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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.)
##### Keywords:
degenerate diffusion; no-flux boundary conditions
Full Text:
##### References:
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