Asymptotic behaviour of fast diffusions on graphs

Abstract

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in L^1 and L^2-type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.