Large time behavior of the solution to the Cauchy problem for the discrete p-Laplacian with density on infinite graphs

TL;DR

This paper investigates the long-term behavior of solutions to the discrete p-Laplacian Cauchy problem on infinite graphs with density, establishing decay rates and bounds based on the density's properties.

Contribution

It provides new results on the decay rates and bounds for solutions when the density function varies, using novel energy inequalities and embedding techniques.

Findings

Established precise decay rates for solutions with non-power density functions.

Proved universal bounds when the density diminishes rapidly.

Extended understanding of the p-Laplacian behavior on infinite graphs.

Abstract

We consider the Cauchy problem for the nonstationary discrete p-Laplacian with inhomogeneous density \r{ho}(x) on an infinite graph which supports the Sobolev inequality. For nonnegative solutions when p > 2, we prove the precise rate of stabilization in time, provided \r{ho}(x) is a non-power function. When p > 2 and \r{ho}(x) goes to zero fast enough, we prove the universal bound. Our technique relies on suitable energy inequalities and a new embedding result.