Behavior of Solutions to Nonlocal Hyperbolic Diffusion and Doubly Nonlocal Cahn-Hilliard Equations

This work focuses on two different type of nonlocal equations, the nonlocal diffusion equation and douby nonlocal Cahn-Hilliard equation. Nonlocal diffusion equation was introduced to model diffusive behavior when Fick’s law is not applicable. The work here introduces nonlocal diffusion and its importance. I then derive a nonlocal hyperbolic diffusion equation and produce an energy identity equation. The resultant equation can be used to study asymptotic behavior of the nonlocal hyperbolic diffusion equation. Next we study the Cahn-Hilliard equation which is one of the most studied PDEs due to its applicability in a variety of fields: phase separation, image processing, alloy, and much more. It was proposed in the 1950s as a model for phase separation in materials made of two components. Recently, a doubly nonlocal Cahn-Hilliard equation has been introduced to allow discontinuous solutions which are physically relevant. The doubly nonlocal model accounts for effects of non-Fickian behavior of the chemical potential. In this work, I will present several results regarding the asymptotic behavior of solutions, which also allow the derivation of exact decay rates of the solutions