Global weak solutions in a chemotaxis system with large singular sensitivity

We study nonnegative radially symmetric solutions of the chemotaxis system equation(⋆){ut=Δu−χ∇⋅(uv∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in a ball Ω⊂RnΩ⊂Rn, n≥2n≥2, with parameter χ>0χ>0 and radially symmetric initial data u0∈C0(Ω̄) and v0∈W1,∞(Ω)v0∈W1,∞(Ω) satisfying u0≥0u0≥0 and v0>0v0>0 in Ω̄.A generalized solution concept is introduced for the Neumann problem associated with (⋆)(⋆), and within this concept global-in-time solutions are shown to exist regardless of the size of χ>0χ>0. This extends previous results which assert global existence of some weak solutions in the case χ0t>0 and each θ