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