On the asymptotic behavior of the Caginalp system with dynamic boundary conditions

We consider a phase-field system of Caginalp type on a three-dimensional bounded domain. The order parameter $\psi $ fulfills a dynamic boundary condition, while the (relative) temperature $\theta $ is subject to a boundary condition of Dirichlet, Neumann, Robin or Wentzell type. The corresponding class of initial and boundary value problems has already been studied by the authors, proving well-posedness results and the existence of global as well as exponential attractors. Here we intend to show first that the previous analysis can be redone for larger phase-spaces, provided that the bulk potential has a fourth-order growth at most whereas the boundary potential has an arbitrary polynomial growth. Moreover, assuming the potentials to be real analytic, we demonstrate that each trajectory converges to a single equilibrium by means of a Łojasiewicz-Simon type inequality. We also obtain a convergence rate estimate.