Solutions of a non-local aggregation equation: Universal bounds, concavity changes, and efficient numerical solutions

Abstract

Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd. We consider a one-dimensional aggregation equation for a non-negative density (Formula presented.) associated with a quartic potential (Formula presented.) ((Formula presented.), (Formula presented.)). We show that for the case of symmetric initial data [(Formula presented.)], the solution of the aggregation equation can be expressed in terms of an explicit function of (Formula presented.), (Formula presented.), and (Formula presented.), where the functions (Formula presented.) and (Formula presented.) are determined by an ordinary differential equation initial value problem, the numerical solution of which is ..