## Abstract

There exists a well defined energy for classical phase-field equations under which the dissipation law is satisfied, i.e., the energy is non-increasing with respect to time. However, it is not clear how to extend the energy definition to time-fractional phase-field equations so that the corresponding dissipation law is still satisfied. In this work, we will try to settle this problem for phase-field equations with Caputo time-fractional derivative, by defining a nonlocal energy as an averaging of the classical energy with a time-dependent weight function. As the governing equation exhibits both nonlocal and nonlinear behavior, the dissipation analysis is challenging. To deal with this, we propose a new theorem on judging the positive definiteness of a symmetric function, that is derived from a special Cholesky decomposition. Then, the nonlocal energy is proved to be dissipative under a simple restriction of the weight function. Within the same framework, the time fractional derivative of classical energy for time-fractional phase-field models can be proved to be always nonpositive.

Original language | English |
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Pages (from-to) | 478-490 |

Number of pages | 13 |

Journal | CSIAM Transactions on Applied Mathematics |

Volume | 1 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sept 2020 |

## Scopus Subject Areas

- Applied Mathematics

## User-Defined Keywords

- Allen-Cahn equations
- Cahn-Hilliard equations
- Caputo fractional derivative
- energy dissipation
- Phase-field equation
- positive definite kernel