Advances in Differential Equations

Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions

Ciprian G. Gal

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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.

Article information

Adv. Differential Equations, Volume 12, Number 11 (2007), 1241-1274.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B41: Attractors 37L30: Attractors and their dimensions, Lyapunov exponents


Gal, Ciprian G. Global well-posedness for the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differential Equations 12 (2007), no. 11, 1241--1274.

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