On the pressureless damped Euler-Poisson equations with non-local forces: Critical thresholds and large-time behavior

TL;DR

This paper investigates the one-dimensional pressureless Euler-Poisson equations with damping and non-local forces, identifying critical thresholds for solution behavior and analyzing long-term dynamics in biological modeling contexts.

Contribution

It establishes a sharp threshold between finite-time breakdown and global existence, and derives explicit solutions to study asymptotic behavior.

Findings

Identified a critical threshold separating finite-time blow-up from global solutions.

Derived explicit solutions in Lagrangian coordinates for subcritical initial data.

Analyzed the long-time behavior of solutions in the subcritical regime.

Abstract

We analyse the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling 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.