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 uτ τ + uτ = uxx + ε(F(u) + F(u)τ ), in which x and τ represent dimensionless distance and time, respectively, and ε > 0 is a parameter related to the relaxation time. Furthermore, the reaction function, F(u), is given by the Arrhenius combustion nonlinearity, F(u) = e−E/u(1 − u), 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 ≤ 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 ε