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Free-boundary regularity for generalized porous medium equations

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  • 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.

    Mathematics Subject Classification: 35Mxx.


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