Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions

We consider a model of non-isothermal phase transition takingplace in a confined container. The order parameter φ is governed by a Cahn-Hilliard type equation which is coupled with a nonlinear heat equation forthe temperature θ. The former is subject to a nonlinear dynamic boundarycondition recently proposed by some physicists to account for interactions ofthe material with the walls. The latter is endowed with a boundary conditionwhich can be a standard one (Dirichlet, Neumann or Robin). We thusformulate a class of initial and boundary value problems whose local existenceand uniqueness is proven by means of a Faedo-Galerkin approximationscheme. The local solution becomes global owing to suitable a priori estimates.Then we analyze the asymptotic behavior of the solutions within the theoryof infinite-dimensional dynamical systems. In particular, we demonstrate theexistence of a finite dimensional global attractor as well as of an exponentialattractor.

Keywords

viscous Cahn-Hilliard equation, dynamic boundary conditions, global attractors, exponential attractors, non-isothermal Cahn-Hilliard equations

2010 Mathematics Subject Classification

35B40, 35B45, 35K55, 37L30, 74N20

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Published 1 January 2008