On a generalized diffusion problem: A complex network approach

Guillaume Cantin; Alexandre Thorel

doi:10.3934/dcdsb.2021135

Article Contents

2022, Volume 27, Issue 4: 2345-2365. Doi: 10.3934/dcdsb.2021135

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On a generalized diffusion problem: A complex network approach

Guillaume Cantin^, and
Alexandre Thorel

Laboratoire de Mathématiques Appliquées du Havre, Université du Havre Normandie, FR CNRS 3335, Institut des Systèmes Complexes de Normandie, 25, rue Philippe Lebon, BP 1123, 76063 Le Havre, Normandie, France

^* Corresponding author: Guillaume Cantin
^* Corresponding author: Guillaume Cantin

Received: February 2020

Revised: April 2021

Early access: May 2021

Published: April 2022

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Abstract

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

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Mathematics Subject Classification: Primary: 35A01, 35A02, 35B40; Secondary: 35K30, 35K57.

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Figure 1. Splitting of a rectangular domain $ \Omega $ into a grid of sub-domains $ \omega_1, \, \dots, \, \omega_n $ in the dimension case $ N = 2 $

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