2025/05/06 by Tréton, Samuel, Zhang, Mingmin
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.03304
To better understand how populations respond to dynamic external pressure, we propose a new diffusion model in the moving half-line z ≥ 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) ∼ ct^β with β ∈ [0, 1]. We prove quantitative convergence results characterized by attraction toward self-similar profiles, based on entropy techniques and Duhamel's principle. When β 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 β = 1, reflecting a delicate balance between diffusion and advection induced by the moving boundary.