Communications on Pure and Applied Analysis (CPAA)

On existence and nonexistence of the positive solutions of non-newtonian filtration equation
Pages: 719 - 730, Volume 10, Issue 2, March 2011

doi:10.3934/cpaa.2011.10.719      Abstract        References        Full text (186.1K)           Related Articles

Emil Novruzov - Department of Mathematics, Hacettepe University, 06800 Beytepe - Ankara, Turkey (email)

1 N. Ahmed and D. K. Sunada, Nonlinear flows in porous media, J. Hydraulics. Div. Proc. Amer. Soc. Civil Eng., 95 (1969), 1847-1857.
2 D. Blanchard and G. A. Francfort, Study of double nonlinear heat equation with no growth assumptions on the parabolic term, SIAM J. Math. Anal., 19 (1988), 1032-1056.       
3 S. P. Degtyarev and A. F. Tedeev, $L_1-L_\infty$- estimates of solutions of the Cauchy problem for an anisotropic degenerate parabolic equation with double non-linearity and growing initial data, Sb. Math., 198 (2007), 639-660.       
4 J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in $R$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-80.       
5 V. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equation of parabolic and hyperbolic types, J. Sov. Math., 10 (1978), 53-70.       
6 H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u) $, Arch. Rational Mech. Anal., 51 (1973), 371-386.       
7 J. L. Lions, "Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires," Dunod, Gauthier Villars, Paris, 1969.       
8 J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," Springer-Verlag, New York, 1972.       
9 A. V. Martynenko, and A. F. Tedeev, The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density, Comput. Math. Math. Phys., 47 (2007), 238-248.       
10 E. Novruzov, On blow-up of solution of nonhomogeneous polytropic equation with source, Nonlinear Anal., 71 (2009), 3992-3998.       
11 L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.       
12 P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Diff. Equ., 150 (1998), 203-214.       
13 G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Communications on Pure and Applied Analysis, 7 (2008), 1275-1294.       
14 A. F. Tedeev, Conditions for the time global existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear degenerate parabolic equations, Siberian Math.J., 45 (2004), 155-164.       
15 A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Applicable Analysis, 86 (2007), 755-782.       
16 M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption, JMMA, 132 (1988), 187-212.       
17 Z. Xiang, Ch. Mu and X. Hu, Support properties of solutions to a degenerate equation with absorption and variable density, Nonlinear Anal., 68 (2008), 1940-1953.       
18 Y. Zhou, Global nonexistence for a quasilinear evolution equation with a general Lewis function, J. for Analysis and its Applications, 24 (2005), 179-187.       

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