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

Abstract

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general formuττ+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 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)=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=12 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.