Stationary States and Asymptotic Behavior of Aggregation Models with Nonlinear Local Repulsion

Abstract

We consider a continuum aggregation model with nonlinear local repulsion given by a degenerate power-law diffusion with general exponent. The steady states and their properties in one dimension are studied both analytically and numerically, suggesting that the quadratic diffusion is a critical case. The focus is on finite-size, monotone, and compactly supported equilibria. We also numerically investigate the long time asymptotics of the model by simulations of the evolution equation. Issues such as metastability and local/global stability are studied in connection to the gradient flow formulation of the model.