Boundedness in a Keller-Segel system with external signal production

TL;DR

This paper investigates the boundedness and global existence of solutions in a Keller-Segel chemotaxis system with external signal production, extending classical results to more general conditions including non-zero external signals.

Contribution

It extends the critical mass result for Keller-Segel systems to cases with external signal production and non-constant external signals, providing new conditions for global bounded solutions.

Findings

Global bounded solutions for 2D with subcritical mass.

Extension of critical mass result to systems with external signals.

Conditions for global boundedness in higher dimensions with small initial data.

Abstract

We study the Neumann initial-boundary problem for the chemotaxis system \begin{align*} \left\{\begin{array}{c@{\,}l@{\quad}l@{\,}c} u_{t}&=\Delta u-\nabla\!\cdot(u\nabla v),\ &x\in\Omega,& t>0,\\ v_{t}&=\Delta v-v+u+f(x,t),\ &x\in\Omega,& t>0,\\ \frac{\partial u}{\partial\nu}&=\frac{\partial v}{\partial\nu}=0,\ &x\in\partial\Omega,& t>0,\\ u(x,0)&=u_{0}(x),\ v(x,0)=v_{0}(x),\ &x\in\Omega& \end{array}\right. \end{align*} in a smooth, bounded domain $\Omega\subset\mathbb{R}^n$ with $n\geq2$ and $f\in\text{L}^\infty\left([0,\infty);\text{L}^{\frac{n}{2}+\delta_0}(\Omega)\right)\cap C^\alpha(\Omega\times(0,\infty))$ with some $\alpha>0$ and $\delta_0\in\left(0,1\right)$. First we prove local existence of classical solutions for reasonably regular initial values. Afterwards we show that in the case of $n=2$ and $f$ being constant in time, requiring the nonnegative initial data $u_0$ to fulfill the property $\smallint_{\Omega} u_0\text{d} x<4\pi$ ensures that the solution is global and remains bounded uniformly in time. Thereby we extend the well known critical mass result by Nagai, Senba and Yoshida for the classical Keller-Segel model (coinciding with $f\equiv 0$ in the system above) to the case $f\not\equiv 0$. Under certain smallness conditions imposed on the initial data and $f$ we furthermore show that for more general space dimension $n\geq2$ and $f$ not necessarily constant in time, the solutions are also global and remain bounded uniformly in time. Accordingly we extend a known result given by Winkler for the classical Keller-Segel system to the present situation.