American Institute of Mathematical Sciences

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

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), 87-106.  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), 663-687.  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), 309-318.  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), 825-830.  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), 199-233.  Google Scholar [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.  Google Scholar [8] S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120. 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), 279-298.  Google Scholar [10] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121.  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), Special Volume on Parabolic Problems, 521-549.  Google Scholar [12] S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogenous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.  Google Scholar [13] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.  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), 337-351.  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), 1275-1294.  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), 493-508.  Google Scholar [17] J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.  Google Scholar [18] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

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References:
 [1] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^n,$ Manuscripta Math., 74 (1992), 87-106.  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), 663-687.  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), 309-318.  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), 825-830.  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), 199-233.  Google Scholar [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.  Google Scholar [8] S. Kamin and R. Kersner, Disappearance of interfaces in finite time, Meccanica, 28 (1993), 117-120. 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), 279-298.  Google Scholar [10] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121.  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), Special Volume on Parabolic Problems, 521-549.  Google Scholar [12] S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogenous medium, Comm. Pure Appl. Math., 34 (1981), 831-852.  Google Scholar [13] S. Kamin and P. Rosenau, Nonlinear diffusion in a finite mass medium, Comm. Pure Appl. Math., 35 (1982), 113-127.  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), 337-351.  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), 1275-1294.  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), 493-508.  Google Scholar [17] J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.  Google Scholar [18] J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar
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