Does a ‘volume-filling effect’ always prevent chemotactic collapse?

@article{Winkler2010DoesA,
  title={Does a ‘volume-filling effect’ always prevent chemotactic collapse?},
  author={M. Winkler},
  journal={Mathematical Methods in The Applied Sciences},
  year={2010},
  volume={33},
  pages={12-24}
}
  • M. Winkler
  • Published 2010
  • Mathematics
  • Mathematical Methods in The Applied Sciences
The parabolic-parabolic Keller-Segel system for chemotaxis phenomena, {u t =∇ · (φ(u)∇u)-∇ · (ψ(u)∇ v ), x∈Ω, t>0 v t =Δv-v+u, x∈Ω, t>0 is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ ℝ n with n≥2. It is proved that if ψ(u)/φ(u) grows faster than u 2/n as u → ∞ and some further technical conditions are fulfilled, then there exist solutions that blow up in either finite or infinite time. Here, the total mass ∫ Ω u(x,t)dx may attain arbitrarily small… Expand
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