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Fundam. Prikl. Mat., 2006, Volume 12, Issue 4, Pages 113–132 (Mi fpm962)  

This article is cited in 4 scientific papers (total in 4 papers)

Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms

L. M. Kozhevnikovaa, F. Kh. Mukminovb

a Sterlitamak State Pedagogical Academy
b Bashkir State Pedagogical University

Abstract: In a cylindric domain $D=(0,\infty)\times\Omega$, where $\Omega\subset \mathbb{R}_{n+1}$ is an unbounded domain, the first mixed problem for a high-order parabolic equation
\begin{gather*} u_t+(-1)^kD_x^k(a(x,\mathbf{y})D_x^ku)+\sum\limits_{i=l}^m\sum\limits_{|\alpha|=|\beta|=i}(-1)^i D_\mathbf{y}^{\alpha}(b_{\alpha\beta}(x,\mathbf{y})D_{\mathbf{y}}^{\beta}u)=0,
l\leq m,\quad k,l,m\in \mathbb{N}, \end{gather*}
is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of new geometrical characteristic of domain, the upper estimate of $L_2$-norm $\|u(t)\|$ of the solution to the problem is established. In particular, in domains $\{(x,\mathbf y)\in\mathbb{R}_{n+1}\mid x>0, |y_1|<x^a\}$, $0<a<q/l$, under the assumption that the upper an lower symbols of the operator $L$ are separated from zero, this estimate takes the form
$$ \|u(t)\|\leq M\exp(-\varkappa_2t^{b})\|\varphi\|,\quad b=\frac{k-la}{k-la+2lak}. $$
This estimate is determined by minor terms of the equation. The sharpness of the estimate for the wide class of unbounded domains is proved in the case $k=l=m=1$.

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English version:
Journal of Mathematical Sciences (New York), 2008, 150:5, 2369–2383

Bibliographic databases:

UDC: 517.956.4

Citation: L. M. Kozhevnikova, F. Kh. Mukminov, “Decay of the solution of the first mixed problem for a high-order parabolic equation with minor terms”, Fundam. Prikl. Mat., 12:4 (2006), 113–132; J. Math. Sci., 150:5 (2008), 2369–2383

Citation in format AMSBIB
\Bibitem{KozMuk06}
\by L.~M.~Kozhevnikova, F.~Kh.~Mukminov
\paper Decay of the solution of the first mixed problem for a~high-order parabolic equation with minor terms
\jour Fundam. Prikl. Mat.
\yr 2006
\vol 12
\issue 4
\pages 113--132
\mathnet{http://mi.mathnet.ru/fpm962}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2314149}
\zmath{https://zbmath.org/?q=an:1151.35380}
\elib{http://elibrary.ru/item.asp?id=11143779}
\transl
\jour J. Math. Sci.
\yr 2008
\vol 150
\issue 5
\pages 2369--2383
\crossref{https://doi.org/10.1007/s10958-008-0136-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42149164384}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. V. F. Gilimshina, F. Kh. Mukminov, “Ob ubyvanii resheniya vyrozhdayuschegosya lineinogo parabolicheskogo uravneniya”, Ufimsk. matem. zhurn., 3:4 (2011), 43–56  mathnet  zmath
    4. L. M. Kozhevnikova, A. A. Leontev, “Otsenki resheniya anizotropnogo parabolicheskogo uravneniya s dvoinoi nelineinostyu”, Ufimsk. matem. zhurn., 3:4 (2011), 64–85  mathnet  zmath
  • Фундаментальная и прикладная математика
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