2011/12/18 by Michael Winkler, Winkler, Michael · 6 citations
Computer Science · Mathematics · #35B44 #35K55 #35Q35 #35Q92 #92C17 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Point processes and geometric inequalities #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1112.4156
openalex publication_date 2011/12/18 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We study the Neumann initial-boundary value problem for the fully parabolic\nKeller-Segel system ut=\Δ u - \∇ \⋅ (u\∇ v), x\∈\Ω,\n t>0, [1mm] vt=\Δ v-v+u, x\∈\Ω, t>0, where \Ω is a\nball in \ℝn with n\≥ 3.\n It is proved that for any prescribed m>0 there exist radially symmetric\npositive initial data (u0,v0) \∈ C0( Ω) \×\nW1,\∞(\Ω) with \∫_\Ω u0=m such that the corresponding\nsolution blows up in finite time.\n Moreover, by providing an essentially explicit blow-up criterion it is shown\nthat within the space of all radial functions, the set of such blow-up\nenforcing initial data indeed is large in an appropriate sense; in particular,\nthis set is dense with respect to the topology of Lp(\Ω) \×\nW1,2(\Ω) for any p \∈ (1,\(2n)/(n+2)).\n