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

2020/11/30 by Nikolaos Michael Ladas, Ladas, Nikolaos Michael, John Christopher Meyer +1
Mathematics · #35A23 #35B50 #35B51 #35K20 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A23 #msc:35B50 #msc:35B51 #msc:35K20

paper · pdf · doi:10.48550/arxiv.2011.15058

12 pages, 0 figures

arxiv created 2020/11/30 · arxiv updated 2020/12/01

Abstract

We present extensions of the comparison and maximum principles available for nonlinear non-local integro-differential operators P:C^2,1(Ω× (0,T])× L^∞ (Ω× (0,T])→ℝ, of the form P[u] = L[u] -f(⋅ ,⋅ ,u,Ju) on Ω× (0,T]. Here, we consider: unbounded spatial domains Ω⊂ ℝⁿ, with T>0; sufficiently regular second order linear parabolic partial differential operators L; sufficiently regular semi-linear terms f:(Ω× (0,T]) × ℝ²→ℝ; and the non-local term Ju= ∫_Ωϕ(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.

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