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

This paper investigates the repulsive chemotaxis-consumption model {align*} ∂_t u &= ∇ · (D(u) ∇ u) + ∇ · (u ∇ v), \\ 0 &= Δ v - uv {align*} in an n-dimensional ball, n ≥ 3, where the diffusion coefficient D is an appropriate extension of the function 0≤ξ↦(1+ξ)ᵐ⁻¹ for some $m>0$. Under the boundary conditions {equation*} ν · (D(u) ∇ u + u ∇ v) = 0 and v = M>0,{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 < 2/n, the system exhibits blow-up behavior for sufficiently large M.