Boundedness in a parabolic-elliptic chemotaxis system with logistic source involving the exponents depending on the spatial variables and nonlinear signal productions

This paper considers the global boundedness of solutions to the parabolic-elliptic chemotaxis system with a logistic source involving exponents that depend on spatial variables and nonlinear signal production: {ut=Δu−χ∇⋅(u∇υ)+u(η−μuα(x)),(x,t)∈Ω×(0,T),0=Δυ−υ+uσ,(x,t)∈Ω×(0,T), where η≥0, μ, σ>0, and α:Ω→(0,∞) is a measurable function, subject to the homogeneous Neumann boundary conditions in a bounded domain RN(N≥1) with smooth boundary. We prove that if either σ0 or essinfx∈Ωα(x)χ>0 or σ=essinfx∈Ωα(x)≥2N with μ>Nσ−2Nσχ, χ>0 or χ<0, then the above system possesses a unique global bounded classical solution.