Large-time solutions of a class of scalar, nonlinear hyperbolic reaction-diffusion equations

摘要

We consider the evolution of the solution of a class of scalar nonlinear hyperbolic reaction-diffusion equations which incorporate a relaxation time and with a reaction function given by a monostable cubic polynomial. An initial-value problem is studied when the prescribed starting data are given by a simple step function. It is established that the large-time structure of the solution is governed by the evolution of a propagating wave-front. The character of this front can be one of three forms, either reaction-diffusion, reaction-relaxation or reaction— relaxation-diffusion, which is relevant and depends on the particular values of the problem parameters that describe the underlying reaction polynomial.