Analysis and numerical methods for nonlocal-in-time Allen-Cahn equation

In this paper, we investigate the nonlocal-in-time Allen-Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well-posedness of the NiTACE by establishing the maximal Lp$$ {L}<^>p $$ regularity for the nonlocal-in-time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy-stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.