Long-time solutions of scalar hyperbolic reaction equations incorporating relaxation and the Arrhenius combustion nonlinearity

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form $$\begin{align*} & u_{\tau\tau}+u_{\tau}=u_{{xx}}+\varepsilon (F(u)+F(u)_{\tau} ), \end{align*}$$ in which ${x}$ and $\tau $ represent dimensionless distance and time, respectively, and $\varepsilon>0$ is a parameter related to the relaxation time. Furthermore, the reaction function, $F(u)$ , is given by the Arrhenius combustion nonlinearity, $$\begin{align*} & F(u)=e^{-{E}/{u}}(1-u), \end{align*}$$ in which $E>0$ is a parameter related to the activation energy. The initial data are given by a simple step function with $u({x},0)=1$ for ${x} \le 0$ and $u({x},0)=0$ for ${x}> 0$ . The above initial-value problem models, under certain simplifying assumptions, combustion waves in premixed gaseous fuels; here, the variable $u$ represents the non-dimensional temperature. It is established that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front, which is of reaction–diffusion or reaction–relaxation type depending on the values of the problem parameters $E$ and $\varepsilon $ .