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}&=Δu-\nabla\!\cdot(u\nabla v),\ &x\in\mathbb{R}^n,& t>0,\\ 0 &=Δv+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^{-α},&\text{ if }\vert x\vert \leq R-ρ,\\ 0,&\text{ if } \vert x\vert\geq R+ρ,\\ \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 $\Vert u_0{\Vert }_{{\text{L}}^{\frac{n}{2}+{\delta }_0}(\Omega )},\Vert f{\Vert }_{{\text{L}}^{\frac{n}{2}+{\delta }_0}(\Omega )}$ and $\Vert v_0{\Vert }_{{\text{L}}^{\theta }(\Omega )}$, for some $\theta >n$, 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.

21 pages

References in corpus (1)

• Boundedness in a Keller-Segel system with external signal production