On the optimality of upper estimates near blow-up in quasilinear Keller--Segel systems

@article{Fuest2020OnTO,
  title={On the optimality of upper estimates near blow-up in quasilinear Keller--Segel systems},
  author={Mario Fuest},
  journal={arXiv: Analysis of PDEs},
  year={2020}
}
  • Mario Fuest
  • Published 2020
  • Mathematics, Physics
  • arXiv: Analysis of PDEs
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ \tau v_t = \Delta v - v + u \end{cases} \end{align*} in a ball $\Omega \subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $\tau \in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(\Omega)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially… Expand
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References

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Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system
Does a ‘volume-filling effect’ always prevent chemotactic collapse?
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