Global boundedness to a chemotaxis system with modified nonlocal and nonlinear reaction

This paper deals with the following Neumann initial-boundary value problem {ut=Δu−∇⋅(u∇υ)+uγ−uα(∫Ωuβ)m,x∈Ω,t∈(0,Tmax),υt=Δυ−υ+uη,x∈Ω,t∈(0,Tmax),where Ω⊂RN (N≥ 2) is a bounded domain with smooth boundary, 0<m≤ 1, mβ≥ 1, β>1, 1<γ≤α, and η>0. The initial data satisfy u0,υ0∈W1,∞(Ω),where u0≥ 0, υ0≥ 0, and ∂νu0=∂νυ0=0 on ∂Ω. We rigorously prove that the problem admits a unique global classical solution which is uniformly bounded in time if one of the following conditions holds: α+mβ>(N+2)γ−N2 when γ≥η+1,or α+mβ>(N+2)η+22 when γ<η+1.