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.
Categories: math.AP