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June  2007, 7(4): 839-857. doi: 10.3934/dcdsb.2007.7.839

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

1. 

Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6

2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

3. 

Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539-2999

Received  June 2006 Revised  January 2007 Published  March 2007

We present some new results on the asymptotic behavior of the periodic solution to a 2$\times$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 it 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.
Citation: Ming Mei, Yau Shu Wong, Liping Liu. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 839-857. doi: 10.3934/dcdsb.2007.7.839
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