On the modified of the one-dimensional Cahn-Hilliard equation with a source term

Dieunel DOR; Dieunel DOR

doi:10.3934/math.2022807

AIMS Mathematics

2022, Volume 7, Issue 8: 14672-14695. doi: 10.3934/math.2022807

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Research article

On the modified of the one-dimensional Cahn-Hilliard equation with a source term

Dieunel DOR ^{1,2
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1.
Universite de Poitiers, Laboratoire de Mathematiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie-Teleport 2 F-86962 Chasseneuil Futuroscope Cedex, France
2.
Departement de Mathematiques, Universite d'Etat d'Haiti, Ecole Normale Superieure, Laboratoire des Sciences pour l'Environnement et l'Energie (LS2E), HT6115 Canape-Vert, Port-au-Prince, Haiti

Received: 03 January 2022 Revised: 21 March 2022 Accepted: 19 April 2022 Published: 08 June 2022
MSC : 35G20, 35L60, 35Q92, 35B41

We consider the modified Cahn-Hilliard equation that govern the relative concentration $ \phi $ of one component of a binary system. This equation is characterized by the presence of the additional inertial term $ \tau_{D}\frac{d^2\phi}{dt^2} $ which stands for the relaxation of the diffusion flux. This equation is associated with Dirichlet boundary conditions. We study the existence, uniqueness and regularity of solutions in one space dimension. We also prove the existence of the global attractor and exponential attractors.
- Cahn-Hilliard equation,
- well posedness,
- strongly continous semigroups,
- globall attractors,
- robust exponential attractors
Citation: Dieunel DOR. On the modified of the one-dimensional Cahn-Hilliard equation with a source term[J]. AIMS Mathematics, 2022, 7(8): 14672-14695. doi: 10.3934/math.2022807

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Abstract

We consider the modified Cahn-Hilliard equation that govern the relative concentration $ \phi $ of one component of a binary system. This equation is characterized by the presence of the additional inertial term $ \tau_{D}\frac{d^2\phi}{dt^2} $ which stands for the relaxation of the diffusion flux. This equation is associated with Dirichlet boundary conditions. We study the existence, uniqueness and regularity of solutions in one space dimension. We also prove the existence of the global attractor and exponential attractors.

References

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