Free-boundary regularity for generalized porous medium equations

Panagiota Daskalopoulos; Eunjai Rhee

doi:10.3934/cpaa.2003.2.481

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2003, Volume 2, Issue 4: 481-494. Doi: 10.3934/cpaa.2003.2.481

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Free-boundary regularity for generalized porous medium equations

Panagiota Daskalopoulos^1, and
Eunjai Rhee^2,

1.
Department of Mathematics, Columbia University
2.
Deaprtment of Mathematics, University of California at Irvine Irvine, CA.

Received: March 31, 2003

Revised: August 31, 2003

Published: November 2003

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Abstract

We consider the Cauchy problem for the generalized porous medium equation

$u_t = \Delta \Phi(u)$ in R$^n \times [0,T]$

$u(x)=u_0(x)$ on R$^n$

with the nonlinearity $\Phi(u)$. For the case of $\Phi(u)=\sum_{i=1}^m c_i u^{\alpha_i}$, we show the existence of a solution which smoothness depends on the exponents $\alpha_i$. Regardless of the regularity of the solution, we show the free-boundary is smooth. We also extend similar results for $\Phi(u)$ as an infinite sum.

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Mathematics Subject Classification: 35Mxx.

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