Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source

In this paper we study the global existence of solutions to the fully parabolic chemotaxis system: u _t =Δu−χ∇⋅([Formula presented]∇v)+f(u), v _t =Δv−v+u in a smooth bounded domain Ω⊂R ⁿ (n≥3) subject to the non-flux boundary conditions, where χ>0 and the logistic function f∈C ¹ [0,∞) satisfies f(s)≤r−μs ^γ with r≥0 and γ,μ>0. It is shown that the problem possesses a global and classical solution as long as γ>2. Moreover, the global existence of the weak solution is also established provided that f(s)=rs−μs ² with any r,μ>0.