### Abstract

We consider the following Keller-Segel system of degenerate type: (KS): ∂u/∂t = ∂/∂x (∂u ^{m}/∂x-u ^{q-1}∂v/∂x), x ∈ R{double-struck}, t > 0,0 = ∂ ^{2}v/∂x ^{2}-γν + u,x ∈ R{double-struck}, t > 0, u(x,0)=u _{0} (x), x ∈ R{double-struck}, where m > 1, γ > 0, q ≥ 2m. We shall first construct a weak solution u(x, t) of (KS) such that u ^{m - 1} is Lipschitz continuous and such that u ^{m-1+δ} for δ > 0 is of class C ^{1} with respect to the space variable x. As a by-product, we prove the property of finite speed of propagation of a weak solution u(x, t) of (KS), i.e., that a weak solution u(x, t) of (KS) has a compact support in x for all t > 0 if the initial data u _{0}(x) has a compact support in R{double-struck}. We also give both upper and lower bounds of the interface of the weak solution u of (KS).

Original language | English |
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Pages (from-to) | 744-757 |

Number of pages | 14 |

Journal | Mathematische Nachrichten |

Volume | 285 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - Apr 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematische Nachrichten*,

*285*(5-6), 744-757. https://doi.org/10.1002/mana.200810258

**Finite speed of propagation in 1-D degenerate Keller-Segel system.** / Sugiyama, Yoshie.

Research output: Contribution to journal › Article

*Mathematische Nachrichten*, vol. 285, no. 5-6, pp. 744-757. https://doi.org/10.1002/mana.200810258

}

TY - JOUR

T1 - Finite speed of propagation in 1-D degenerate Keller-Segel system

AU - Sugiyama, Yoshie

PY - 2012/4/1

Y1 - 2012/4/1

N2 - We consider the following Keller-Segel system of degenerate type: (KS): ∂u/∂t = ∂/∂x (∂u m/∂x-u q-1∂v/∂x), x ∈ R{double-struck}, t > 0,0 = ∂ 2v/∂x 2-γν + u,x ∈ R{double-struck}, t > 0, u(x,0)=u 0 (x), x ∈ R{double-struck}, where m > 1, γ > 0, q ≥ 2m. We shall first construct a weak solution u(x, t) of (KS) such that u m - 1 is Lipschitz continuous and such that u m-1+δ for δ > 0 is of class C 1 with respect to the space variable x. As a by-product, we prove the property of finite speed of propagation of a weak solution u(x, t) of (KS), i.e., that a weak solution u(x, t) of (KS) has a compact support in x for all t > 0 if the initial data u 0(x) has a compact support in R{double-struck}. We also give both upper and lower bounds of the interface of the weak solution u of (KS).

AB - We consider the following Keller-Segel system of degenerate type: (KS): ∂u/∂t = ∂/∂x (∂u m/∂x-u q-1∂v/∂x), x ∈ R{double-struck}, t > 0,0 = ∂ 2v/∂x 2-γν + u,x ∈ R{double-struck}, t > 0, u(x,0)=u 0 (x), x ∈ R{double-struck}, where m > 1, γ > 0, q ≥ 2m. We shall first construct a weak solution u(x, t) of (KS) such that u m - 1 is Lipschitz continuous and such that u m-1+δ for δ > 0 is of class C 1 with respect to the space variable x. As a by-product, we prove the property of finite speed of propagation of a weak solution u(x, t) of (KS), i.e., that a weak solution u(x, t) of (KS) has a compact support in x for all t > 0 if the initial data u 0(x) has a compact support in R{double-struck}. We also give both upper and lower bounds of the interface of the weak solution u of (KS).

UR - http://www.scopus.com/inward/record.url?scp=84858689765&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858689765&partnerID=8YFLogxK

U2 - 10.1002/mana.200810258

DO - 10.1002/mana.200810258

M3 - Article

AN - SCOPUS:84858689765

VL - 285

SP - 744

EP - 757

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

IS - 5-6

ER -