### Abstract

Let us consider (KS)_{m} below for all N ≥ 2 and general exponents m and q. In particular, the 2-D semi-linear case such as N = 2, m = 1 and q = 2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in C ([0, T] ; L^{frac(N (q - m), 2)} (R^{N})) which coincides with a scaling invariant class of weak solutions associated with (KS)_{m}. In addition, the Hausdorff dimension of its singular set is zero if u ∈ L^{∞} (0, T ; L^{frac(N (q - m), 2)} (R^{N})) and u^{frac(N (q - m), 2)} ∈ C_{w} ([0, T] ; L^{1} (R^{N})).

Original language | English |
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Pages (from-to) | 51-70 |

Number of pages | 20 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 364 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

**ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions.** / Sugiyama, Yoshie.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 364, no. 1, pp. 51-70. https://doi.org/10.1016/j.jmaa.2009.11.019

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TY - JOUR

T1 - ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions

AU - Sugiyama, Yoshie

PY - 2010/4/1

Y1 - 2010/4/1

N2 - Let us consider (KS)m below for all N ≥ 2 and general exponents m and q. In particular, the 2-D semi-linear case such as N = 2, m = 1 and q = 2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in C ([0, T] ; Lfrac(N (q - m), 2) (RN)) which coincides with a scaling invariant class of weak solutions associated with (KS)m. In addition, the Hausdorff dimension of its singular set is zero if u ∈ L∞ (0, T ; Lfrac(N (q - m), 2) (RN)) and ufrac(N (q - m), 2) ∈ Cw ([0, T] ; L1 (RN)).

AB - Let us consider (KS)m below for all N ≥ 2 and general exponents m and q. In particular, the 2-D semi-linear case such as N = 2, m = 1 and q = 2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in C ([0, T] ; Lfrac(N (q - m), 2) (RN)) which coincides with a scaling invariant class of weak solutions associated with (KS)m. In addition, the Hausdorff dimension of its singular set is zero if u ∈ L∞ (0, T ; Lfrac(N (q - m), 2) (RN)) and ufrac(N (q - m), 2) ∈ Cw ([0, T] ; L1 (RN)).

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U2 - 10.1016/j.jmaa.2009.11.019

DO - 10.1016/j.jmaa.2009.11.019

M3 - Article

VL - 364

SP - 51

EP - 70

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -