A reaction-diffusion approximation of a semilinear wave equation with damping

A reaction-diffusion approximation is a method that solutions of multi-component reaction-diffusion systems approximate those of differential equations. We introduce the reaction-diffusion approximations of a semilinear wave equation and a semilinear damped wave equation under some assumptions of a reaction term. These approximation systems consist of a two-component reaction-diffusion system with a small parameter. In this paper, we prove that a first component of a solution for the system converges to a solution for the semilinear damped wave equation as the parameter tends to zero. Moreover, let us show the numerical results of reaction-diffusion approximation for the wave equation and the damped wave equation, respectively.