A critical exponent for blow-up in a two-dimensional chemotaxis-consumption system

The repulsive chemotaxis-consumption system {ut=∇·(D(u)∇u)+∇·(u∇v),0=Δv-uv, is considered along with the boundary conditions (D(u) ∇ u+ u∇ v) · ν| _∂Ω= 0 and v| _∂Ω= M in a ball Ω ⊂ R² . Under the assumption that D suitably generalizes the function 0 ≤ ξ↦ (ξ+ 1) ^-α for some α> 0 , it is firstly shown that for each nontrivial radially symmetric u∈ W^1,∞(Ω) , one can find M_⋆(u) > 0 with the property that whenever M> M_⋆(u) , a corresponding initial-boundary value problem admits a classical solution blowing up in finite time. This is complemented by a second statement which asserts that when inf D and M are positive, for any such initial data a global bounded classical solution exists.