##
**Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions.**
*(English)*
Zbl 1162.35386

Summary: We consider a model of non-isothermal phase transition taking place in a confined container. The order parameter \(\phi\) is governed by a Cahn-Hilliard-type equation which is coupled with a heat equation for the temperature \(\theta\). The former is subject to a nonlinear dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or even dynamic. We thus formulate a class of initial- and boundary-value problems whose local existence and uniqueness is proven by means of the contraction mapping principle. The local solution becomes global owing to suitable a priori estimates.

### MSC:

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35K55 | Nonlinear parabolic equations |

74N20 | Dynamics of phase boundaries in solids |

35B40 | Asymptotic behavior of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |

37L30 | Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems |