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 & 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 $\mathbbR^n,$, Manuscripta Math., 74 (1992), 87.   Google Scholar

[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.   Google Scholar

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,'', Springer-Verlag, (1993).   Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[7]

S. Kamin, Heat propagation in an inhomogeneous medium,, Progress in Partial Differential Equations: the Metz Surveys \textbf{4}, 4 (1996), 229.   Google Scholar

[8]

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

[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.   Google Scholar

[10]

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

[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), 521.   Google Scholar

[12]

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

[13]

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

[14]

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

[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.   Google Scholar

[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.   Google Scholar

[17]

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

[18]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs. The Clarendon Press, (2007).   Google Scholar

show all references

References:
[1]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^n,$, Manuscripta Math., 74 (1992), 87.   Google Scholar

[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.   Google Scholar

[3]

E. DiBenedetto, "Degenerate Parabolic Equations,'', Springer-Verlag, (1993).   Google Scholar

[4]

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

[5]

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

[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.   Google Scholar

[7]

S. Kamin, Heat propagation in an inhomogeneous medium,, Progress in Partial Differential Equations: the Metz Surveys \textbf{4}, 4 (1996), 229.   Google Scholar

[8]

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

[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.   Google Scholar

[10]

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

[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), 521.   Google Scholar

[12]

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

[13]

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

[14]

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

[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.   Google Scholar

[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.   Google Scholar

[17]

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

[18]

J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'', Oxford Mathematical Monographs. The Clarendon Press, (2007).   Google Scholar

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