Geodesic
The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 8 (2015) no. 2, pp. 192-200

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We investigate the solutions for the following nonlinear degenerate parabolic equation in non-divergent form with density $$ \left|x\right|^{n} \frac{\partial u}{\partial t} =u^{m} div\left(\left|\nabla u\right|^{p-2} \nabla u\right). $$ 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, self-similar solution, asymptotic behavior of solutions.
Mots-clés : non-divergent form 
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     title = {The properties of the solutions for {Cauchy} problem of nonlinear parabolic equations in non-divergent form with density},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
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Jakhongir R. Raimbekov. The properties of the solutions for Cauchy problem of nonlinear parabolic equations in non-divergent form with density. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 8 (2015) no. 2, pp. 192-200. http://geodesic.mathdoc.fr/item/JSFU_2015_8_2_a8/