Families of traveling impulses and fronts in some models with cross-diffusion

Details Families of traveling impulses and fronts in some models with cross-diffusion Families of traveling impulses and fronts in some models with cross-diffusion

2007-06-06

arxiv.org/abs/0706.0903v1

Computer Science

Numerical Analysis

20 pages, 9 figures; submitted to Journal of Nonlinear Analysis: Real World Applications

Scientific paper

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller-Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse-impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.

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