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On a generalized diffusion problem: A complex network approach

  • * Corresponding author: Guillaume Cantin

    * Corresponding author: Guillaume Cantin 
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  • 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.

    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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