American Institute of Mathematical Sciences

April  2010, 26(2): 521-549. doi: 10.3934/dcds.2010.26.521

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

 1 School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel 2 Departamento de Matemática Aplicada, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid. 28040 Madrid 3 Departamento de Matemáticas and ICMAT. Universidad Autónoma de Madrid, Cantoblanco. 28049 Madrid, Spain

Received  March 2009 Revised  June 2009 Published  October 2009

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

$\rho(x)\, \partial_t u= \Delta u^m$ in $Q$:$=\mathbb{R}^N\times\mathbb{R}_+$
$u(x, 0)=u_0$ in $\mathbb{R}^N$

in dimensions $N\ge 3$. We assume that $m> 1$ and $\rho(x)$ is positive and bounded with $\rho(x)\le C|x|^{-\gamma}$ as $|x|\to\infty$ with $\gamma>2$. The initial data $u_0$ are nonnegative and have finite energy, i.e., $\int \rho(x)u_0 dx< \infty$.
We show that in this case nontrivial solutions to the problem have a long-time universal behavior in separate variables of the form

$u(x,t)$~$t^{-1/(m-1)}W(x),$

where $V=W^m$ is the unique bounded, positive solution of the sublinear elliptic equation $-\Delta V=c\,\rho(x)V^{1/m}$, in $\mathbb{R}^N$ vanishing as $|x|\to\infty$; $c=1/(m-1)$. Such a behavior of $u$ is typical of Dirichlet problems on bounded domains with zero boundary data. It strongly departs from the behavior in the case of slowly decaying densities, $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$ with 0 ≤ $\gamma<2$, previously studied by the authors.
If $\rho(x)$ has an intermediate decay, $\rho$~$|x|^{-\gamma}$ as $|x|\to\infty$ with $2<\gamma<\gamma_2$:$=N-(N-2)/m$, solutions still enjoy the finite propagation property (as in the case of lower $\gamma$). In this range a more precise description may be given at the diffusive scale in terms of source-type solutions $U(x,t)$ of the related singular equation $|x|^{-\gamma}u_t= \Delta u^m$. Thus in this range we have two different space-time scales in which the behavior of solutions is non-trivial. The corresponding results complement each other and agree in the intermediate region where both apply, thus providing an example of matched asymptotics.

Citation: 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 - A, 2010, 26 (2) : 521-549. doi: 10.3934/dcds.2010.26.521
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