Evolution problems with perturbed $1$-Laplacian type operators on random walk spaces

TL;DR

This paper investigates evolution problems involving perturbed 1-Laplacian operators within random walk spaces, encompassing graph and nonlocal PDE frameworks, focusing on functionals with varying growth conditions across structures.

Contribution

It introduces a novel analysis of evolution problems with dual random walk structures and different growth behaviors, expanding the understanding of PDEs in complex random walk spaces.

Findings

Analysis of evolution problems with mixed growth conditions

Extension of PDE frameworks to combined random walk structures

Insights into the behavior of perturbed 1-Laplacian operators

Abstract

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $\mathbb{R}^N$. We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.