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

Laura M White, University of Nebraska - Lincoln

Abstract

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.^

Subject Area

Mathematics

Recommended Citation

White, Laura M, "Behavior of Solutions to Nonlocal Hyperbolic Diffusion and Doubly Nonlocal Cahn-Hilliard Equations" (2018). ETD collection for University of Nebraska - Lincoln. AAI10792754.
https://digitalcommons.unl.edu/dissertations/AAI10792754

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