A method of moving planes for a mixed local-nonlocal parabolic system with quenching

Abstract

In this paper we develop a parabolic version of the method of moving planes on systems with a mixed local and nonlocal diffusion operator, that being L=−(−Δ)s+Δ, where (−Δ)s is the integral fractional laplacian. We then use this result to study the quenching phenomena occurring in said systems with a singular absorption term of the type (1−u)−p. We prove that there exists a range of multiplicative parameters for which every solution presents quenching, and that for any initial datum u0, there exists a small enough multiplicative parameter such as the solution is globally defined. We also prove that the quenching phenomena cannot happen arbitrarily close to the boundary, but they are contained in a compact subset of the domain.