<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.</p>
</abstract>

In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Khawarizmi Laboratory for Mathematics and Applications

International Journal of Mathematics and Computer in Engineering

This work is licensed under the Creative Commons Attribution 4.0 International License.

In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term...

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Article Category: Original Study

Publicado en línea: 02 jun 2024

Páginas: -

Recibido: 12 nov 2023

Aceptado: 20 ene 2024

DOI: https://doi.org/10.2478/ijmce-2025-0008

Palabras clave
Cahn–Hilliard equation, existence, Blow up, logarithmic nonlinear term, formation of islands

© 2025 Hussein Fakih et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Article Category: Original Study

Publicado en línea: 02 jun 2024

Páginas: -

Recibido: 12 nov 2023

Aceptado: 20 ene 2024

DOI: https://doi.org/10.2478/ijmce-2025-0008

Palabras claveCahn–Hilliard equation, existence, Blow up, logarithmic nonlinear term, formation of islands

© 2025 Hussein Fakih et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Palabras clave
Cahn–Hilliard equation, existence, Blow up, logarithmic nonlinear term, formation of islands