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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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/

[1] M. Aripov, S. A. Sadulaeva, “To properties of solutions to reaction diffusion equation with double nonlinearity with distributed parameters”, Jour. of Siberian Fed. Univer. Math. Phys., 6:2 (2013), 157–167

[2] M. Aripov, Standard Equation's Methods for Solutions to Nonlinear problems, FAN, Tashkent, 1988 | MR

[3] M. Bertsh, R. D. Passo, M. Ughi, “Nonuniqueness of solutions of a degenerate parabolic equation”, Ann. Mat. Pura Appl. (IV), 161 (1992), 57–81 | DOI | MR

[4] M. Bertsch, P. Bisegna, “Blow-up of solutions of a nonlinear parabolic equation in damage mechanics”, Eur. J. Appl. Math., 8 (1997), 89–123 | MR | Zbl

[5] R. D. Passo, S. Luckhaus, “A degenerate diffusion problem not in divergence form”, Jour. of Diff. Eq., 69 (1987), 1–14 | DOI | MR | Zbl

[6] V. A. Galaktionov, S. P. Kurdyumov, A. A. Samarskii, “On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation”, Math. USSR Sbornik, 54 (1986), 421–455 | DOI | MR | Zbl

[7] C. Jin, J. Yin, “Self-similar solutions for a class of non-divergence form equations”, Nonlinear Differ. Equ. Appl. Nodea, 20:3 (2013), 873–893 | DOI | MR | Zbl

[8] R. G. Iagar, J. L. Vazquez, Asymptotic analysis for a p-Laplacian evolution equation in an exterior domain, Almeria, San Jose, 2007

[9] S. Kamin, J. L. Vazquez, “Fundamental solutions and asymptotic behaviour for the p-Laplacian equation”, Revista Matematica Iberoamericana, 4:2 (1988), 339–354 | DOI | MR | Zbl

[10] L. A. Peletier, Z. Junning, “Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case”, Nonlinear Analysis, Theory, Methods Applications, 17:10 (1991), 991–1009 | DOI | MR | Zbl

[11] W. Shu, W. Mingxin, X. C. Hong, “A nonlinear degenerate diffusion equation not in divergence form”, Z. Angew. Math. Phys., 51 (2000), 149–159 | DOI | MR | Zbl

[12] M. Ughi, “A degenerate parabolic equation modeling spread of an epidemic”, Ann. Mat. Pura Appl., 143 (1986), 385–400 | DOI | MR | Zbl

[13] C. Wang, J. Yin, “Shrinking self-similar solutions of a nonlinear diffusion equation with nondivergence form”, J. Math. Anal. Appl., 289 (2004), 387–404 | DOI | MR | Zbl

[14] W. Zhou, Z. Yao, “Cauchy problem for a degenerate parabolic equation with non-divergence form”, Acta. Mathematica Scienta, 30B:5 (2010), 1679–1686 | MR | Zbl

[15] M. Wiegner, “Blow-up for solutions of some degenerate parabolic equations”, Differ. Integral Eqs., 7 (1994), 1641–1647 | MR | Zbl

[16] Z. Wenshu, Y. Zheng-an, “Behaviors of solutions for a singular diffusion equation”, J. Math. Anal. Appl., 327 (2007), 611–619 | DOI | MR | Zbl

[17] Z. Yao, Z. Wenshu, “Nonuniqueness of solutions for a singular diffusion problem”, J. Math. Anal. Appl., 325 (2007), 183–204 | DOI | MR | Zbl

[18] W. Zhou, Z. Wu, “Some results on a class of degenerate parabolic equations not in divergence form”, Nonlinear Analysis: Theory, Methods Appl., 60:5 (2005), 863–886 | DOI | MR | Zbl

[19] Ya. B. Zel'dovich, A. S. Kompaneets, “Towards a theory of heat conduction with thermal conductivity depending on the temperature”, Collection of Papaers Dedicated to 70th Anniversary of A. F. Ioffe, Izd. Akad. Nauk SSSR, M., 1950, 61–72 (in Russian)