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Sbornik: Mathematics, 1995, Volume 186, Issue 12, Pages 1843–1864
DOI: https://doi.org/10.1070/SM1995v186n12ABEH000096
(Mi sm96)
 

This article is cited in 2 scientific papers (total in 2 papers)

Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order

A. E. Shishkov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
References:
Abstract: The Cauchy problem for a quasi-linear degenerate parabolic equation in divergence from with energy space $L_p\bigl(0,T;W_{p,\operatorname{loc}}^m(\mathbb R^n)\bigr)$, $m\geqslant 1$, $p>2$, $n\geqslant 1$ and with initial function $u_0(x)\in L_{2,\operatorname{loc}}(\mathbb R^n)$ is considered. The existence of a generalized solution $u(x,t)$ is proved for $u_0(x)$ growing at infinity at the rate:
$$ \int_{|x|<\tau}u_0(x)^2\,dx<c\tau^{n+\frac{2mp}{p-2}} \qquad \forall\,\tau>\tau'>0, \quad c<\infty. $$
For more sever constraints on the growth of $u_0(x)$ several fairly wide uniqueness classes for the above-mentioned solution are discovered. The question of describing the geometry of the domain $\Omega(t)\equiv\mathbb R^n\setminus\operatorname{supp}_xu(x,t)$ for $\Omega_0\equiv\mathbb R^n\setminus\operatorname{supp}u_0(x)\ne\varnothing$ is considered. In case when the domain $\Omega_0$ is unbounded, estimates in terms of the global properties of the initial function $u_0(x)$ are established that characterize the geometry of $\Omega(t)$ as $t\to\infty$.
Received: 03.11.1994
Russian version:
Matematicheskii Sbornik, 1995, Volume 186, Number 12, Pages 151–172
Bibliographic databases:
UDC: 517.9
MSC: 35K55, 35K65
Language: English
Original paper language: Russian
Citation: A. E. Shishkov, “Evolution of the support of a solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order”, Mat. Sb., 186:12 (1995), 151–172; Sb. Math., 186:12 (1995), 1843–1864
Citation in format AMSBIB
\Bibitem{Shi95}
\by A.~E.~Shishkov
\paper Evolution of the support of a~solution with unbounded energy of quasi-linear degenerate parabolic equation of arbitrary order
\jour Mat. Sb.
\yr 1995
\vol 186
\issue 12
\pages 151--172
\mathnet{http://mi.mathnet.ru/sm96}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1376096}
\zmath{https://zbmath.org/?q=an:0864.35061}
\transl
\jour Sb. Math.
\yr 1995
\vol 186
\issue 12
\pages 1843--1864
\crossref{https://doi.org/10.1070/SM1995v186n12ABEH000096}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1995UL00600014}
Linking options:
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  • https://doi.org/10.1070/SM1995v186n12ABEH000096
  • https://www.mathnet.ru/eng/sm/v186/i12/p151
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник - 1992–2005 Sbornik: Mathematics
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    Abstract page:361
    Russian version PDF:93
    English version PDF:12
    References:62
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