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 Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 2, Pages 245–255 (Mi zvmmf332)

Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

A. V. Martynenko, A. F. Tedeev

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, ul. R. Luxemburg 74, Donetsk, 83114, Ukraine

Abstract: The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered:
$$\rho(x)\frac{\partial u}{\partial t}=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+u^p.$$
The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.

Key words: inhomogeneous density, degenerate parabolic equation, blowup regime.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:2, 238–248

Bibliographic databases:

UDC: 519.633

Citation: A. V. Martynenko, A. F. Tedeev, “Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density”, Zh. Vychisl. Mat. Mat. Fiz., 47:2 (2007), 245–255; Comput. Math. Math. Phys., 47:2 (2007), 238–248

Citation in format AMSBIB
\Bibitem{MarTed07} \by A.~V.~Martynenko, A.~F.~Tedeev \paper Cauchy problem for a~quasilinear parabolic equation with a~source term and an inhomogeneous density \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2007 \vol 47 \issue 2 \pages 245--255 \mathnet{http://mi.mathnet.ru/zvmmf332} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2351815} \zmath{https://zbmath.org/?q=an:05200978} \transl \jour Comput. Math. Math. Phys. \yr 2007 \vol 47 \issue 2 \pages 238--248 \crossref{https://doi.org/10.1134/S096554250702008X} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33947126955} 

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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:
1. A. V. Martynenko, A. F. Tedeev, “On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source”, Comput. Math. Math. Phys., 48:7 (2008), 1145–1160
2. Zheng Sining, Wang Chunpeng, “Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms”, Nonlinearity, 21:9 (2008), 2179–2200
3. Cao Yang, Yin Jingxue, Wang Chunpeng, “Cauchy problems of semilinear pseudo-parabolic equations”, J. Differential Equations, 246:12 (2009), 4568–4590
4. Cianci P., Martynenko A.V., Tedeev A.F., “The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source”, Nonlinear Anal., 73:7 (2010), 2310–2323
5. Novruzov E., “On existence and nonexistence of the positive solutions of non-Newtonian filtration equation”, Commun. Pure Appl. Anal., 10:2 (2011), 719–730
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8. Martynenko A.V., “Global Solvability for Quasilinear Parabolic Equation with Inhomogeneous Density and a Source”, Appl. Anal., 92:9 (2013), 1863–1888
9. Martynenko A.V. Tedeev A.F. Shramenko V.N., “On the Behavior of Solutions of the Cauchy Problem for a Degenerate Parabolic Equation with Source in the Case Where the Initial Function Slowly Vanishes”, Ukr. Math. J., 64:11 (2013), 1698–1715
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