
This article is cited in 1 scientific paper (total in 1 paper)
On largetime behavior of solutions of parabolic equations
V. N. Denisov^{} ^{} M. V. Lomonosov Moscow State University, Moscow, Russia
Abstract:
We study the stabilization of solutions of the Cauchy problem for
secondorder parabolic equations depending on the behavior of the
lowerorder coefficients of equations at the infinity and on the growth
rate of initial functions. We also consider the stabilization of
solution of the first boundaryvalue problem for a parabolic
equation without lowerorder coefficients depending on the
domain $Q$ where the initial function is defined for $t=0.$
In the first chapter, we study sufficient conditions for uniform
in $x$ on a compact $K\subset\mathbb{R}^N$ stabilization to zero of the
solution of the Cauchy problem with divergent elliptic operator
and coefficients independent of $t$ and depending only on $x.$ We
consider classes of initial functions:
 bounded in $\mathbb{R}^N$,
 with power growth rate at the infinity in $\mathbb{R}^N$,
 with exponential order at the infinity.
\noindent Using examples, we show that sufficient conditions are
sharp and, moreover, do not allow the uniform in $\mathbb{R}^N$
stabilization to zero of the solution of the Cauchy problem.
In the second chapter, we study the Cauchy problem with elliptic
nondivergent operator and coefficients depending on $x$ and $t.$
In different classes of growing initial functions we obtain exact
sufficient conditions for stabilization of solutions of the
corresponding Cauchy problem uniformly in $x$ on any compact $K$
in $\mathbb{R}^N$. We consider examples proving the sharpness of these
conditions.
In the third chapter, for the solution of the first boundaryvalue
problem without lowerorder terms, we obtain necessary and
sufficient conditions of uniform in $x$ on any compact in $Q$
stabilization to zero in terms of the domain $\mathbb{R}^N \setminus Q$
where $Q$ is the definitional domain of the initial function for
$t=0.$ We establish the power estimate for the rate of
stabilization of the solution of the boundaryvalue problem with
bounded initial function in the case where $\mathbb{R}^N \setminus Q$ is a
cone for $t=0$.
DOI:
https://doi.org/10.22363/2413363920206611155
Full text:
PDF file (1123 kB)
References:
PDF file
HTML file
UDC:
517.9
Citation:
V. N. Denisov, “On largetime behavior of solutions of parabolic equations”, Partial differential equations, CMFD, 66, no. 1, RUDN University, M., 2020, 1–155
Citation in format AMSBIB
\Bibitem{Den20}
\by V.~N.~Denisov
\paper On largetime behavior of solutions of parabolic equations
\inbook Partial differential equations
\serial CMFD
\yr 2020
\vol 66
\issue 1
\pages 1155
\publ RUDN University
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd397}
\crossref{https://doi.org/10.22363/2413363920206611155}
Linking options:
http://mi.mathnet.ru/eng/cmfd397 http://mi.mathnet.ru/eng/cmfd/v66/i1/p1
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:

Jenaliyev M.T., Ramazanov M.I., Attaev A.Kh., Gulmanov N.K., “Stabilization of a Solution For TwoDimensional Loaded Parabolic Equation”, Bull. Karaganda UnivMath., 100:4 (2020), 55–70

Number of views: 
This page:  187  Full text:  95  References:  11 
