 98735 J. A. Carrillo, G. Toscani
 Exponential $L^1$decay of solutions of the porous medium equation
to selfsimilarity
(298K, Postscript)
Nov 30, 98

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Abstract. We consider the flow of gas in an $N$dimensional porous medium with initial density $v_0(x)\geq 0$.
The density $v(x,t)$ then satisfies the nonlinear degenerate parabolic equation $v_t = \Delta v^m$
where $m>1$ is a physical constant. Assuming that $\int (1 + x^2)v_0(x)dx <\infty$, we prove that
$v(x,t)$ behaves asymptotically, as $t \to \infty$, like the BarenblattPattle solution $V(x,t)$.
We prove that the $L^1$distance decays at a rate $t^{1/((N+2)mN)}$ which is sharp. Moreover, if $N
=1$, we obtain an explicit time decay for the $L^\infty$distance at a suboptimal rate. The method we
use is based on recent results we obtained for the FokkerPlanck equation.
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