2009/02/11 by Yoshie Sugiyama, Sugiyama, Yoshie
Mathematics · #35K45 #35K57 #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K45 #msc:35K57 #msc:35K65
paper · pdf · doi:10.48550/arxiv.0902.1878
arxiv created 2009/02/11 · arxiv updated 2009/12/01
We consider the following Keller-Segel system of degenerate type: ∂ u / ∂ t = ∂ / ∂ x (∂ um / ∂ x - uq-1 ⋅ ∂ v / ∂ x), x ∈ \R, t>0, ∂2 v / ∂ x2 - γv + u, x ∈ \R, t>0, u(x,0) = u0(x), x ∈ \R, where m>1, γ> 0, q ≥ 2m. We shall first construct a weak solution u(x,t) of (KS) such that um-1 is Lipschitz continuous and such that um-1+δ for δ>0 is of class C1 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), \it 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 u0(x) has a compact support in \R. We also give both upper and lower bounds of the interface of the weak solution u of (KS).