Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions

This paper investigates the repulsive chemotaxis-consumption model \begin{align*} \partial_t u &= \nabla \cdot (D(u) \nabla u) + \nabla \cdot (u \nabla v), \\ 0 &= \Delta v - uv \end{align*} in an $n$-dimensional ball, $n \ge 3$, where the diffusion coefficient $D$ is an appropriate extension of the function $0\le\xi\mapsto(1+\xi)^{m-1}$ for some $m>0$. Under the boundary conditions \begin{equation*} \nu \cdot (D(u) \nabla u + u \nabla v) = 0 \quad\text{ and }\quad v = M>0,\end{equation*} we first demonstrate that for $m > 1$, or $m = 1$ with $0 < M < 2/(n-2)$, the system admits globally defined classical solutions that are uniformly bounded in time for any choice of sufficiently smooth radial initial data. This result is further extended to the case $0<m<1$ when $M$ is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for $0 < m < \frac{2}{n}$, the system exhibits blow-up behavior for sufficiently large $M$.