March  2009, 8(2): 493-508. doi: 10.3934/cpaa.2009.8.493

Long time behavior for the inhomogeneous PME in a medium with slowly decaying density


Departamento de Matemática Aplicada, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid. 28040 Madrid


Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received  February 2008 Revised  August 2008 Published  December 2008

We study the long-time behavior of non-negative solutions to the Cauchy problem

(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow diffusion) and $\rho(x)$ is positive, bounded and behaves like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with $0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative and such that $\int \rho(x)u_0 dx< \infty$.
Our asymptotic analysis leads to the associated singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which admits a one-parameter family of selfsimilar solutions $ U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty $ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

Citation: Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493

Edoardo Mainini. On the signed porous medium flow. Networks & Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525


Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337


Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355


Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123


Anna Marciniak-Czochra, Andro Mikelić. A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1065-1077. doi: 10.3934/dcdss.2014.7.1065


Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083


Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021


Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93


María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012


Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093


Magnus Aspenberg, Fredrik Ekström, Tomas Persson, Jörg Schmeling. On the asymptotics of the scenery flow. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2797-2815. doi: 10.3934/dcds.2015.35.2797


Fang Zeng, Xiaodong Liu, Jiguang Sun, Liwei Xu. The reciprocity gap method for a cavity in an inhomogeneous medium. Inverse Problems & Imaging, 2016, 10 (3) : 855-868. doi: 10.3934/ipi.2016024


Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393


Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457


R.E. Showalter, Ning Su. Partially saturated flow in a poroelastic medium. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 403-420. doi: 10.3934/dcdsb.2001.1.403


Michiel Bertsch, Carlo Nitsch. Groundwater flow in a fissurised porous stratum. Networks & Heterogeneous Media, 2010, 5 (4) : 765-782. doi: 10.3934/nhm.2010.5.765


Shoshana Kamin, Guillermo Reyes, Juan Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 521-549. doi: 10.3934/dcds.2010.26.521


Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068


Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783


Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

2020 Impact Factor: 1.916


  • PDF downloads (92)
  • HTML views (0)
  • Cited by (26)

Other articles
by authors

[Back to Top]