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Mathematics of the USSR-Sbornik, 1984, Volume 49, Issue 1, Pages 125–149
DOI: https://doi.org/10.1070/SM1984v049n01ABEH002701
(Mi sm2158)
 

This article is cited in 11 scientific papers (total in 11 papers)

Methods of constructing approximate self-similar solutions of nonlinear heat equations. IV

V. A. Galaktionov, A. A. Samarskii
References:
Abstract: Asymptotic properties of nonnegative solutions of quasilinear parabolic equations
$$ \frac{\partial u}{\partial t}=\frac\partial{\partial x}\bigg(k(u)\frac{\partial u}{\partial x}\bigg);\qquad k(u)>0\quad\text{for}\quad u>0 $$
with coefficients $k(u)$ of rather general form are studied in the paper. The investigation is carried out by constructing approximate self-similar solutions which do not satisfy the original equation but nevertheless correctly describe the asymptotic behavior of solutions of the boundary value or Cauchy problems considered. On the basis of a unified method “transformation laws” are established for well-known self-similar solutions of an equation with a power nonlinearity $\dfrac{\partial u}{\partial t}=\dfrac\partial{\partial x}\biggl(u^\sigma\dfrac{\partial u}{\partial x}\biggr)$ (the cases $\sigma=0$ and $\sigma>0$ are considered separately) which result from small changes of the coefficient $u^\sigma\to k(u)$ (for example, transformations of the form $u^\sigma\to u^\sigma\ln(1+u)$, $ u^\sigma\to u^\sigma\exp[|\ln u|^{1/2}]$, etc.).
Figures: 1.
Bibliography: 24 titles.
Received: 20.09.1982
Bibliographic databases:
UDC: 517.956
MSC: 35K05, 35K55, 35B40
Language: English
Original paper language: Russian
Citation: V. A. Galaktionov, A. A. Samarskii, “Methods of constructing approximate self-similar solutions of nonlinear heat equations. IV”, Math. USSR-Sb., 49:1 (1984), 125–149
Citation in format AMSBIB
\Bibitem{GalSam83}
\by V.~A.~Galaktionov, A.~A.~Samarskii
\paper Methods of constructing approximate self-similar solutions of nonlinear heat equations.~IV
\jour Math. USSR-Sb.
\yr 1984
\vol 49
\issue 1
\pages 125--149
\mathnet{http://mi.mathnet.ru//eng/sm2158}
\crossref{https://doi.org/10.1070/SM1984v049n01ABEH002701}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=703321}
\zmath{https://zbmath.org/?q=an:0574.35049}
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  • https://doi.org/10.1070/SM1984v049n01ABEH002701
  • https://www.mathnet.ru/eng/sm/v163/i2/p131
    Cycle of papers
    This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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