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

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}$$,oftheform$P[u] = L[u] -f(\cdot ,\cdot ,u,Ju)$on$\Omega \times (0,T]$. Here, we consider: unbounded spatial domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$, with $T>0$; sufficiently regular second order linear parabolic partial differential operators $L$; sufficiently regular semi-linear terms $f:(\Omega \times (0,T]) \times $\mathbb{R}$^2\to$\mathbb{R}$$;andthenon-localterm$Ju= \int_{{\Omega}}\phi(x-y)u(y,t)dy$,with$\phi$ in a class of non-negative sufficiently summable kernels. We also provide examples illustrating the limitations and applicability of our results.