On spectra of linearized operators for Keller-Segel models of chemotaxis

PHYSICA D-NONLINEAR PHENOMENA

卷:241

Article

Dejak, S. I.²  Lushnikov, P. M.¹  Ovchinnikov, Yu N.^3,4  Sigal, I. M.² 

[1] Univ New Mexico, Dept Math & Stat, Albuquerque, NM 87131 USA

[2] Univ Toronto, Dept Math, Toronto, ON M5S 1A1, Canada

[3] LD Landau Theoret Phys Inst, Chernogolovka 142432, Russia

[4] Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany

关键词: Matched asymptotics;    Critical Keller-Segel equation;    Collapse and formation of singularities;    Linearized operators;   

DOI  :  10.1016/j.physd.2012.04.003

来源: Elsevier

【 摘 要 】

We consider the phenomenon of collapse in the critical Keller-Segel equation (KS) which models chemotactic aggregation of micro-organisms underlying many social activities, e.g. fruiting body development and biofilm formation. Also KS describes the collapse of a gas of self-gravitating Brownian particles. We find the fluctuation spectrum around the collapsing family of steady states for these equations, which is instrumental in the derivation of the critical collapse law. To this end we develop a rigorous version of the method of matched asymptotics for the spectral analysis of a class of second order differential operators containing the linearized Keller-Segel operators (and as we argue linearized operators appearing in nonlinear evolution problems). We explain how the results we obtain are used to derive the critical collapse law, as well as for proving its stability. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free