik: Mathematics, 2005, 196:7, 999–1032

Bibliographic databases:

UDC: 517.956.4
MSC: 35K35, 35B35, 35B40
Received: 25.10.2004

Citation: L. M. Kozhevnikova, “Stabilization of a solution of the first mixed problem for a quasi-elliptic evolution equation”, Mat. Sb., 196:7 (2005), 67–100; Sb. Math., 196:7 (2005), 999–1032

Citation in format AMSBIB
\by L.~M.~Kozhevnikova
\paper Stabilization of a~solution of the first mixed problem for a~quasi-elliptic evolution equation
\jour Mat. Sb.
\yr 2005
\vol 196
\issue 7
\pages 67--100
\jour Sb. Math.
\yr 2005
\vol 196
\issue 7
\pages 999--1032

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    This publication is cited in the following articles:
    1. L. M. Kozhevnikova, “Anisotropic classes of uniqueness of the solution of the Dirichlet problem for quasi-elliptic equations”, Izv. Math., 70:6 (2006), 1165–1200  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. L. M. Kozhevnikova, “Uniqueness classes for solutions in unbounded domains of the first mixed problem for the equation $u_t=Au$ with quasi-elliptic operator $A$”, Sb. Math., 198:1 (2007), 55–96  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. L. M. Kozhevnikova, “Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains”, Sb. Math., 199:8 (2008), 1169–1200  mathnet  crossref  crossref  mathscinet  isi  elib
    4. L. M. Kozhevnikova, “O suschestvovanii i edinstvennosti reshenii zadachi Dirikhle dlya psevdodifferentsialnykh ellipticheskikh uravnenii v oblastyakh s nekompaktnymi granitsami”, Ufimsk. matem. zhurn., 1:1 (2009), 38–68  mathnet  zmath  elib
    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, “On the decay of a solution of a nonuniformly parabolic equation”, Differential Equations, 46:2 (2010), 239–254  crossref  mathscinet  zmath  isi  elib  elib
    9. V. F. Gilimshina, F. Kh. Mukminov, “On the decay of solutions of non-uniformly elliptic equations”, Izv. Math., 75:1 (2011), 53–71  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. L. M. Kozhevnikova, A. A. Leontev, “Otsenki resheniya anizotropnogo parabolicheskogo uravneniya s dvoinoi nelineinostyu”, Ufimsk. matem. zhurn., 3:4 (2011), 64–85  mathnet  zmath
    11. 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  elib  elib
    12. 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
    13. 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
    14. 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
    15. 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  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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