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

Abstract

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for the fully parabolic chemotaxis-growth system $(u_{\epsilon }{)}_t$ $=\Delta u_{\epsilon }-\epsilon \nabla \cdot (u_{\epsilon }\nabla v_{\epsilon })+\mu u_{\epsilon }(1-u_{\epsilon })$, $(v_{\epsilon }{)}_t=\Delta v_{\epsilon }-v_{\epsilon }+u_{\epsilon },$ with positive small parameter $\epsilon >0$ in a bounded convex domain $\Omega \subset {\mathbb{R}}^n$ ($n\ge 1$) 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 \[ \sup_{t>0}\|u_\varepsilon(\cdot,t)-u(\cdot,t)\|_{L^\infty(Ω)} \leq C\varepsilon \quad for\ all\ \varepsilon\in(0,\varepsilon_0). \]