The Cahn–Hilliard system

is usually rewritten, equivalently, as the fourth-order-in-space parabolic equation

which is precisely the equation known as the Cahn–Hilliard equation. It was proposed by J.W. Cahn^{1} and J.E. Hilliard in 1958 (see [81]). These equations play an essential role in materials science and describe important qualitative features of two-phase systems related to phase separation processes, assuming isotropy and a constant temperature. This can be observed, e.g., when a binary alloy (e.g., aluminum/zinc (see [467]) or iron/chromium (see [364, 365, 366])) is cooled down sufficiently. We then observe a partial nucleation (i.e., the appearance of nuclides in the material) or a total nucleation, known as spinodal decomposition: the initially homogeneous material quickly becomes inhomogeneous, resulting in a very finely dispersed microstructure. In a second stage, which occurs at a slower time scale, these microstructures coarsen (hence the term “coarsening”). See, e.g., YouTube, https://www.youtube.com/watch?v=wWXS52OFo7w, for an animation. Such phenomena play an essential role in the mechanical properties of the material, e.g., strength, hardness, fracture, toughness, and ductility. We refer the reader to, e.g., [79, 81, 328, 333, 357, 358, 415, 417] for more details.