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


School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel


Departamento de Matemática Aplicada, E.T.S.I. de Caminos, Canales y Puertos, Universidad Politécnica de Madrid. 28040 Madrid


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

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