A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system

This paper deals with time-global solutions to the parabolic system {τu_t = Δu - ∇ · (u∇ χ (ν)) in Ω × (0, ∞), ν_t = Δν - ν + u in Ω × (0,∞), under the homogeneous Neumann boundary conditions in a bounded and convex domain Ω ⊂ ℝⁿ (n ≥ 2) with smooth boundary ∂Ω. Here τ is a positive parameter, χ is a smooth function on (0,∞) satisfying χ' > 0 and (u₀, ν₀) is a pair of nonnegative initial data. We will consider the above system as a perturbation of a nonlocal parabolic equation and establish a sufficient condition of the sensitivity function ? for the global existence of solutions under the assumption of smallness of the constant τ.