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Mat. Sb. (N.S.), 1984, Volume 124(166), Number 2(6), Pages 163–188 (Mi msb2046)  

This article is cited in 13 scientific papers (total in 13 papers)

On approximate self-similar solutions of a class of quasilinear heat equations with a source

V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii


Abstract: Quasilinear parabolic equations of the form
$$ \frac{\partial u}{\partial t}=\nabla(k(u)\nabla u)+Q(u),\qquad\nabla( \cdot ) =\operatorname{grad}_x( \cdot ),\quad k\geqslant0, $$
are considered; here $k(u)$ and $Q(u)$ are sufficiently smooth given functions (respectively, the coefficient of thermal conductivity and the power of heat sources depending on the temperature $u=u(t,x)\geqslant0$). A family of coefficients $\{k\}$ and corresponding functions $\{Q_k\}$ is distinguished for which the properties of the solution of the boundary value problem for the equation in question are described by invariant solutions $v_A(t,x)$ of a first-order equation of Hamilton–Jacobi type
$$ \frac{\partial v}{\partial t}=\frac{k(v)}{v+1}(\nabla v)^2 +G(t)\nabla\mathbf{vx}+H(t)Q_k(v). $$
The function $u_A$ is an approximate self-similar solution of the original equation.
Tables: 1.
Figures: 1.
Bibliography: 70 titles.

Full text: PDF file (1392 kB)
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English version:
Mathematics of the USSR-Sbornik, 1985, 52:1, 155–180

Bibliographic databases:

UDC: 517.95
MSC: 35K05, 35K55, 35A35
Received: 18.11.1983

Citation: V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii, “On approximate self-similar solutions of a class of quasilinear heat equations with a source”, Mat. Sb. (N.S.), 124(166):2(6) (1984), 163–188; Math. USSR-Sb., 52:1 (1985), 155–180

Citation in format AMSBIB
\Bibitem{GalKurSam84}
\by V.~A.~Galaktionov, S.~P.~Kurdyumov, A.~A.~Samarskii
\paper On approximate self-similar solutions of a~class of quasilinear heat equations with a~source
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 124(166)
\issue 2(6)
\pages 163--188
\mathnet{http://mi.mathnet.ru/msb2046}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=746066}
\zmath{https://zbmath.org/?q=an:0573.35049}
\transl
\jour Math. USSR-Sb.
\yr 1985
\vol 52
\issue 1
\pages 155--180
\crossref{https://doi.org/10.1070/SM1985v052n01ABEH002883}


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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. V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii, “On asymptotic “eigenfunctions” of the Cauchy problem for a nonlinear parabolic equation”, Math. USSR-Sb., 54:2 (1986), 421–455  mathnet  crossref  mathscinet  zmath
    2. Galaktionov V., “Asymptotic-Behavior of Unbounded Solutions of the Nonlinear Parabolic Equation Ut=(Usigmaux)X+Usigma+1”, Differ. Equ., 21:7 (1985), 751–758  mathscinet  zmath  isi
    3. A. S. Kalashnikov, “Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations”, Russian Math. Surveys, 42:2 (1987), 169–222  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. Shestakov A., “Generalized Direct Lyapunov Method for Abstract Semidynamical Processes .3. Localization of Limit-Sets of Compact Dispersive Semidynamical Processes - Applications to Evolution-Equations”, Differ. Equ., 23:6 (1987), 611–622  mathscinet  zmath  isi
    5. Bakirova M., Dimova S., Dorodnitsyn V., Kurdiumov S., Samarskii A., Svirshchevksii S., “Invariant Solutions of Heat-Conduction Equation Describing the Directed Propagation of Combustion and Spiral Waves in a Nonlinear Medium”, 299, no. 2, 1988, 346–350  mathscinet  zmath  isi
    6. Akhromeyeva TS., Kurdyumov S., Malinetskii G., Samarskii A., “Nonstationary Dissipative Structures and Diffusion-Induced Chaos in Nonlinear Media”, Phys. Rep.-Rev. Sec. Phys. Lett., 176:5-6 (1989), 189–370  crossref  mathscinet  isi
    7. Galaktionov V., “On Blow-Up and Degeneracy for the Semilinear Heat-Equation with Source”, Proc. R. Soc. Edinb. Sect. A-Math., 115:Part 1-2 (1990), 19–24  crossref  mathscinet  zmath  isi
    8. Comput. Math. Math. Phys., 33:2 (1993), 217–227  mathnet  mathscinet  zmath  isi
    9. Galaktionov V. Vazquez J., “Regional Blow-Up in a Semilinear Heat-Equation with Convergence to a Hamilton–Jacobi Equation”, SIAM J. Math. Anal., 24:5 (1993), 1254–1276  crossref  mathscinet  zmath  isi
    10. Bebernes J. Bricher S. Galaktionov V., “Asymptotics of Blowup for Weakly Quasi-Linear Parabolic Problems”, Nonlinear Anal.-Theory Methods Appl., 23:4 (1994), 489–514  crossref  mathscinet  zmath  isi
    11. Comput. Math. Math. Phys., 35:3 (1995), 303–319  mathnet  mathscinet  zmath  isi
    12. Hatem Zaag, “Blow-up results for vector-valued nonlinear heat equations with no gradient structure”, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 15:5 (1998), 581  crossref  mathscinet  zmath
    13. Juntang Ding, Shengjia Li, “Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions”, Nonlinear Analysis: Theory, Methods & Applications, 68:3 (2008), 507  crossref  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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