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

• DOI: 10.3934/dcds.2010.26.521
• Publication Date: 2009-11-04T23:14:20Z
• Authors: Shoshana Kamin; ; Guillermo Reyes; Juan Luis Vázquez
• Keywords: 0101 mathematics; 01 natural sciences

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 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.