On the Logarithmic Cahn-Hilliard Equation with General Proliferation Term

Rim Mheich; Madalina Petcu; Raafat Talhouk

doi:10.3934/cpaa.2024016

Article Contents

2024, Volume 23, Issue 3: 383-403. Doi: 10.3934/cpaa.2024016

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On the Logarithmic Cahn-Hilliard Equation with General Proliferation Term

1.
Université de Poitiers, Laboratoire de Mathématiques et Applications, France
2.
Léonard de Vinci Pôle Universitaire, Research Center, 92 916 Paris La Défense, France
3.
Université Libanaise, Laboratoire de Mathématiques - EDST, Liban

^*Corresponding author: Rim Mheich
^*Corresponding author: Rim Mheich

Received: October 2023

Revised: February 2024

Early access: March 2024

Published: March 2024

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Abstract

Our aim in this article is to study the well-posedness of the generalized logarithmic nonlinear Cahn-Hilliard equation with regularization and proliferation terms. We are interested in studying the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem and majorate the rate of convergence between the solutions of the Cahn-Hilliard equation and the regularized one. Additionally, we present some further regularity results and subsequently prove a strict separation property of the solution. Finally, we provide some numerical simulations to compare the solution with and without the regularization term, and more.

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Mathematics Subject Classification: Primary: 35B40, 35B41, 35B45; Secondary: 35K55.

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Figure 1. The solution $ u $ for $ \varepsilon = 0 $

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Figure 2. The solution $ u $ for $ \varepsilon = 0.01 $.

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Figure 3. The solution $ u $ for $ \varepsilon = 0.1 $.

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Figure 4. The solution $ u $ for $ \varepsilon = 1 $.

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Figure 5. The solution $ u $ for $ \lambda = 0 $

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Figure 6. The solution $ u $ for $ \lambda = 5 $

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Figure 7. The solution $ u $ for $ \lambda = 10 $

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Figure 8. The solution $ u $ for $ \lambda = 15 $

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Figure 9. The solution $ u $ for $ \lambda = 0 $

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Figure 10. The solution $ u $ for $ \lambda = 5 $

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Figure 11. The solution $ u $ for $ \lambda = 10\;at\;t=0.65 $

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