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

Abstract

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 $\rho\in\left(0,\frac{R}{2}\right)$, 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 $\Omega$, where $f\in\text{L}^{\frac{n}{2}+\delta_0}(\Omega)\cap C^{\alpha}(\Omega)$ 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 $\|v_0\|_{\text{L}^{\theta}(\Omega)}$, for some $\theta>n$, are all bounded by some $\varepsilon>0$ 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.