Memory relaxation of the one-dimensional Cahn-Hilliard equation

We consider the memory relaxation of the one-dimensionalCahn-Hilliard equation endowed with the no-flux boundaryconditions. The resulting integrodifferential equation ischaracterized by a memory kernel which is the rescaling of a given positive decreasing function. The Cahn-Hilliard equation is then viewed as the formal limit of the relaxed equation, when thescaling parameter (or relaxation time) ε tends to zero. Inparticular, if the memory kernel is the decreasing exponential,then the relaxed equation is equivalent to the standard hyperbolicrelaxation. The main result of this note is the existence of afamily of robust exponential attractors for the one-parameterdissipative dynamical system generated by the relaxed equation,which is stable with respect to the singular limit ε→0.This theorem is obtained as a nontrivial application of a recentabstract result.