Boundedness in a Keller-Segel system with external signal production

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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 2$ and $f\in$\text{L}$^\infty\left([0,\infty);$\text{L}$^{\frac{n}{2}+\delta_0}(\Omega)\right)\cap C^\alpha(\Omega\times(0,\infty))$withsome$\alpha>0$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 ${\int }_{\mathrm{\Omega }}{u}_0\text{d}x<4\pi$ 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 $f\equiv ̸0$. Under certain smallness conditions imposed on the initial data and $f$ we furthermore show that for more general space dimension $n\ge 2$ 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.