# Analysis of a chemotaxis model with indirect signal absorption

## TL;DR

This paper proves the global existence and convergence to equilibrium of solutions for a chemotaxis model with indirect signal absorption in certain dimensions and initial conditions, introducing a novel functional inequality.

## Contribution

It establishes the first global existence results for this chemotaxis model with indirect absorption, using a new sublinear functional inequality.

## Key findings

- Global classical solutions exist for n ≤ 2 or small initial v.
- Solutions converge to a spatially constant equilibrium.
- A new functional inequality is derived for the analysis.

## Abstract

We consider the chemotaxis model \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - vw, \\ w_t = -\delta w + u \end{cases} \end{align*} in smooth, bounded domains $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, where $\delta \gt 0$ is a given parameter. If either $n \le 2$ or $\|v_0\|_{L^\infty(\Omega)} \le \frac1{3n}$ we show the existence of a unique global classical solution $(u, v, w)$ and convergence of $(u(\cdot, t), v(\cdot, t), w(\cdot, t))$ towards a spatially constant equilibrium, as $t \to \infty$. The proof of global existence for the case $n \le 2$ relies on a bootstrap procedure. As a starting point we derive a functional inequality for a functional being sublinear in $u$, which appears to be novel in this context.