Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network

Francesca R. Guarguaglini

doi:10.3934/nhm.2018003

Article Contents

2018, Volume 13, Issue 1: 47-67. Doi: 10.3934/nhm.2018003

This issue Previous Article Stochastic homogenization of maximal monotone relations and applications Next Article On a vorticity-based formulation for reaction-diffusion-Brinkman systems

Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network

Francesca R. Guarguaglini

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi di L'Aquila, Via Vetoio, I-67100 Coppito (L'Aquila), Italy

Received: October 31, 2016

Revised: September 30, 2017

Published: March 2018

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Abstract

This paper approaches the question of existence and uniqueness of stationary solutions to a semilinear hyperbolic-parabolic system and the study of the asymptotic behaviour of global solutions. The system is a model for some biological phenomena evolving on a network composed by a finite number of nodes and oriented arcs. The transmission conditions for the unknowns, set at each inner node, are crucial features of the model.

Keywords:

Nonlinear hyperbolic systems,
transmission conditions,
stationary solutions,
asymptotic behaviour of global solutions,
chemotaxis.

Mathematics Subject Classification: Primary: 35R02; Secondary: 35M33, 35L50, 35B40, 35Q92.

Citation:

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Figure 1. Example of acyclic network; the highlighted arcs form the path linking the nodes $N_4$ and $N_5$.

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Figure 2. Example: the highlighted arcs form the path from the outer point $e_1$ to the inner node $N_4$ and $I_5$ is an arc incident with $N_4$, not belonghing to the path.

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