In this paper, we derive several differential Harnack estimates (also known as Li–Yau–Hamilton type estimates) for nonnegative solutions and positive solutions of the porous medium equation $f_t=f{f}_{xx}+f_x^2+f^2$, which is a nonlinear parabolic equation. Our derivations rely on an idea related to the parabolic maximum principle. We prove these estimates by different methods. As the applications of the estimates, Harnack inequalities are also obtained.

中文翻译：

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Harnack估计一种多孔介质方程

在本文中，我们导出了多孔介质方程的非负解和正解的几个差分Harnack估计（也称为Li–Yau–Hamilton型估计）。 $F_{Ť}=F{F}_{XX}+F_X^2+F^2$，这是一个非线性的抛物线方程。我们的推论基于与抛物线最大原理有关的思想。我们用不同的方法证明这些估计。作为估计的应用，还获得了Harnack不等式。