March  2013, 12(2): 1123-1139. doi: 10.3934/cpaa.2013.12.1123

Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density

1. 

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

2. 

Depto. de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés 28911, Madrid

Received  July 2011 Revised  May 2012 Published  September 2012

We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem \begin{eqnarray*} \rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, \\ u(x,0)=u_{0}(x), \quad in\quad R^n, \end{eqnarray*} where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$. We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in $R^n$. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation $|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.
Citation: Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123
References:
[1]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^n,$ Manuscripta Math., 74 (1992), 87-106.

[2]

J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, (English summary) Ann. Scuola Norm. Sup. Pisa Cl. Sci, 26 (1998), 663-687.

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer-Verlag, 1993.

[4]

D. Eidus, The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium, J. Diff. Eqns., 84 (1990), 309-318.

[5]

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830.

[6]

V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233.

[7]

S. Kamin, Heat propagation in an inhomogeneous medium, Progress in Partial Differential Equations: the Metz Surveys 4, 229-237, Pitman Res. Notes Math. Ser. 345, Longman, Harlow, 1996.

[8]

S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120.

[9]

S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298.

[10]

S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121.

[11]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, DCDS-A, 26 (2010), Special Volume on Parabolic Problems, 521-549.

[12]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogenous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.

[13]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.

[14]

G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Networks and Heterogeneous Media NHM, 1 (2006), 337-351.

[15]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294.

[16]

G. Reyes and J. L. Vázquez, Long time behavior for the inohomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.

[17]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.

[18]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

References:
[1]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^n,$ Manuscripta Math., 74 (1992), 87-106.

[2]

J. W. Dold, V. A. Galaktionov, A. A. Lacey and J. L. Vázquez, Rate of approach to a singular steady state in quasilinear reaction-diffusion equations, (English summary) Ann. Scuola Norm. Sup. Pisa Cl. Sci, 26 (1998), 663-687.

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer-Verlag, 1993.

[4]

D. Eidus, The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium, J. Diff. Eqns., 84 (1990), 309-318.

[5]

D. Eidus and S. Kamin, The filtration equation in a class of functions decreasing at infinity, Proc. Amer. Math. Soc., 120 (1994), 825-830.

[6]

V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents, J. Differential Equations, 189 (2003), 199-233.

[7]

S. Kamin, Heat propagation in an inhomogeneous medium, Progress in Partial Differential Equations: the Metz Surveys 4, 229-237, Pitman Res. Notes Math. Ser. 345, Longman, Harlow, 1996.

[8]

S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120.

[9]

S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298.

[10]

S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121.

[11]

S. Kamin, G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density, DCDS-A, 26 (2010), Special Volume on Parabolic Problems, 521-549.

[12]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogenous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.

[13]

S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.

[14]

G. Reyes and J. L. Vázquez, The Cauchy problem for the inhomogeneous porous medium equation, Networks and Heterogeneous Media NHM, 1 (2006), 337-351.

[15]

G. Reyes and J. L. Vázquez, The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions, Commun. Pure Appl. Anal., 7 (2008), 1275-1294.

[16]

G. Reyes and J. L. Vázquez, Long time behavior for the inohomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.

[17]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.

[18]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[1]

Mohammad Asadzadeh, Anders Brahme, Jiping Xin. Galerkin methods for primary ion transport in inhomogeneous media. Kinetic and Related Models, 2010, 3 (3) : 373-394. doi: 10.3934/krm.2010.3.373

[2]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843

[3]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

[4]

Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794

[5]

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

[6]

Fioralba Cakoni, Anne Cossonnière, Houssem Haddar. Transmission eigenvalues for inhomogeneous media containing obstacles. Inverse Problems and Imaging, 2012, 6 (3) : 373-398. doi: 10.3934/ipi.2012.6.373

[7]

María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks and Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004

[8]

Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations and Control Theory, 2020, 9 (2) : 375-398. doi: 10.3934/eect.2020010

[9]

Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644

[10]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks and Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63

[11]

Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307

[12]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897

[13]

Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems and Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725

[14]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks and Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009

[15]

Martial Agueh, Guillaume Carlier, Reinhard Illner. Remarks on a class of kinetic models of granular media: Asymptotics and entropy bounds. Kinetic and Related Models, 2015, 8 (2) : 201-214. doi: 10.3934/krm.2015.8.201

[16]

Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445

[17]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305

[18]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281

[19]

Ting Zhang. The modeling error of well treatment for unsteady flow in porous media. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2171-2185. doi: 10.3934/dcdsb.2015.20.2171

[20]

T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (178)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]