Global boundedness to a chemotaxis system with singular sensitivity and logistic source

We consider the parabolic-parabolic Keller–Segel system with singular sensitivity and logistic source: $$ u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v) +ru-\mu u^2$$ut=Δu-χ∇·(uv∇v)+ru-μu2, $$v_t=\Delta v-v+u$$vt=Δv-v+u under the homogeneous Neumann boundary conditions in a smooth bounded domain $$\Omega \subset \mathbb {R}^2$$Ω⊂R2, $$\chi ,\mu >0$$χ,μ>0 and $$r\in \mathbb {R}$$r∈R. It is proved that the system exists globally bounded classical solutions if $$r>\frac{\chi ^2}{4}$$r>χ24 for $$0<\chi \le 2$$0\chi -1$$r>χ-1 for $$\chi >2$$χ>2.