On the large time behavior of a phase-field system with memory

Abstract

We consider a nonconserved phase-field model of Caginalp type which has been proposed by H.G. Rotstein et al. This model consists of two coupled integro-partial differential equations, one for the temperature ϑ, the other for the order parameter χ, and each one is characterized by a nonnegative decreasing memory kernel. Such equations, endowed with homogeneous Dirichlet (for ϑ) and Neumann (for χ) boundary conditions, generate a dissipative dynamical system which possesses a global attractor 𝒜. This fact was proven by V. Pata and the author. Here we construct a compact exponentially attracting set which, in particular, entails a smoothness result for 𝒜. Then we also demonstrate the convergence of any trajectory to a single equilibrium, via the Łojasiewicz–Simon approach.