A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound that decreases with respect to time and coincides original at $t = 0$ as $t$ tends $\infty$. This can also be viewed nonlocal-in-time modified energy, summation an accumulation term due memory effect fractional derivative. particular, indicates indeed decays w.r.t. in small neighborhood $t=0$. illustrate theory mainly equation, but it could applied other phase-field models such Cahn-Hilliard On discrete level, first-order L1 second-order L2 schemes for equation have similar energies, so stability established. Some numerical results are provided behavior verify our theoretical results.