### Abstract

We shall show an exact time interval for the existence of local strong solutions to the Keller-Segel system with the initial data u_{0} in L ^{n}
_{2w} (R^{n}), the weak L ^{n}
_{2} -space on Rn. If ||u_{0}||L ^{n}
_{2} w (R^{n}) is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L ^{n}
_{2w} (Rn) stems from obtaining a self-similar solution which does not belong to any usual L^{p}(R^{n}). Furthermore, the characterization of local existence of solutions gives us an explicit blow-up rate of ||u(t)||L^{p}(R^{n}) for n 2 < p≤∞as t → Tmax, where Tmax denotes the maximal existence time.

Original language | English |
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Pages (from-to) | 732-751 |

Number of pages | 20 |

Journal | Mathematische Nachrichten |

Volume | 283 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2010 May |

Externally published | Yes |

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### Keywords

- Blow-up rate
- Global and local existence
- Keller-Segel system
- Weak L-L estimate

### ASJC Scopus subject areas

- Mathematics(all)