## 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 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)