A framework for non-local, non-linear initial value problems

TL;DR

This paper develops a comprehensive framework for analyzing non-local, non-linear initial value problems, establishing existence and uniqueness of solutions, and introducing new examples of such operators.

Contribution

It introduces a general framework for non-local, non-linear operators, including cases with solution-dependent jump intensities, and proves fundamental properties like existence and uniqueness.

Findings

Existence and uniqueness of bounded solutions for the studied problems.

Introduction of new examples of non-local, non-linear operators.

Analysis of properties of solutions under the proposed framework.

Abstract

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.