Phase-field crystal equation with memory

Phase-field crystal models are used to describe several pattern formation phenomena like crystallization of liquid, diffusion defects and glass formation. The prototypical equation is obtained as the conserved gradient flow associated with a free-energy functional of Swift–Hohenberg type. Here we consider a variant of the phase-field crystal equation proposed by P. Galenko et al. to account for fast dynamics. This version is characterized by the fact that the gradient flow depends on the past history of the particle density through a memory kernel kεkε, ε>0ε>0 being a relaxation time. Therefore the resulting nonlinear evolution equation is an integro-differential equation of sixth order which is equipped with the physically relevant periodic boundary conditions. We show that this problem generates a dissipative dynamical system on a suitable phase space. Moreover, we prove the existence of a robust family of exponential attractors which is Hölder continuous with respect to ε.