Finite-time blow-up in a degenerate chemotaxis system with flux limitation

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 \[ (\star )\qquad \qquad \left \{ \begin {array}{l} \displaystyle u_t=\nabla \cdot \Big (\frac {u\nabla u}{\sqrt {u^2+|\nabla u|^2}}\Big ) - \chi \nabla \cdot \Big (\frac {u\nabla v}{\sqrt {1+|\nabla v|^2}}\Big ), \\[1mm] 0=\Delta v - \mu + u, \end {array} \right .\qquad \qquad \] under the initial condition $u|_{t=0}=u_0>0$ and no-flux boundary conditions in a ball $\Omega \subset \mathbb {R}^n$, where $\chi >0$ and $\mu :=\frac {1}{|\Omega |} \int _\Omega u_0$. A previous result of the authors [Comm. Partial Differential Equations 42 (2017), 436–473] 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 satisfies $\chi >1$, then for any choice of \[ \left \{ \begin {array}{ll} m>\frac {1}{\sqrt {\chi ^2-1}} \quad & \mbox {if } n=1, \\[2mm] m>0 \mbox { is arbitrary } \quad & \mbox {if } n\ge 2, \end {array} \right . \] 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 $\limsup _{t\nearrow T} \|u(\cdot ,t)\|_{L^\infty (\Omega )}=\infty$.

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 ($\star$).