On the pressureless damped Euler–Poisson equations with quadratic confinement: Critical thresholds and large-time behavior

Abstract

We analyze the one-dimensional pressureless Euler–Poisson equations with linear damping and nonlocal interaction forces. These equations are relevant for modeling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.