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}&=Δu-\nabla\!\cdot(u\nabla v),\ &x\inΩ,& t>0,\\ v_{t}&=Δv-v+u+f(x,t),\ &x\inΩ,& t>0,\\ \frac{\partial u}{\partialν}&=\frac{\partial v}{\partialν}=0,\ &x\in\partialΩ,& t>0,\\ u(x,0)&=u_{0}(x),\ v(x,0)=v_{0}(x),\ &x\inΩ& \end{array}\right. \end{align*} in a smooth, bounded domain $Ω\subset\mathbb{R}^n$ with $n\geq2$ and $f\in\text{L}^\infty\left([0,\infty);\text{L}^{\frac{n}{2}+δ_0}(Ω)\right)\cap C^α(Ω\times(0,\infty))$ with some $α>0$ and $δ_0\in\left(0,1\right)$. 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 $\smallint_Ω u_0\text{d} x<4π$ 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.