Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition. II: convergence (Q2467584)

The authors present new results on the asymptotic behaviour of the periodic solution to a viscous-capillarity mixed type system \[ v_t-u_x=\varepsilon_1 v_{xx},\;u_t-(\sigma(v)_x=\varepsilon_2 u_{xx},\;(x,t)\in \mathbb{R}\times \mathbb{R}_+ \] with the initial condition \((v,u)| _t=0=(v_0,u_0)(x),\;x\in R\) and the \(2L\)-periodic boundary condition \[ (v,u)(x,t)=(v,u)(x+2L,t),\;(x,t)\in \mathbb{R}\times \mathbb{R}_+. \] They 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 data is in the hyperbolic regions, the solution converges to the trivial stationary solution with any large initial data. The convergence rates are received. The numerical simulations confirming the theoretical results are presented.