The Stability of Traveling Wave Front Solutions of a Reaction-Diffusion System

• Published in: SIAM journal on applied mathematics Vol. 41; no. 1; pp. 145 - 167
• Main Authors: Klaasen, Gene A., Troy, William C.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01-08-1981
• Subjects: Applied mathematics; Axons; Chemical reactions; College mathematics; Eigenvectors; Mathematical functions; Nerves; Neurosciences; Polynomials; Population genetics; Roots of functions; Traveling waves; Uniqueness; Values
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0141011

We investigate the behavior of solutions of the problem $\frac{\partial x}{\partial t} = F(x, y) + \frac{D \partial^2x}{\partial \zeta^2},\quad \frac{\partial y}{\partial t} = G(x, y),$ $x(\zeta, 0) = \varphi (\zeta), \quad y(\zeta, 0) = \psi (\zeta),$ where t ≥ 0 and $-\infty < \zeta < \infty$. Under appropriate assumptions on F and G this system is a model similar to the degenerate forms (i.e., recovery variables kept at equilibrium values) of the Hodgkin-Huxley nerve conduction equations and the Field-Noyes model of the Belousov-Zhabotinskii chemical reaction. We prove the existence and uniqueness of a traveling wave front solution. Secondly, we demonstrate the stability of the traveling wave solution for a general class of initial data.