GLOBAL BOUNDEDNESS IN A CHEMOTAXIS MODEL WITH SIGNAL-DEPENDENT MOTILITY AND LOGISTIC SOURCE WITH VARIABLE COEFFICIENTS

This paper deals with the following parabolic-parabolic chemotaxis system with growth source { u _ t = ∇· (γ (ε) ∇ u) - ∇· (χ (ε) u ∇v) + η (x) u - μ (x) u ^ α, x ∈Ω, t > 0,; v _ t = Δ v - v + u ^ β, x ∈Ω, t > 0, ]. under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂ℝ^N (N≥ 2), and the parameters β >0, α >1, and γ ,χ are positive motility functions. Moreover, we assume that η (x) ^ m + α - 1/α - 1/(μ (x) - μ_ *) ^ m/α - 1 is integrable for some m ≥ 1, where η(·) and μ(·) are positive measurable functions satisfying μ(x) > μ_* > 0 for almost every x ∈Ω. Under these conditions, if α > max{2β, 2}, then the system possesses a unique global bounded classical solution.