Approximation of the Heaviside function by sigmoidal functions in reaction–diffusion equations

Abstract

A reaction–diffusion system with a Heaviside reaction term on a smooth bounded domain in Rd is studied by using sigmoidal functions to approximate the Heaviside term. First, the global well-posedness of solutions to the sigmoidal reaction–diffusion system and the existence of solutions to the Heaviside reaction–diffusion system are proved. Then the lattice systems resulting from Galerkin expansions of the solutions to the sigmoidal system and the Heaviside system are studied. In particular, solutions of the sigmoidal lattice system are shown to converge to the solution of the Heaviside lattice system as the steepness parameter ϵ goes to 0, through an inflated lattice system. Moreover, dynamics of the lattice systems are utilized to show that solutions of the sigmoidal reaction–diffusion system tend to the solution of the Heaviside reaction–diffusion system as ϵ goes to 0. Finally, relations between the attractors for the sigmoidal system and the Heaviside system are established.