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 ($\star$).
Categories: math.AP