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Mat. Sb., 2006, Volume 197, Number 5, Pages 125–160 (Mi msb1561)  

This article is cited in 5 scientific papers (total in 5 papers)

Asymptotic behaviour of supports of solutions of quasilinear many-dimensionsal parabolic equations of non-stationary diffusion-convection type

D. A. Sapronova, A. E. Shishkovb

a Donetsk National University
b Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: We study the phenomenon of the finiteness of the rate of propagation of the supports of generalized energy solutions of mixed problems for a broad class of doubly degenerate parabolic equations of high order; a model example here is the equation
$$ (|u|^{q-1}u)_t+(-1)^m \sum_{|\alpha|=m} D_x^\alpha(|D_x^\alpha u|^{p-1} D_x^\alpha u)+(|u|^{\lambda-1}u)_{x_1}=0, $$
$m \geqslant 1$, $p>0$, $q>0$, $\lambda>0$.
Bounds (that are sharp in a certain sense) for the early evolution of the supports of solutions (in particular, of the ‘right’ and the ‘left’ fronts of the solutions), which depend on local properties of the initial function and the parameters of the equation, are established. The behaviour of the supports for large times is also studied.
Bibliography: 31 titles.

DOI: https://doi.org/10.4213/sm1561

Full text: PDF file (747 kB)
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English version:
Sbornik: Mathematics, 2006, 197:5, 753–790

Bibliographic databases:

UDC: 517.9
MSC: Primary 35K55, 35B05; Secondary 35K30, 35K35
Received: 04.01.2003 and 13.05.2005

Citation: D. A. Sapronov, A. E. Shishkov, “Asymptotic behaviour of supports of solutions of quasilinear many-dimensionsal parabolic equations of non-stationary diffusion-convection type”, Mat. Sb., 197:5 (2006), 125–160; Sb. Math., 197:5 (2006), 753–790

Citation in format AMSBIB
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\paper Asymptotic behaviour of supports of solutions of
quasilinear many-dimensionsal parabolic equations of
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\vol 197
\issue 5
\pages 125--160
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\pages 753--790
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    1. Namlyeyeva Yu.V., Taranets R.M., “Backward motion and waiting time phenomena for degenerate parabolic equations with nonlinear gradient absorption”, Manuscr. Math., 136:3-4 (2011), 475–500  crossref  mathscinet  zmath  isi
    2. de Loubens R., Ramakrishnan T.S., “Asymptotic solution of a nonlinear advection-diffusion equation”, Quart. Appl. Math., 69:2 (2011), 389–401  crossref  mathscinet  zmath  isi
    3. Chunhua Jin, Jingxue Yin, Sining Zheng, “Propagation profile of support for evolution p-Laplacian with convection in half space”, Journal of Mathematical Analysis and Applications, 2014  crossref  mathscinet
    4. Pan Zheng, Chunlai Mu, Fuchen Zhang, Iftikhar Ahmed, “Waiting time phenomena for the porous medium equation with gradient absorption”, J. Appl. Math. Comput, 2014  crossref  mathscinet
    5. Hailong Ye, Jingxue Yin, “Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection”, Journal of Mathematical Analysis and Applications, 2014  crossref  mathscinet
  • Математический сборник Sbornik: Mathematics (from 1967)
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