On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn-Hilliard type inpainting model

In this paper, we study the analytical and numerical long-time stability of a nonlocal Cahn-Hilliard model with a fidelity term for image inpainting introduced in our previous work Jiang et al. (2024). First, we establish the uniform boundedness of the continuous problem in both L2 and H1 spaces, which is obtained by using the Gagliardo–Nirenberg inequality and the uniform Grönwall lemma. Then, for the temporal semi-discrete scheme, the uniform estimates in L2 and H1 spaces are derived with the aid of the discrete uniform Grönwall lemma under a suitable assumption on the nonlinear potential. This demonstrates the long-time stability of the proposed scheme in L2 and H1 spaces. Finally, we validate the long-time stability and the applicability of our method in signal reconstruction and image inpainting. These numerical experiments demonstrate the high effectiveness of our proposed model.