GLOBAL SOLUTIONS OF REACTION-DIFFUSION SYSTEMS ON 1D-NETWORKS

Abstract : The purpose of this paper is to assess some results and the associated techniques for global existence of solutions of reaction-diffusion systems on networks. The motivation comes from the fact that phenomena can occur on ramified physical structures, of which one-dimensional networks are the simplest examples. We work in the setting of solutions provided by the classical semigroup theory. Local existence and uniqueness in this setting is ensured by the fixed-point argument, which is detailed. Construction of diffusion operators on networks via bilinear forms, generation of an analytic semigroup, ultracontractivity and maximal regularity properties, essential for the global existence analysis, are recalled or proved in detail, following in particular Mugnolo [25]. With these tools at hand, we exemplify the fact that fundamental results available in the literature on global existence and time asymptotics of reaction-diffusion systems extend from open domains of R n to networks. To be specific, here we deal with one dimensional networks with Kirchhoff conditions at nodes. In this setting, we first implement on a basic 2 × 2 example the celebrated L p-method of Martin-Pierre [22] which provided a large number of existence results in past 30 years. To give a second example, we revisit the system studied in Haraux-Youkana [17] where global existence results from construction of a Lyapunov function. Compactness properties of solutions, as t → +∞, are obtained adapting the arguments of Haraux-Kirane [16]. This permits to establish convergence of solutions to equilibria of the Haraux-Youkana systems on networks.
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Frédéric Kuczma. GLOBAL SOLUTIONS OF REACTION-DIFFUSION SYSTEMS ON 1D-NETWORKS. 2019. ⟨hal-02373355⟩

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