On a Parabolic–Elliptic system with chemotaxis and logistic type growth

We consider a nonlinear PDEs system of two equations of Parabolic–Elliptic type with chemotactic terms. The system models the movement of a biological population “u” towards a higher concentration of a chemical agent “w  ” in a bounded and regular domain Ω⊂RNΩ⊂RN for arbitrary N∈NN∈N. After normalization, the system is as followsut−Δu=−div(umχ∇w)+μu(1−uα), in ΩT=Ω×(0,T),−Δw+w=uγ, in ΩT, for some positive constants m, χ, μ, α and γ  , with positive initial datum u0u0 and Neumann boundary conditions.We study the range of parameters and constrains for which the solution exists globally in time. If eitherα>m+γ−1α>m+γ−1, orα=m+γ−1α=m+γ−1 and μ>Nα−22(m−1)+Nαχ, the solution exists globally in time. Moreover, ifα≥m+γ−1 and μ>2χ, and there exist positive constants u‾0 and u_0 such that 0