Decay of Mass of the Solution to the Cauchy Problem of the p-Laplacian with Absorption on Infinite Graphs

TL;DR

This paper investigates the decay behavior of solutions to the nonstationary discrete p-Laplacian with absorption on infinite graphs, establishing decay rates and bounds based on the properties of the inhomogeneous density function.

Contribution

It provides new decay rate results for solutions of the p-Laplacian on infinite graphs with inhomogeneous density, using energy inequalities and a novel embedding theorem.

Findings

Established decay rates for solutions when p > 2 and ta(x) is non-power.

Proved universal bounds when ta(x) decays rapidly.

Developed new energy inequalities and embedding results for analysis.

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.