Nonlinear Tempering of Subdiffusion with Chemotaxis, Volume Filling, and Adhesion

Abstract

The purpose of this paper is to implement nonlinear particle interactions into subdiffusive transport, involving chemotaxis and nonlinear effects such as volume filling and adhesion. We systematically derive nonlinear subdiffusive fractional equations with chemoattractant dependent forcing. We consider the diffusion limit of the master equation and analyse the role of nonlinear tempering in the stationary case. We study the interaction between attractive forces of anomalous aggregation and chemotaxis, with repulsive forces induced by nonlinear reactions. We show that this nonlinear interaction can prevent the phenomenon of anomalous aggregation when the local particle concentration grows too high. We also show that the effect of nonlinear tempering is to suppress the intermediate subdiffusive behaviour which results in a nonlinear advection diffusion equation in the long time limit.