The non-isothermal Allen-Cahn equation with dynamic boundary conditions

Abstract

We consider a model of nonisothermal phase transitions takingplace in a bounded spatial region. The order parameter $\psi$ is governed byan Allen-Cahn type equation which is coupled with the equation for thetemperature $\theta$. The former is subject to a dynamic boundary conditionrecently proposed by some physicists to account for interactions with thewalls. The latter is endowed with a boundary condition which can be astandard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzelltype. We thus formulate a class of initial and boundary value problems whoselocal existence and uniqueness is proven by means of a fixed point argument.The local solution becomes global owing to suitable a priori estimates. Thenwe analyze the asymptotic behavior of the solutions within the theory ofinfinite-dimensional dynamical systems. In particular, we demonstrate theexistence of the global attractor as well as of an exponential attractor.