Asymptotics behaviour in one dimensional model of interacting particles

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional interacting particle model, identifying conditions under which solutions evolve into self-similar profiles, thus advancing understanding of particle interaction dynamics.

Contribution

It introduces a class of interaction kernels for which the solutions exhibit self-similar asymptotic behavior, providing new insights into the model's long-term dynamics.

Findings

Solutions tend to self-similar profiles over time

The asymptotic behavior depends on the interaction kernel class

Self-similar profiles are compactly supported

Abstract

We consider the equation u_t=\epsilon u_{xx}+(u\ K'*u)_x for x\in\mathbb{R}, t>0 and with \epsilon\geq 0, supplemented with a nonnegative, integrable initial datum. We present a class of interaction kernels K' such that the large time behaviour of solutions to this initial value problem is described by a compactly supported self-similar profile.