Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

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Abstract

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.

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 (0,1) is a function u:[0,1]↦R which satisfies −uxx=f,x∈(0,1),ux|x=0,1=0,for given function f∈L2 satisfying f¯=0.

Lemma 2.1

For given function f∈L2 with f¯=0, 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 S≡0. In this case, ‖S‖L∞=0, 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 γ>12, ε>0 and (1.6) is fulfilled. Then the solution (ϱ,ψ,m) to the problem (1.5) with S≡0 satisfies that ϱ(t)≥0 and there exists C>0 independent of ε and t, such that ‖ϱ(t)‖L1(Ω)=‖ϱ0‖L1,t>0,∫01ϱlnϱ+12ψx2+1edx+∫0t∫012(ϱ−μ0)2+ϱψx2+ϱ

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 ε→0. Hereafter, without loss of generality, we set ε∈(0,1) 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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