Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence

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 viscoelastic 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 solution converges exponentially to the trivial stationary solution with any large initial datum. The location of the initial datum and the amplitude of viscosity 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.