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 J. Sib. Fed. Univ. Math. Phys., 2015, Volume 8, Issue 2, Pages 192–200 (Mi jsfu421)

The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density

Jakhongir R. Raimbekov

National University of Uzbekistan, Yunus Abad-17, 3/66, 10037, Tashkent, Uzbekistan

Abstract: We investigate the solutions for the following nonlinear degenerate parabolic equation in non-divergent form with density
$$|x|^{n} \frac{\partial u}{\partial t} =u^{m} div(|\nabla u|^{p-2} \nabla u).$$
We discuss the properties, which are different from those for the equations in divergence form, thus generalizing various known results. Then getting a self-similar solution we show the asymptotic behavior of solutions at $t \to \infty$. Slow and fast diffusion cases are investigated. Finally, we present the results of some numerical experiments.

Keywords: nonlinear degenerate parabolic equation, non-divergent form, self-similar solution, asymptotic behavior of solutions.

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UDC: 517.955.8
Accepted: 03.04.2014
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Citation: Jakhongir R. Raimbekov, “The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density”, J. Sib. Fed. Univ. Math. Phys., 8:2 (2015), 192–200

Citation in format AMSBIB
\Bibitem{Rai15} \by Jakhongir~R.~Raimbekov \paper The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density \jour J. Sib. Fed. Univ. Math. Phys. \yr 2015 \vol 8 \issue 2 \pages 192--200 \mathnet{http://mi.mathnet.ru/jsfu421} 

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This publication is cited in the following articles:
1. M. M. Aripov, A. S. Matyakubov, “Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behaviour”, Nanosyst.-Phys. Chem. Math., 8:1 (2017), 5–12
2. Mersaid M. Aripov, Jakhongir R. Raimbekov, “The critical curves of a doubly nonlinear parabolic equation in non-divergent form with a source and nonlinear boundary flux”, Zhurn. SFU. Ser. Matem. i fiz., 12:1 (2019), 112–124
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