Boundedness in a fully parabolic chemotaxis system with singular sensitivity

This paper deals with a fully parabolic chemotaxis system ut=Δu−χ∇⋅(uv∇v), vt=Δv−v+uvt=Δv−v+u with singular sensitivity χv (χ>0χ>0) on a bounded domain Ω⊂RnΩ⊂Rn, n≥2n≥2. The main result solves the open problem of uniform-in-time boundedness of solutions for χ<2n, which was conjectured by Winkler [16].