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 $(u_{\varepsilon})_t$ $=Δu_{\varepsilon} - \varepsilon \nabla \cdot ( u_\varepsilon \nabla v_\varepsilon) + μu_\varepsilon(1 - u_\varepsilon)$, $(v_{\varepsilon})_t=Δv_\varepsilon -v_\varepsilon+u_\varepsilon,$ with positive small parameter $\varepsilon>0$ in a bounded convex domain $Ω\subset\mathbb{R}^n$ ($n\geq 1$) with smooth boundary. The solutions converge to the solution $u$ to the Fisher-KPP equation as $\varepsilon\to 0$. It is shown that for all $μ>0$ and any suitably regular nonnegative initial data $(u_{init},v_{init})$ there are some constants $\varepsilon_0>0$ and $C>0$ such that $\sup_{t>0}\|u_\varepsilon(\cdot,t)-u(\cdot,t)\|_{L^\infty(Ω)} \leq C\varepsilon \quad for\ all\ \varepsilon\in(0,\varepsilon_0).$

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