Long term behaviour of a perturbed Cahn-Hilliard equation

The Cahn–Hilliard equation is one of the most important models in phase-field theory. It was first proposed by J. W. Cahn and J. E. Hilliard to describe the evolution of phase separation in binary alloys. The aim of this thesis is to study the long-term behavior of a perturbed Cahn–Hilliard equation with a singular potential. In the first chapter, we present the derivation of the physical model proposed by F. Duda, A. Sarmiento, and E. Fried, which is based on a microforce balance originally introduced by M. E. Gurtin. In the second chapter, we discuss the well-posedness and key properties of the equation, following the work of M. Conti, S. Gatti, and A. Miranville. We address the existence, uniqueness, and continuous dependence on initial data for both weak and strong solutions. Additionally, the strict separation property over finite time intervals is examined. In the third chapter, we present original results concerning the long-term behavior of the solutions. We prove that the solutions generate a closed semigroup of operators in a suitable phase space. We also establish the dissipativity of this semigroup, namely the existence of a bounded absorbing set, as well as new regularity results and the uniform strict separation property with respect to time. Furthermore, we demonstrate the existence of the universal attractor for the semigroup. The thesis concludes with three appendices. In these, we discuss the theory of Yosida approximations of the singular potential, present several technical results used throughout the work, and provide regularity results for the elliptic problem employed in the proof of the strict separation property.