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

Sophia Hertrich, Tao Huang, Diego Yépez, Kun Zhao

TL;DR

The paper analyzes stationary solutions with vacuum for a 2D hyperbolic-parabolic chemotaxis model with nonlinear pressure under radial symmetry, reducing the PDE system to ODEs on $ (0,\infty) $ whose solutions are expressed through Bessel-type functions. It constructs two nontrivial states: a half-bump near the origin and a full nonsymmetric bump away from the origin, and proves nonexistence results for a full bump starting at the origin and for fully symmetric bumps away from the origin. The approach hinges on classifying vacuum versus nonvacuum regions, solving linear ODEs with Bessel-type fundamental solutions, and carefully matching across transition radii via Wronskian-based continuity. Energy considerations show the constructed stationary states have negative energy, aligning with a variational perspective on pattern formation in vasculogenesis. Overall, the work provides explicit analytic configurations that reflect vascular-network-like structures and connects to prior numerical observations.

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 $r=0$.

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

TL;DR

The paper analyzes stationary solutions with vacuum for a 2D hyperbolic-parabolic chemotaxis model with nonlinear pressure under radial symmetry, reducing the PDE system to ODEs on whose solutions are expressed through Bessel-type functions. It constructs two nontrivial states: a half-bump near the origin and a full nonsymmetric bump away from the origin, and proves nonexistence results for a full bump starting at the origin and for fully symmetric bumps away from the origin. The approach hinges on classifying vacuum versus nonvacuum regions, solving linear ODEs with Bessel-type fundamental solutions, and carefully matching across transition radii via Wronskian-based continuity. Energy considerations show the constructed stationary states have negative energy, aligning with a variational perspective on pattern formation in vasculogenesis. Overall, the work provides explicit analytic configurations that reflect vascular-network-like structures and connects to prior numerical observations.

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 . 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 and a region of vacuum on the right. Another one is a full nonsymmetric bump away from . 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 and nonexistence of full symmetric bump away from .
Paper Structure (10 sections, 4 theorems, 167 equations)

This paper contains 10 sections, 4 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. History
  3. Results
  4. Derivation of equations
  5. General solutions
  6. Transitions between vacuum and positivity regions
  7. Half bump solutions at Lg
  8. Single whole bump solutions touching Lg
  9. Single whole bump solutions in Lg
  10. Energy estimates

Key Result

Proposition 4.1

For any solutions $\rho\in C^0$ and $\phi\in C^2$ of system steqn, suppose that there is transition point $\bar{r}>0$ between vacuum and positivity regions of density $\rho(r)$. Then, the solution $\phi(r)$ is $C^2$ at $\bar{r}$ if and only if $\phi(r)$ is $C^1$ at $\bar{r}$ and

Theorems & Definitions (6)

  • Proposition 4.1
  • Remark 4.2
  • Proposition 7.1
  • Proposition A.1
  • Remark A.2
  • Proposition A.3