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

This paper investigates the repulsive chemotaxis-consumption model ∂_tu=∇⋅(D(u)∇u)+∇⋅(u∇v),0=Δv−uv, 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 m>0. Under the boundary conditions ν⋅(D(u)∇u+u∇v)=0 and v=M>0, we demonstrate that for m>1, or m=1 and 0<M<2/(n−2), the system admits globally bounded classical solutions 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 [Formula presented], the system exhibits blow-up behavior for sufficiently large M.