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

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.Comment: 22 pages, 1 figur

On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated.Comment: 26 page

Stable singularity formation for the Keller-Segel system in three dimensions

We consider the parabolic-elliptic Keller-Segel system in dimensions $d \geq 3$, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example has been found more than two decades ago by Brenner et al. \cite{BCKSV99}, and is conjectured to be nonlinearly radially stable. We prove this conjecture for $d=3$. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we crucially rely on a technique we recently developed in \cite{GloSch20}. To our knowledge, this provides the first result on stable self-similar blowup for the Keller-Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.Comment: 30 page

Reduction of critical mass in a chemotaxis system by external application of a chemoattractant

In this paper we study non-negative radially symmetric solutions of a parabolic-elliptic Keller-Segel system. The system describes the chemotactic movement of cells under the additional circumstance that an external application of a chemo attractant at a distinguished point is introduced

Stationary States and Asymptotic Behaviour of Aggregation Models with Nonlinear Local Repulsion

We consider a continuum aggregation model with nonlinear local repulsion given by a degenerate power-law diffusion with general exponent. The steady states and their properties in one dimension are studied both analytically and numerically, suggesting that the quadratic diffusion is a critical case. The focus is on finite-size, monotone and compactly supported equilibria. We also investigate numerically the long time asymptotics of the model by simulations of the evolution equation. Issues such as metastability and local/ global stability are studied in connection to the gradient flow formulation of the model