Time Discretization of Transition Layer Dynamics in One-Dimensional Viscoelastic Systems

Abstract

We investigate how evolution occurs as the strain ux of a viscoelastic system $u_{tt}-(\sigma (u_x)+u_{xt})_x+u=0$ goes towards a state of equilibrium. The time limit of ux eventually shows a finite number of discontinuous interfaces if the strain starts from the continuous initial data whose transition layers are steep enough and the initial energy is sufficiently small. The number of phases is conserved and the transition layers stay in the initial position of interfaces. The results are obtained by using the implicit time discretization method and the Andrews--Pego transformed equations.