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Finite-time blow-up in a repulsive chemotaxis-consumption system
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 4 / August 2023
- Published online by Cambridge University Press:
- 06 June 2022, pp. 1150-1166
- Print publication:
- August 2023
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- Article
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In a ball $\Omega \subset \mathbb {R}^{n}$
with $n\ge 2$
, the chemotaxis system
\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]is considered along with no-flux boundary conditions for $u$
and with prescribed constant positive Dirichlet boundary data for $v$
. It is shown that if $D\in C^{3}([0,\infty ))$
is such that $0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$
for all $\xi >0$
with some ${K_D}>0$
and $\alpha >0$
, then for all initial data from a considerably large set of radial functions on $\Omega$
, the corresponding initial-boundary value problem admits a solution blowing up in finite time.
