Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity

We consider the chemotaxis system with nonlinear diffusion and singular sensitivity ut=Δum−∇(uv∇v),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ωin a smooth bounded domain Ω⊂Rn, n≥2. In this work it is shown that for all reasonably regular initial data u0≥0 and v0>0, the corresponding Neumann initial-boundary value problem possesses a global generalized solution provided that m>1+n−22n.