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

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.