Phragmèn-Lindelöf type theorems for parabolic equations on infinite graphs

TL;DR

This paper extends Phragmèn-Lindelöf type principles to the parabolic equation $\rho\partial_t u-\Delta u=f$ on infinite weighted graphs with density $\rho>0$, establishing a PL principle via a time-dependent supersolution $Z$ and proving optimality of the corresponding density decay conditions tied to the outer degree. It then derives both existence of uniqueness for possibly unbounded solutions under these growth constraints and sharp nonuniqueness results when the decay is too weak, with concrete thresholds on lattices $\\mathbb Z^n$ and on spherically symmetric trees. The methodology blends a weak parabolic maximum principle, barrier constructions, and barrier-based expansion arguments to extend Euclidean PL theory to discrete networks, including explicit barriers on $\\mathbb Z^n$ and special cases like $\\mathbb Z^2$ and anti-trees. Overall, the work provides explicit geometric-growth criteria and optimality statements for the uniqueness of discrete parabolic diffusion with density on infinite graphs, advancing both the theory and potential applications to network dynamics and discrete spectral problems.

Abstract

We obtain the Phragmèn-Lindelöf principle on combinatorial infinite weighted graphs for the Cauchy problem associated to a certain class of parabolic equations with a variable density. We show that the hypothesis made on the density is optimal.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• Theorem 3.4
• Theorem 3.5
• Corollary 3.6
• Corollary 3.7
• Corollary 3.8