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.


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