Analysis of a chemotaxis model with indirect signal absorption

Abstract

We consider the chemotaxis model \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v), \\ v_t = Δv - vw, \\ w_t = -δw + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \in \mathbb N$, where $δ\gt 0$ is a given parameter. If either $n \le 2$ or $\|v_0\|_{L^\infty(Ω)} \le \frac1{3n}$ 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.