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 $(u_{\epsilon }{)}_t$ $=\Delta u_{\varepsilon} - \varepsilon \nabla \cdot ( u_\varepsilon \nabla v_\varepsilon) + \mu u_\varepsilon(1 - u_\varepsilon)$,$ (v_{\varepsilon})t=\Delta v\varepsilon -v_\varepsilon+u_\varepsilon,$ with positive small parameter $\epsilon >0$ in a bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary. The solutions converge to the solution $u$ to the Fisher-KPP equation as $\epsilon \to 0$. It is shown that for all $\mu >0$ and any suitably regular nonnegative initial data $(u_{init},v_{init})$ there are some constants ${\epsilon }_0>0$ and $C>0$ such that $$\underset{t>0}{\sup }\mathrm{\parallel }{u}_{\epsilon }(\cdot ,t)-u(\cdot ,t){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\epsilon forall\epsilon \in (0,{\epsilon }_0)\mathrm{.}$$