In the vicinity of the threshold of a continuous pattern-forming instability a description of the basic periodic solution and its slow modulations is possible by perturbation theory [Newell & Whitehead, 1969; Segel, 1969; Stuart & Di Prima, 1978]. For quasi-one-dimensional, stationary, and periodic patterns formed from an unstructured state through a continuous soft-mode finite-q instability, a...

As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by cl...

Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get stability for the Log-HLS inequality. Finally, using all these estimates, we prove a quantitative ...

In the limit of a large mass M À 1, and on a finite interval of length 2L, an equilibrium spike solution to the classical Keller-Segel chemotaxis model with a linear chemotactic function is constructed asymptotically. By calculating an asymptotic formula for the translational eigenvalue for M À 1, it is shown that the equilibrium spike solution is unstable to translations of the spike profile. ...

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller...

In this work, we describe a hyperbolic model with cell–cell repulsion dynamics in the population of cells. More precisely, consider cells producing field (which call “pressure”) which induces motion following opposite gradient. The indicates local density and assume that try to avoid crowded areas prefer locally empty spaces are far away from carrying capacity. We analyze well-posedness propert...

We study stationary solutions to the Keller--Segel equation on curved planes. prove necessity of mass being $8 \pi$ and a sharp decay bound. Notably, our results do not require have finite second moment, thus are novel already in flat case. Furthermore, we provide correspondence between static planes positively Riemannian metrics sphere. use this duality show nonexistence certain situations. In...