98-735 J. A. Carrillo, G. Toscani

Exponential $L^1$-decay of solutions of the porous medium equation to self-similarity (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 Barenblatt-Pattle solution $V(|x|,t)$. We prove that the $L^1$-distance decays at a rate $t^{1/((N+2)m-N)}$ 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 Fokker-Planck equation.

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