Propagation for KPP bulk-surface systems in a general cylindrical domain

TL;DR

This paper studies the propagation speed of KPP bulk-surface systems in cylindrical domains with heterogeneous coefficients, providing new theoretical insights and numerical shape optimization results that extend previous radial homogeneous cases.

Contribution

It introduces a general framework for analyzing spreading speeds in complex domains and coefficients, extending prior radial homogeneous results with new asymptotic and numerical analyses.

Findings

Spreading speed characterized by principal eigenvalues of elliptic operators

Dependence of speed on diffusion rates and domain shape analyzed

Numerical shape optimization shows disks as extremizers under certain conditions

Abstract

In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer.