Asymptotic estimates for the p-Laplacian on infinite graphs with decaying initial data

TL;DR

This paper studies the long-term behavior and bounds of solutions to the discrete p-Laplacian on infinite graphs with decaying initial data, providing optimal estimates and insights into mass confinement and propagation speed.

Contribution

It introduces new optimal bounds and techniques for analyzing the p-Laplacian on infinite graphs, extending continuous PDE methods to discrete settings.

Findings

Established optimal sup and gradient bounds for solutions.

Derived sharp estimates for mass confinement and propagation speed.

Extended analysis to cases with infinite initial mass.

Abstract

We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and also sharp evaluation for the confinement of mass, i.e., the effective speed of propagation. We provide estimates for some moments of the solution, defined using the distance from a given vertex. Our technique relies on suitable inequalities of Faber-Krahn type, and looks at the local theory of continuous nonlinear partial differential equations. As it is known, however, not all of this approach can have a direct counterpart in graphs. A basic tool here is a result connecting the supremum of the solution at a given positive time with the measure of its level sets at previous times. We also consider the case of slowly decaying initial data, where the total mass is infinite.