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

Yoshie Sugiyama

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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 = ∂ 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).

Original languageEnglish
Pages (from-to)744-757
Number of pages14
JournalMathematische Nachrichten
Volume285
Issue number5-6
DOIs
Publication statusPublished - Apr 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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