Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence
Ming Mei, Yau Shu Wong, Liping Liu · Discrete and Continuous Dynamical Systems - B · 2007
We present some new results on the asymptotic behavior of theperiodic solution to a 2$\times$2 mixed-type system ofviscosity-capillarity in a viscoelastic material. We prove thatthe solution converges to a certain stationary solution as timeapproaches to infinity, in particular, when the viscosity is largeenough or the mean of the initial datum is in the hyperbolicregions, the solution converges exponentially to the trivialstationary solution with it any large initial datum. Thelocation of the initial datum and the amplitude of viscosity playa key role for the phase transitions. Furthermore, we obtain theconvergence rate to the stationary solutions. Finally, we carryout numerical simulations to confirm the theoreticalpredictions.