The aim of this talk is to present theoretical analysis and numerical simulations on higher-order (in space) anisotropic phase separation models [2], corresponding to a modified free energy introduced in [1] in which the temperature is omitted. In particular, we will discuss a priori estimates and the well-posedness results, also the dissipativity of semigroup, as well as the existence of the global attractor. Moreover, for the Allen-Cahn models with periodic boundary conditions, numerical simulations which illustrate the effects of the higher-order terms and the anisotropy, are displayed. We will further discuss the Cahn-Hilliard cases and show the numerical simulations.

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References
[1] G. Caginalp and E. Esenturk, Anisotropic phase field equations of arbitrary order, DCDS-S, 4 (2011), 311-350.
[2] S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465 (2009), 1337-1359.
[3] L. Cherfils, A. Miranville and S. Peng, Higher-order models in phase separation, J. Appl. Anal. Comput., 7(1) (2016), 39-56.
[4] L. Cherfils, A. Miranville and S. Peng, Higher-order Allen-Cahn models with logarithmic nonlinear terms, Advances in dynamical systems and control, 247-263, Stud. Syst. Decis. Control, 69, Springer, 2016.
[5] L. Cherfils, A. Miranville and S. Peng, Higher-order anisotropic models in phase separation, Advances in Nonlinear Analysis, 2017.