## Abstract

This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates the global-in-time solution. In view of our theorem one sees that the one-dimensional Keller-Segel system is essentially different from its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known classical Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem.

Original language | English |
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Pages (from-to) | 2529-2540 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

- Chemotaxis
- Global existence
- Lyapunov functional