Global classical solution to a chemotaxis consumption model with singular sensitivity

In this paper, we are concerned with the chemotaxis consumption model with singular sensitivity ut=Δu−∇⋅(f(u)v∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥1), for 00 and α<2. It is shown that for any sufficiently smooth initial data, the above system admits a global classical solution when either n=1andα<2, or n≥2andα<1−n4.