Comparison principles for a class of nonlinear non-local integro-differential operators on unbounded domains

TL;DR

This paper extends comparison and maximum principles to nonlinear non-local integro-differential operators on unbounded domains, broadening their applicability in analysis of such operators.

Contribution

It introduces new comparison principles for nonlinear non-local operators on unbounded domains, including conditions on kernels and operators, with illustrative examples.

Findings

Extended comparison principles for unbounded domains.

Applicable to operators with specific kernel conditions.

Demonstrated limitations and potential applications.

Abstract

We present extensions of the comparison and maximum principles available for nonlinear non-local integro-differential operators $P:\mathcal{C}^{2,1}(\Omega \times (0,T])\times L^\infty (\Omega \times (0,T])\to\mathbb{R}$, of the form $P[u] = L[u] -f(\cdot ,\cdot ,u,Ju)$ on $\Omega \times (0,T]$. Here, we consider: unbounded spatial domains $\Omega \subset \mathbb{R}^n$, with $T>0$; sufficiently regular second order linear parabolic partial differential operators $L$; sufficiently regular semi-linear terms $f:(\Omega \times (0,T]) \times \mathbb{R}^2\to\mathbb{R}$; and the non-local term $Ju= \int_{{\Omega}}\phi(x-y)u(y,t)dy$, with $\phi$ in a class of non-negative sufficiently summable kernels. We also provide examples illustrating the limitations and applicability of our results.