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

Sofía Nieto; Guillermo Reyes

doi:10.3934/cpaa.2013.12.1123

Article Contents

2013, Volume 12, Issue 2: 1123-1139. Doi: 10.3934/cpaa.2013.12.1123

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

Sofía Nieto^1, and
Guillermo Reyes^2,

1.
Departamento de Matemáticas. Universidad Autónoma de Madrid, Cantoblanco. 28049 Madrid
2.
Depto. de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés 28911, Madrid

Received: June 30, 2011

Revised: April 30, 2012

Published: February 2013

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Abstract

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.

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Mathematics Subject Classification: 35K15, 35K65, 35B40.

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