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March  2011, 10(2): 719-730. doi: 10.3934/cpaa.2011.10.719

On existence and nonexistence of the positive solutions of non-newtonian filtration equation

1. 

Department of Mathematics, Hacettepe University, 06800 Beytepe - Ankara, Turkey

Received  June 2010 Revised  September 2010 Published  December 2010

The subject of this investigation is existence and nonexistence of positive solutions of the following nonhomogeneous equation

$ \rho (|x|) \frac{\partial u}{\partial t}- \sum_{i=1}^N D_i(u^{m-1}|D_i u|^{\lambda -1}D_i u)+g(u)+lu=f(x)$ (1)

or, after the change $v=u^{\sigma}$, $\sigma =\frac{m+\lambda -1}{\lambda }, $ of equation

$\rho (|x|) \frac{\partial v^{\frac{1}{ \sigma }}}{\partial t}-\sigma ^{-\lambda }\sum_{i=1} ^N D_i(|D_i v|^{\lambda -1}D_i v)+g(v^{\frac{1}{\sigma }}) +lv^{\frac{1}{ \sigma }}=f(x),$ (1')

in unbounded domain $R_+\times R^N,$ where the term $g(s)$ is supposed to satisfy just a lower polynomial growth condition and $g'(s) > -l_1$. The existence of the solution in $ L^{1+1/\sigma}(0, T; L^{1+1/\sigma}(R^N))\cap L^{\lambda +1}(0, T; W^{1,\lambda +1}(R^N))$ is proved. Also, under some condition on $g(s)$ and $u_0$ is shown a nonexistence of the solution.

Citation: Emil Novruzov. On existence and nonexistence of the positive solutions of non-newtonian filtration equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 719-730. doi: 10.3934/cpaa.2011.10.719
References:
[1]

N. Ahmed and D. K. Sunada, Nonlinear flows in porous media,, J. Hydraulics. Div. Proc. Amer. Soc. Civil Eng., 95 (1969), 1847.   Google Scholar

[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.  doi: doi:10.1137/0519070.  Google Scholar

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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.  doi: doi:10.1070/SM2007v198n05ABEH003853.  Google Scholar

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J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in $R$ with power-like nonlinear diffusivity,, Arch. Rational Mech. Anal., 103 (1988), 39.  doi: doi:10.1007/BF00292920.  Google Scholar

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V. Kalantarov and O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equation of parabolic and hyperbolic types,, J. Sov. Math., 10 (1978), 53.  doi: doi:10.1007/BF01109723.  Google Scholar

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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.  doi: doi:10.1007/BF00263041.  Google Scholar

[7]

J. L. Lions, "Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires,", Dunod, (1969).   Google Scholar

[8]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Springer-Verlag, (1972).   Google Scholar

[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.  doi: doi:10.1134/S096554250702008X.  Google Scholar

[10]

E. Novruzov, On blow-up of solution of nonhomogeneous polytropic equation with source,, Nonlinear Anal., 71 (2009), 3992.  doi: doi:10.1016/j.na.2009.02.069.  Google Scholar

[11]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: doi:10.1007/BF02761595.  Google Scholar

[12]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy,, J. Diff. Equ., 150 (1998), 203.  doi: doi:10.1006/jdeq.1998.3477.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2008.7.1275.  Google Scholar

[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.  doi: doi:10.1023/B:SIMJ.0000013021.66528.b6.  Google Scholar

[15]

A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations,, Applicable Analysis, 86 (2007), 755.  doi: doi:10.1080/00036810701435711.  Google Scholar

[16]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, JMMA, 132 (1988), 187.   Google Scholar

[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.  doi: doi:10.1016/j.na.2007.01.021.  Google Scholar

[18]

Y. Zhou, Global nonexistence for a quasilinear evolution equation with a general Lewis function,, J. for Analysis and its Applications, 24 (2005), 179.   Google Scholar

show all references

References:
[1]

N. Ahmed and D. K. Sunada, Nonlinear flows in porous media,, J. Hydraulics. Div. Proc. Amer. Soc. Civil Eng., 95 (1969), 1847.   Google Scholar

[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.  doi: doi:10.1137/0519070.  Google Scholar

[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.  doi: doi:10.1070/SM2007v198n05ABEH003853.  Google Scholar

[4]

J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in $R$ with power-like nonlinear diffusivity,, Arch. Rational Mech. Anal., 103 (1988), 39.  doi: doi:10.1007/BF00292920.  Google Scholar

[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.  doi: doi:10.1007/BF01109723.  Google Scholar

[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.  doi: doi:10.1007/BF00263041.  Google Scholar

[7]

J. L. Lions, "Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires,", Dunod, (1969).   Google Scholar

[8]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", Springer-Verlag, (1972).   Google Scholar

[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.  doi: doi:10.1134/S096554250702008X.  Google Scholar

[10]

E. Novruzov, On blow-up of solution of nonhomogeneous polytropic equation with source,, Nonlinear Anal., 71 (2009), 3992.  doi: doi:10.1016/j.na.2009.02.069.  Google Scholar

[11]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.  doi: doi:10.1007/BF02761595.  Google Scholar

[12]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy,, J. Diff. Equ., 150 (1998), 203.  doi: doi:10.1006/jdeq.1998.3477.  Google Scholar

[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.  doi: doi:10.3934/cpaa.2008.7.1275.  Google Scholar

[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.  doi: doi:10.1023/B:SIMJ.0000013021.66528.b6.  Google Scholar

[15]

A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations,, Applicable Analysis, 86 (2007), 755.  doi: doi:10.1080/00036810701435711.  Google Scholar

[16]

M. Tsutsumi, On solutions of some doubly nonlinear degenerate parabolic equations with absorption,, JMMA, 132 (1988), 187.   Google Scholar

[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.  doi: doi:10.1016/j.na.2007.01.021.  Google Scholar

[18]

Y. Zhou, Global nonexistence for a quasilinear evolution equation with a general Lewis function,, J. for Analysis and its Applications, 24 (2005), 179.   Google Scholar

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