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 $\Omega \subset {\mathbb{R}}^n$, $n\in \mathbb{N}$, where $\delta >0$ is a given parameter. If either $n\le 2$ or $\Vert v_0{\Vert }_{L^{\infty }(\Omega )}\le \frac{1}{3n}$ 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.

22 pages

References in corpus (1)

• Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system