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 &= Δv - 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*} ν\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 $0<M<2\mathrm{/}(n-2)$, 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 $M$.