Cell polarisation model : the 1D case

TL;DR

This paper analyzes a one-dimensional non-linear, non-local drift-diffusion model, revealing how initial mass influences long-term behavior, including convergence to profiles or finite-time blow-up, using advanced inequalities and comparison principles.

Contribution

It introduces a comparison principle for the integrated form of the equation, enabling blow-up proofs without monotonicity assumptions, and provides convergence rates using Sobolev inequalities.

Findings

Subcritical and critical mass cases show convergence to profiles.

Supercritical mass leads to finite-time blow-up.

Comparison principle aids in analyzing blow-up without monotonicity.

Abstract

We study the dynamics of a one-dimensional non-linear and non-local drift-di usion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self similar pro le, to a steady state of nite time blow up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the cases subcritical and critical mass. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.