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 $Ju={\int }_{\Omega }\varphi (x-y)u(y,t)dy$, with $\varphi$ in a class of non-negative sufficiently summable kernels. We also provide examples illustrating the limitations and applicability of our results.

12 pages, 0 figures