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 &= v - uv align* in an n-dimensional ball, n 3, where the diffusion coefficient D is an appropriate extension of the function 0 (1+) ^m-1 for some m>0. Under the boundary conditions equation* (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 < 2n, the system exhibits blow-up behavior for sufficiently large M.