In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation \[ u_t= \Delta u^m, \quad m > 1, \] usually called the porous medium equation. More precisely, we provide sharp regularity estimates for bounded nonnegative weak solutions along the free boundary $\partial\{u > 0\}$, when the equation is universally close to the heat equation. As a consequence, local Lipschitz estimates are also established for this scenario.

中文翻译：

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热方程附近多孔介质方程的更高正则性估计

在本文中，我们研究了非线性抛物线方程 \[ u_t= \Delta u^m, \quad m > 1, \] 通常称为多孔介质方程的解的规律性方面。更准确地说，当方程普遍接近热方程时，我们为沿自由边界 $\partial\{u > 0\}$ 的有界非负弱解提供了清晰的正则性估计。因此，也为这种情况建立了局部 Lipschitz 估计值。