# Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model

## TL;DR

This paper studies the stability of traveling shock waves in a hyperbolic-parabolic chemotaxis model, introducing a relative entropy method that demonstrates contraction properties even for large initial disturbances when shock strength is small.

## Contribution

It develops a relative entropy framework for analyzing shock stability in a chemotaxis system, showing contraction properties independent of diffusion strength for small shocks.

## Key findings

- Relative entropy captures shock proximity in $L^2$-sense.
- Non-increasing entropy for large perturbations with small shocks.
- Contraction property holds regardless of diffusion strength.

## Abstract

We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.