On existence and nonexistence of thepositive solutions of non-newtonian filtration equation

Emil Novruzov

doi:10.3934/cpaa.2011.10.719

Article Contents

2011, Volume 10, Issue 2: 719-730. Doi: 10.3934/cpaa.2011.10.719

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

Emil Novruzov^1,

1.
Department of Mathematics, Hacettepe University, 06800 Beytepe - Ankara

Received: May 31, 2010

Revised: August 31, 2010

Published: February 2011

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Abstract

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.

Keywords:

Nonlinear degenerate equation,
FSP(finite speed of propagation of perturbations),
existence,
nonexistence.

Mathematics Subject Classification: Primary: 35K15, 35K65; Secondary: 35B30.

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