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Finite-time blow-up in a degenerate chemotaxis system with flux limitation. (English) Zbl 1367.35044
Summary: This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \[ \begin{cases} u_t=\nabla\cdot\left( \frac{u\nabla u}{\sqrt{u^2+|\nabla u|^2}} \right)-\chi\nabla\cdot\left( \frac{u\nabla v}{\sqrt{1+|\nabla v|^2}} \right) \end{cases}{(\star)} \] 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 [Commun. Partial Differ. Equations 42, No. 3, 436–473 (2017; Zbl 1430.35166)] has asserted global existence of bounded classical solutions for arbitrary positive radial initial data \( u_0\in C^3(\bar \Omega )\) when either \( n\geq 2\) 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 \[ \begin{cases} m>\frac{1}{\sqrt{\chi^2-1}}\quad & \text{if }n=1, \\ m>0 \text{ is arbitrary} \quad & \text{if } n\geq2, \end{cases} \] 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 (\( \star \)).

MSC:
35B44 Blow-up in context of PDEs
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C17 Cell movement (chemotaxis, etc.)
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