Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks
Section snippets
Introduction and main results
Biological transport networks, such as leaf venation in plants, mammalian circulatory systems or neural networks, play a vital role in water and nutrients delivery or electronic charge transfer for living systems. In recent years, the mathematical modeling and in-depth analysis for biological transport networks have attracted wide attention [1], [2], [3], [4], [5], [6], [7], [8]. Particularly, based on the model of fluid transport networks [5], [9] and the Poisson–Nernst–Planck (PNP) model—see,
Global classical solution of the fully diffusive problem
The purpose of this section is to explore the global existence of classical solutions to the initial–boundary value problem (1.5). To this end, we begin with recalling the existence and regularity of solutions to the (elliptic) Neumann problem, see for instance [34], [35], [36]. Herein, the solution to the (elliptic) Neumann problem on is a function which satisfies for given function satisfying .
Lemma 2.1 For given function with , the
Proof of Theorem 1.1
In this section, we will present the boundedness and the asymptotic behavior of solutions to the initial–boundary value problem (1.5) with . In this case, , from Lemma 2.3, Lemma 2.4, Lemma 2.5, Lemma 2.6, we can easily get the following estimates.
Lemma 3.1 Assume that , and (1.6) is fulfilled. Then the solution to the problem (1.5) with satisfies that and there exists independent of and , such that
Proof of Theorem 1.2
In this section, we aim to prove the global existence of weak solutions to the partially diffusive problem (1.12), and to compute the convergence rates of the solution of the fully diffusive problem (1.5) toward the solution of the partially diffusive problem (1.12), as . Hereafter, without loss of generality, we set for convenience.
Acknowledgments
The authors are deeply indebted to the anonymous referees for their many useful comments which led to the present improved version of the paper as it stands. The research of BL is supported by China Postdoctoral Science Foundation (No. 2020M671282). The research of LX is supported by National Natural Science Foundation of China (No. 11701461), China Postdoctoral Science Foundation (No. 2017M622990, No. 2018T110956) and Chongqing Science and Technology Commission Project, PR China (No.
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