### Abstract

We consider the Keller-Segel system of degenerate type (KS)_{m} with m > 1 below. We establish a uniform estimate of ∂^{2} _{x}u^{m-1} from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)_{m} In addition, we find that the set D(t) := {x ∈ ℝ;u(x, t) > 0} of positive region to the solution u is monotonically non-decreasing with respect to t.

Original language | English |
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Pages (from-to) | 891-913 |

Number of pages | 23 |

Journal | Revista Matematica Iberoamericana |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2010 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

**Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems.** / Sugiyama, Yoshie.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 26, no. 3, pp. 891-913. https://doi.org/10.4171/RMI/620

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TY - JOUR

T1 - Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems

AU - Sugiyama, Yoshie

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We consider the Keller-Segel system of degenerate type (KS)m with m > 1 below. We establish a uniform estimate of ∂2 xum-1 from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)m In addition, we find that the set D(t) := {x ∈ ℝ;u(x, t) > 0} of positive region to the solution u is monotonically non-decreasing with respect to t.

AB - We consider the Keller-Segel system of degenerate type (KS)m with m > 1 below. We establish a uniform estimate of ∂2 xum-1 from below. The corresponding estimate to the porous medium equation is well-known as an Aronson-Bénilan type. We apply our estimate to prove the optimal Hölder continuity of weak solutions of (KS)m In addition, we find that the set D(t) := {x ∈ ℝ;u(x, t) > 0} of positive region to the solution u is monotonically non-decreasing with respect to t.

UR - http://www.scopus.com/inward/record.url?scp=78649646813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649646813&partnerID=8YFLogxK

U2 - 10.4171/RMI/620

DO - 10.4171/RMI/620

M3 - Article

AN - SCOPUS:78649646813

VL - 26

SP - 891

EP - 913

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 3

ER -