Boundedness in a Keller-Segel system with external signal production

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.