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

2025/05/12 by Yuanyang Hu, Hu, Yuanyang
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.07565

openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

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Related

• Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs
• Heat kernel estimates and related inequalities on metric graphs
• Solvability of the L^p Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph
• Semilinear heat equations and parabolic variational inequalities on graphs
• Life span of solutions to a semilinear parabolic equation on locally finite graphs