Gradient estimates for the weighted porous medium equation on graphs

Abstract

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$\Delta u^m=\delta (x)u_t+\psi u^m$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].

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