Stationary solutions with vacuum for a hyperbolic-parabolic chemotaxis model in dimension two

TL;DR

This paper investigates stationary solutions with vacuum in a 2D hyperbolic-parabolic chemotaxis model, identifying specific solution types that resemble vascular networks and proving nonexistence results for others.

Contribution

It provides the first analysis of vacuum solutions in a chemotaxis model with nonlinear pressure, including explicit solutions and nonexistence results.

Findings

Found two nontrivial stationary solutions with vacuum regions.

Identified solutions resembling vascular network structures.

Proved nonexistence of certain symmetric bump solutions.

Abstract

In this research, we study the existence of stationary solutions with vacuum to a hyperbolic-parabolic chemotaxis model with nonlinear pressure in dimension two that describes vasculogenesis. We seek solutions in the radial symmetric class of the whole space, in which the system will be reduced to a system of ODE's on $(0,\infty)$. The fundamental solutions to the ODE system are the Bessel functions of different types. We find two nontrivial solutions. One is formed by half bump (positive density region) starting at $r=0$ and a region of vacuum on the right. Another one is a full nonsymmetric bump away from $r=0$. These solutions bear certain resemblance to in vitro vascular network and the numerically produced structure by Gamba et al arXiv:cond-mat/0303468v1. We also show the nonexistence of full bump starting at $r=0$ and nonexistence of full symmetric bump away from $r=0$.