A degenerate chemotaxis system with flux limitation: Finite-time blow-up

Abstract

This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \begin{equation} \left\{ \begin{array}{l} \displaystyle u_t=\nabla \cdot \Big(\frac{u\nabla u}{\sqrt{u^2+|\nabla u|^2}}\Big) - \chi \, \nabla \cdot \Big(\frac{u\nabla v}{\sqrt{1+|\nabla v|^2}}\Big), \\[1mm] 0=\Delta v - \mu + u, \end{array} \right. \qquad \qquad (\star) \end{equation} under the initial condition $u|_{t=0}=u_0>0$ and no-flux boundary conditions in a ball $\Omega\subset R^n$, where $\chi>0$ and $\mu:=\frac{1}{|\Omega|} \int_\Omega u_0$. A previous result [3] has asserted global existence of bounded classical solutions for arbitrary positive radial initial data $u_0\in C^3(\bar\Omega)$ when either $n\ge 2$ and $\chi<1$, or $n=1$ and $ \int_\Omega u_0<\frac{1}{\sqrt{(\chi^2-1)_+}}$. This present paper shows that these conditions are essentially optimal: Indeed, it is shown that if the taxis coefficient is large enough in the sense that $\chi>1$, then for any choice of \begin{equation} \left\{ \begin{array}{ll} m>\frac{1}{\sqrt{\chi^2-1}} \quad & \mbox{if } n=1, \\[2mm] m>0 \mbox{ is arbitrary } \quad & \mbox{if } n\ge 2, \end{array} \right. \end{equation} there exist positive initial data $u_0\in C^3(\bar\Omega)$ satisfying $ \int_\Omega u_0=m$ which are such that for some $T>0$, ($\star$) possesses a uniquely determined classical solution $(u,v)$ in $\Omega\times (0,T)$ blowing up at time $T$ in the sense that $\limsup_{t\nearrow T} \|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty$.\abs This result is derived by means of a comparison argument applied to the doubly degenerate scalar parabolic equation satisfied by the mass accumulation function associated with ($\star$).