$L^{p}$-$L^{q}$ estimates of the heat kernels on graphs with applications to a parabolic system

TL;DR

This paper derives sharp $L^{p}$-$L^{q}$ estimates for heat kernels on graphs with specific curvature and volume growth conditions, and applies these results to establish global solutions for a semilinear parabolic system.

Contribution

It provides the first systematic derivation of $L^{p}$-$L^{q}$ heat kernel estimates on graphs with curvature-dimension conditions and applies these to solve a nonlinear parabolic system.

Findings

Established sharp $L^{p}$ bounds for heat operators on graphs.

Proved decay estimates for heat kernels under geometric conditions.

Demonstrated existence of global solutions to a semilinear parabolic system.

Abstract

Let $G=(V, E)$ be a locally finite connected graph satisfying curvature-dimension conditions ($CDE(n, 0)$ or its strengthened version $CDE'(n, 0))$) and polynomial volume growth conditions of degree $m$. We systematically establish sharp $L^{p}$-bounds and decay-type $L^{p}$-$L^{q}$ estimates for heat operators on $G$, accommodating both bounded and unbounded Laplacians. The analysis utilizes Li-Yau-type Harnack inequalities and geometric completeness arguments to handle degenerate cases. As a key application, we prove the existence of global solutions to a semilinear parabolic system on $G$ under critical exponents governed by volume growth dimension $m$.