Article Contents
Article Contents

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

• 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.

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

 Citation:

•  [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.doi: doi:10.1137/0519070. [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.doi: doi:10.1070/SM2007v198n05ABEH003853. [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.doi: doi:10.1007/BF00292920. [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.doi: doi:10.1007/BF01109723. [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.doi: doi:10.1007/BF00263041. [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.doi: doi:10.1134/S096554250702008X. [10] E. Novruzov, On blow-up of solution of nonhomogeneous polytropic equation with source, Nonlinear Anal., 71 (2009), 3992-3998.doi: doi:10.1016/j.na.2009.02.069. [11] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.doi: doi:10.1007/BF02761595. [12] P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Diff. Equ., 150 (1998), 203-214.doi: doi:10.1006/jdeq.1998.3477. [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.doi: doi:10.3934/cpaa.2008.7.1275. [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.doi: doi:10.1023/B:SIMJ.0000013021.66528.b6. [15] A. F. Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Applicable Analysis, 86 (2007), 755-782.doi: doi:10.1080/00036810701435711. [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.doi: doi:10.1016/j.na.2007.01.021. [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.