Global boundedness of solutions in a parabolic–parabolic chemotaxis system with singular sensitivity

We consider a parabolic–parabolic Keller–Segel system of chemotaxis model with singular sensitivity: ut=Δu−χ∇⋅(uv∇v), vt=kΔv−v+uvt=kΔv−v+u under the homogeneous Neumann boundary condition in a smooth bounded domain Ω⊂RnΩ⊂Rn(n≥2)(n≥2), with χ,k>0χ,k>0. It is proved that for any k>0k>0, the problem admits global classical solutions, whenever χ∈(0,−k−12+12(k−1)2+8kn). The global solutions are moreover globally bounded if n≤8n≤8. This shows a way the size of the diffusion constant k of the chemicals v affects the behavior of solutions.