QUASICONVEXITY PRESERVING PROPERTY FOR FIRST ORDER NONLOCAL EVOLUTION EQUATIONS (Innovation of the theory for evolution equations: developments via cross-disciplinary studies)

Abstract

This note is a companion of our earlier paper [Kagaya-Liu-Mitake, 2023] to study the quasiconvexity preserving property of positive, spatially coercive viscosity solutions to a class of first order evolution equations with monotone nonlocal terms. We show that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. In contrast to our results in [Kagaya-Liu-Mitake, 2023], we focus only on the first order case, but slightly change our assumptions to allow more general dependence of the operator on the nonlocal term.