Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form

$u_{\tau \tau }+u_{\tau }=u_{xx}+\epsilon (F(u)+F{(u)}_{\tau }),$

in which x and τ represent dimensionless distance and time respectively and $\epsilon >0$ is a parameter related to the relaxation time. Furthermore the reaction function, $F(u)$, is given by the bistable cubic polynomial,

$F(u)=u(1?u)(u?\mu ),$

in which $0<\mu <1/2$ is a parameter. The initial data is given by a simple step function with $u(x,0)=1$ for $x\le 0$ and $u(x,0)=0$ for $x>0$. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when $\mu =\frac{1}{2}$ in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin.