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_{τ τ} + 𝓊_τ = 𝓊_𝓍𝓍 + ɛ(F(𝓊) + F(𝓊)_τ),

in which 𝓍 and τ represent dimensionless distance and time respectively and ɛ > 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 - μ),

in which 0 < μ < 1—2 is a parameter. The initial data is given by a simple step function with 𝓊(𝓍, 0) = 1 for 𝓍 ≤ 0 and 𝓊(𝓍, 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 μ = ½ 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.