Calculus of Variations and Geometric Measure Theory

P. Miller - D. Fortunato - M. Novaga - S. Shvartsman - C. Muratov

Generation and motion of interfaces in a mass-conserving reaction-diffusion system

created by novaga on 29 Sep 2022
modified on 04 Oct 2022


Submitted Paper

Inserted: 29 sep 2022
Last Updated: 4 oct 2022

Year: 2022

ArXiv: 2210.00585 PDF


Reaction-diffusion models with nonlocal constraints naturally arise as limiting cases of coupled bulk-surface models of intracellular signalling. In this paper, a minimal, mass-conserving model of cell-polarization on a curved membrane is analyzed in the limit of slow surface diffusion. Using the tools of formal asymptotics and calculus of variations, we study the characteristic wave-pinning behavior of this system on three dynamical timescales. On the short timescale, generation of an interface separating high- and low-concentration domains is established under suitable conditions. Intermediate timescale dynamics is shown to lead to a uniform growth or shrinking of these domains to sizes which are fixed by global parameters. Finally, the long time dynamics reduces to area-preserving geodesic curvature flow that may lead to multi-interface steady state solutions. These results provide a foundation for studying cell polarization and related phenomena in biologically relevant geometries.