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

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of solutions to a hyperbolic-parabolic chemotaxis model on a network, emphasizing the role of transmission conditions at network nodes.

Contribution

It provides new results on stationary solutions and asymptotic behavior for a chemotaxis model on networks, considering the impact of transmission conditions at nodes.

Findings

Existence and uniqueness of stationary solutions established.

Asymptotic behavior of solutions analyzed.

Transmission conditions at nodes are crucial for solution properties.

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.