The field-road diffusion model: fundamental solution and asymptotic behavior

TL;DR

This paper derives explicit fundamental solutions and decay estimates for a linear PDE system modeling fast diffusion in population dynamics, involving complex geometry and boundary conditions.

Contribution

It provides an explicit fundamental solution and decay estimates for the linear field-road PDE system, advancing understanding of its asymptotic behavior.

Findings

Explicit fundamental solution derived using Fourier-Laplace transform.

Decay rate estimates for the $L^{ abla}^\infty$ norm of solutions.

Analysis applicable to complex geometries in population models.

Abstract

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the $L^{\infty}$ norm of these solutions.