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Asymptotic behaviour of fast diffusions on graphs. (English) Zbl 07286441

Summary: 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.

MSC:

47-XX Operator theory
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