Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) existence and uniform boundedness, Discrete Continuous Dynam

(Communicated by Yanping Lin) Abstract. We present some new results on the asymptotic behavior of the periodic solution to a 2×2 mixed-type system of viscosity-capillarity in a vis-coelastic material. We prove that the solution converges to a certain stationary solution as time approaches to infinity, in particular, when the viscosity is large enough or the mean of the initial datum is in the hyperbolic regions, the so-lution converges exponentially to the trivial stationary solution with any large initial datum. The location of the initial datum and the amplitude of vis-cosity play a key role for the phase transitions. Furthermore, we obtain the convergence rate to the stationary solutions. Finally, we carry out numerical simulations to confirm the theoretical predictions. 1. Introduction. Th