On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis

Abstract

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 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.

22 pages, 1 figure

References in corpus (5)

• Continuous macroscopic limit of a discrete stochastic model for interaction of living cells
• Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description
• Critical dynamics of self-gravitating Langevin particles and bacterial populations
• Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills problems
• Blow up of the critical norm for some radial L^2 super critical nonlinear Schrodinger equations

Cited by in corpus (5)

• Beyond leading order logarithmic scaling in the catastrophic self-focusing (collapse) of a laser beam in Kerr media
• Stable blow-up dynamic for the parabolic-parabolic Patlak-Keller-Segel model
• Logarithmic scaling of the collapse in the critical Keller-Segel equation
• Existence and stability of infinite time blow-up in the Keller-Segel system
• On the stability of critical chemotactic aggregation