DYNAMICS IN A PARABOLIC-ELLIPTIC CHEMOTAXIS SYSTEM WITH LOGISTIC SOURCE INVOLVING EXPONENTS DEPENDING ON THE SPATIAL VARIABLES

We consider the parabolic-elliptic chemotaxis system with the exponents depending on the spatial variables logistic source and nonlinear signal production: ut = ∆u−χ∇·(u∇υ)+f(x, u), (x, t) ∈ Ω×(0, T), 0 = ∆υ−υ+uγ in a bounded domain Ω ⊂ RN (N ≥ 1) with smooth boundary, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ > 0, γ ≥ 1 and ∂ν∂ denotes the outward normal derivative on ∂Ω. The logistic function f fulfilling f(x, s) ≤ ηs− µsα(x)+1, η ≥ 0, µ > 0 for all s > 0 with f(x, 0) ≥ 0 ∀x ∈ Ω, where α : Ω → [1, ∞) is a measurable function. It is proved that if 1 ≤ α(x) < ∞ for all x ∈ Ω such that essinfx∈Ω α(x) > γ or essinfx∈Ω α(x) = γ with µ > χ, then there exists a nonnegative classical solution (u, υ) that is global-in-time and bounded. In addition, under the particular conditions γ = 1 and f(x, s) = µ(s − sα(x)+1), if µ is sufficiently large, the global bounded solution (u, υ) satisfies ∥u(·, t) − 1∥L∞(Ω) + ∥υ (·, t) − 1∥L∞(Ω) ≤ Ce− Nk+2 t for all t > 0 with k = min n χ42 12o , C > 0. The global-in-time existence and uniform-in-time boundedness of solutions are established under specific parameter conditions, which improves the known results.