### 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 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

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

**Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system.** / Kozono, Hideo; Sugiyama, Yoshie; Yahagi, Yumi.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 253, no. 7, pp. 2295-2313. https://doi.org/10.1016/j.jde.2012.06.001

}

TY - JOUR

T1 - Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system

AU - Kozono, Hideo

AU - Sugiyama, Yoshie

AU - Yahagi, Yumi

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84864001480&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864001480&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2012.06.001

DO - 10.1016/j.jde.2012.06.001

M3 - Article

AN - SCOPUS:84864001480

VL - 253

SP - 2295

EP - 2313

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -