Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source

This paper deals with a fully parabolic chemotaxis system with nonlinear logistic source(0.1){ut=Δu−χ∇⋅(u∇v)+u(1−μur−1),vt=Δv−v+u, under homogeneous Neumann boundary conditions in a smooth bounded convex domain RN, with parameters μ,χ>0,r≥2. It is shown that if r>2 or r=2 and μ>Nχ4, then for all sufficiently smooth initial data, the associated initial-boundary-value problem (0.1) possesses a unique global-in-time classical solution that is bounded in Ω×(0,∞), which satisfieslimsupt→∞‖u(⋅,t)‖L∞(Ω)≤μ(maxt≥0⁡(Nχ4μt2+rμt−tr))min⁡{r−1,2}. Moreover, with the assumption u0≢0 and appropriate growth assumptions, the globally asymptotical stability of ((1μ)1r−1,(1μ)1r−1) is established.