The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.