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

TL;DR

This work develops sharp $L^{p}$-$L^{q}$ estimates for the heat semigroup on locally finite graphs under curvature-dimension conditions $CDE(n,0)$ or $CDE'(n,0)$ with polynomial volume growth of dimension $m$, encompassing both bounded and unbounded Laplacians. By employing Li–Yau-type gradient bounds and completeness arguments, the authors derive Gaussian-type heat-kernel bounds and temporal decay rates $t^{-m/2(1/p-1/q)}$, enabling robust $L^{p}$-$L^{q}$ control. As a key application, they study a weakly coupled semilinear parabolic system $u_t=\Delta u+v^{p}$, $v_t=\Delta v+u^{q}$ on graphs, proving local existence and global existence for subcritical exponents determined by $m$, and establishing a Fujita-type blow-up/global-dichotomy governed by volume growth. The results extend discrete diffusion analysis to graphs with unbounded diffusion, offering tools for global behavior of nonlinear parabolic dynamics in discrete spaces and contributing to the broader understanding of heat kernels on graphs.

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 $m$.

Theorems & Definitions (39)

• Theorem 1.1
• Definition 2.1
• Lemma 2.1: LCE
• Lemma 2.2: HL
• Lemma 2.3: HL
• Proposition 2.1: RW
• Proposition 2.2: BHLLMY
• Definition 2.2
• Definition 2.3
• Remark 2.1: BHLLMYHLLY