Gradient estimates for the heat equation on graphs

Let $G(V, E)$ be an infinite (locally finite) graph which satisfies $CDE (m, -K)$ condition for some $m \gt 0, K \gt 0$. In this paper we mainly establish a generalized gradient estimate for positive solutions to the following heat equation\[(\partial_t - \Delta_{\mu}) u = 0 \quad \textrm{,}\]this gradient estimate includes Davies’ estimate, Hamilton’s estimate, Bakry–Qian’s estimate and Li–Xu’s estimate, these four estimates for positive solutions to the linear heat equation had been established on complete manifolds with Ricci curvature bounded from below by a negative number. When $t \searrow 0$, the dominant terms of Hamilton, Bakry–Qian and Li–Xu’s estimates are consistent with the corresponding term on the case that $K = 0$. We can also derive the Harnack inequalities from the gradient estimates, and then obtain the heat kernel estimates.

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The authors were supported by the NSF of Jiangsu Province (BK20141235).

Received 21 June 2016

Accepted 24 April 2017

Published 8 October 2019