Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant

We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \begin{align*} \left{\begin{array}{c@{,}l@{\quad}l@{,}c} u_{t}&=\Delta u-\nabla!\cdot(u\nabla v),\ &x\in$\mathbb{R}$^n,& t>0,\ 0 &=\Delta v+u+f(x),
&x\in$\mathbb{R}$^n,& t>0,\ u(x,0)&=u_{0}(x),\ &x\in$\mathbb{R}$^n,& \end{array}\right. \end{align*} with prototypical external signal production \begin{align*} f(x):=\begin{cases} f_0\vert x\vert^{-\alpha},&$\text{ if }$\vert x\vert \leq R-\rho,\ 0,&$\text{ if }$ \vert x\vert\geq R+\rho,\ \end{cases} \end{align*} for $R\in (0,1)$ and $\rho \in (0,\frac{R}{2})$, which is still integrable but not of class ${\text{L}}^{\frac{n}{2}+{\delta }_0}({\mathbb{R}}^n)$ for some ${\delta }_0\in [0,1)$. For corresponding parabolic-parabolic Neumann-type boundary-value problems in bounded domains $\mathrm{\Omega }$, where $f\in {\text{L}}^{\frac{n}{2}+{\delta }_0}(\mathrm{\Omega })\cap C^{\alpha }(\mathrm{\Omega })$ for some ${\delta }_0\in (0,1)$ and $\alpha \in (0,1)$, it is known that the system does not emit blow-up solutions if the quantities $|u_0|{$\text{L}$^{\frac{n}{2}+\delta_0}(\Omega)}, |f|{$\text{L}$^{\frac{n}{2}+\delta_0}(\Omega)}$and$|v_0|_{$\text{L}$^{\theta}(\Omega)}$,forsome$\theta>n$, are all bounded by some $\epsilon >0$ small enough. We will show that whenever $f_0>\frac{2n}{\alpha }(n-2)(n-\alpha )$ and $u_0\equiv c_0>0$ in $\overline{B_1(0)}$, a measure-valued global-in-time weak solution to the system above can be constructed which blows up immediately. Since these conditions are independent of $R\in (0,1)$ and $c_0>0$, we will thus prove the criticality of ${\delta }_0=0$ for the existence of global bounded solutions under a smallness conditions as described above.