A piston to counteract diffusion: The influence of an inward-shifting boundary on the heat equation in half-space

TL;DR

This paper studies a diffusion process in a half-space with a moving boundary acting as a piston, analyzing how the boundary's speed influences the long-term behavior and shape of the population distribution.

Contribution

It introduces a new diffusion model with a moving boundary and provides quantitative convergence results, revealing a phase transition in asymptotic profiles based on boundary speed.

Findings

Convergence to self-similar profiles characterized by entropy methods.

Profile shape switches from Gaussian to exponential at critical boundary speed.

Stationary profile occurs when boundary moves at maximum speed.

Abstract

To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line {z $\ge$ b(t)}, where the boundary position b(t) is a given nondecreasing function of time. A Robin boundary condition is imposed at z = b(t) to prevent individuals from leaving the domain, so that the shifting boundary acts as an impermeable wall-a ''piston''-that sweeps the individuals it encounters. Our analysis focuses on the cases where b(t) $\sim$ ct^$\beta$ with $\beta$ $\in$ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When $\beta$ goes through the critical value 1/2, the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when $\beta$ = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.