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) { u t = ∇ ⋅ ( u ∇ u u 2 + | ∇ u | 2 ) − χ ∇ ⋅ ( u ∇ v 1 + | ∇ v | 2 ) , 0 = Δ v − μ + u , \begin{equation}\tag {\star } \begin {cases} 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 ), \\[3pt] 0=\Delta v - \mu + u, \end{cases} \end{equation} under the initial condition u | t = 0 = u 0 > 0 u|_{t=0}=u_0>0 and no-flux boundary conditions in a ball Ω ⊂ R n \Omega \subset \mathbb {R}^n , where χ > 0 \chi >0 and μ := 1 | Ω | ∫ Ω u 0 \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 ∈ C 3 ( Ω ¯ ) u_0\in C^3(\bar \Omega ) when either n ≥ 2 n\ge 2 and χ > 1 \chi >1 , or n = 1 n=1 and ∫ Ω u 0 > 1 ( χ 2 − 1 ) + \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 χ > 1 \chi >1 , then for any choice of \[ { m > 1 χ 2 − 1 a m p ; if n = 1 , m > 0 is arbitrary a m p ; if n ≥ 2 , \begin {cases} m>\frac {1}{\sqrt {\chi ^2-1}} & \text {if $n=1$}, \\ \text {$m>0$ is arbitrary} & \text {if $n\ge 2$}, \end {cases} \] there exist positive initial data u 0 ∈ C 3 ( Ω ¯ ) u_0\in C^3(\bar \Omega ) satisfying ∫ Ω u 0 = m \int _\Omega u_0=m which are such that for some T > 0 T>0 , ( ⋆ \star ) possesses a uniquely determined classical solution ( u , v ) (u,v) in Ω × ( 0 , T ) \Omega \times (0,T) blowing up at time T T in the sense that lim sup t ↗ T ‖ u ( ⋅ , t ) ‖ L ∞ ( Ω ) = ∞ \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 ).