How far does small chemotactic interaction perturb the Fisher-KPP dynamics?

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for the fully parabolic chemotaxis-growth system,{(uε)t=Δuε−ε∇⋅(uε∇vε)+μuε(1−uε),x∈Ω,t>0,(vε)t=Δvε−vε+uε,x∈Ω,t>0, with positive small parameter ε>0 in a bounded convex domain Ω⊂Rn (n≥1) with smooth boundary. The solutions converge to the solution u to the Fisher-KPP equation as ε→0. It is shown that for all μ>0 and any suitably regular nonnegative initial data (uinit,vinit) there are some constants ε0>0 and C>0 such thatsupt>0⁡‖uε(⋅,t)−u(⋅,t)‖L∞(Ω)≤Cεforallε∈(0,ε0).