A Piston to Counteract Diffusion: The Influence of an Inward-Shifting Boundary on the Heat Equation in Half-Space

Abstract. To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line [Formula: see text], where the boundary position [Formula: see text] is a given nondecreasing function of time. A Robin boundary condition is imposed at [Formula: see text] 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 [Formula: see text] with [Formula: see text]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel’s principle. When [Formula: see text] goes through the critical value [Formula: see text], the shape of the self-similar asymptotic profile switches from Gaussian to exponential. In particular, this profile turns out to be stationary when [Formula: see text], reflecting a delicate balance between diffusion and advection induced by the moving boundary.