Theory and Applications of Reaction-diffusion Equations: Patterns and Waves (Oxford Applied Mathematics & Computing Science)

Chapter 5. Nonlinear Dispersal mechanisms

1. Introduction

In this chapter, nonlinear convection and nonlinear diffusion work together to generate solutions exhibitiong pattern structures.

The first example is

from a swarm model, in which the density u is transported with a nonlocal velocity w and diffuses. This equation can be transformed into a Burgers' like equation and there is a steady state with finite initial total mass.

Normally a second order Fokker-Planck equation (possibly nonlinear) is preferred compared higher order equations like

because the solution may become negative even the initial data is nonnegative.

For the equation from chemotaxis,

The long time asymptotics depends on the bifurcation of the solution max|u_x| and the total mass of u.