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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.