### Abstract

In Rn (n≥ 3), we first define a notion of weak solutions to the Keller-Segel system of parabolic-elliptic type in the scaling invariant class Ls(0,T;L ^{r}(R ^{n})) for 2/ s + n/ r = 2 with n/2 < r< n. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in L _{n/2}(R _{n}). We prove also their uniqueness. As for the marginal case when r= n/2, we show that if n≥ 4, then the class C([0,T);L _{n/2}(R _{n})) enables us to obtain the only weak solution.

Original language | English |
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Pages (from-to) | 2295-2313 |

Number of pages | 19 |

Journal | Journal of Differential Equations |

Volume | 253 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2012 Oct 1 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Kozono, H., Sugiyama, Y., & Yahagi, Y. (2012). Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system.

*Journal of Differential Equations*,*253*(7), 2295-2313. https://doi.org/10.1016/j.jde.2012.06.001