For any α > 0, we study k α -type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in C ∞ -norm. Other relevant k α -type nonlocal flow is also discussed when α ≥ 1.
A classical nonlocal curvature flow preserving the enclosed area is reinvestigated. The uniform upper bound and lower bound of curvature are established for the first time. As a result, a detailed proof is presented for the asymptotic behavior of the flow.
We investigate a quasi-linear parabolic problem with nonlocal absorption, for which the comparison principle is not always available. The sufficient conditions are established via energy method to guarantee solution to blow up or not, and the long time behavior is also characterized for global solutions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.