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