# American Institute of Mathematical Sciences

December  2003, 2(4): 481-494. doi: 10.3934/cpaa.2003.2.481

## Free-boundary regularity for generalized porous medium equations

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

Received  April 2003 Revised  September 2003 Published  October 2003

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.

Citation: Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481
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