American Institute of Mathematical Sciences

March  2009, 8(2): 493-508. doi: 10.3934/cpaa.2009.8.493

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

Received  February 2008 Revised  August 2008 Published  December 2008

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

(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We assume that $m> 1$ (slow diffusion) and $\rho(x)$ is positive, bounded and behaves like $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with $0\le \gamma<2$. The data $u_0$ are assumed to be nonnegative and such that $\int \rho(x)u_0 dx< \infty$.
Our asymptotic analysis leads to the associated singular equation $|x|^{-\gamma}u_t= \Delta u^m,$ which admits a one-parameter family of selfsimilar solutions $U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which are source-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. We show that these solutions provide the first term in the asymptotic expansion of generic solutions to problem (P) for large times, both in the weighted $L^1$ sense

$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$

and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty$ as $t\to \infty$ for the explicit rate $\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decay rate of $U_E$. For a given solution, the proper choice of the parameter is $E=\int \rho(x)u_0 dx$.

Citation: Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493
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