Existence and Uniqueness of Solution for Cahn-Hilliard Hyperbolic Phase-Field System with Dirichlet Boundary Condition and Regular Potentials

Our aim in this paper is to study the existence and the uniqueness of the solutions for hyperbolic Cahn-Hilliard phase-field system, with initial conditions, Dirichlet boundary condition and regular potentials.


Introduction
G. Caginalp introduced in [1] the following phase-field system ( ) where u is the order parameter and θ is the (relative) temperature.These equations model phase transition processes such as melting-solidification processes and have been studied, see [2]- [6], for a similar phase-field model with a nonlinear term.These Cahn-Hilliard phase-fiel system are known as the conserved phase-field system (see [7]- [9]) based on type III heat conduction and with two temperatures (see [10]).The authors have proved the existence and the uniqueness of the solutions, the existence of global attractor and of exponential attractors with singularly or regular In [11], Ntsokongo and Batangouna have studied the following Cahn-Hilliard phasefield system ( ) where 1 β = , u is the order parameter and α is the (relative) temperature, they have proved the existence and the uniqueness solution with Dirichlet boundary condition and regular potentials.
In this paper, we consider the following Cahn-Hilliard hyperbolic phase-fiel system ( ) ( ) which is the perturbed phase-field system of Cahn-Hilliard phase-field system (3)-( 4) with 0 β = .In the above hyperbolic system Ω is a bounded and regular domain of n  with 2 n = or 3 and f is the nonlinear regular potentials.
The hyperbolic system has been extensively studied for Dirichlet boundary conditions and regular or singular potentials (see [12]- [14]).Whose certain have to end at existence of global attractor or at the existence of exponential attractors (see [15]).
In this paper we prove the existence and the uniqueness of solutions of ( 5)- (8).We consider the regular potential ( ) 3 f s s s = − which satisfies the following properties:

A Priori Estimates
We multiply (5) by ( ) , integrate over Ω and add the two resulting differential equalities.We find where ( ) ( ) and integrate over Ω .We get.
In this study, we have three main results; existence theorem, uniqueness theorem and existence theorem with more regularity.( )

Existence and Uniqueness of Solutions
then the system ( 5) -( 8) possesses at least one solution ( ) ( ) , for all 0 T > .
The proof is based on a priori estimates obtained in the previous section and on a standard Galerkin scheme.Theorem 4.2.(Uniqueness) Let the assumpptions of Theorem 4.1 hold.Then, the system (5) -( 8) possesses a unique solution ( ) ( ) ( ) , u α and , u α be two solutions of the system ( 5)-( 8) with initial data ) respectively.We set We multiply ( 12) by ( ) and integrate over Ω .We find Now summing ( 14) and ( 15) we obtain where Inserting the above estimate into (16), we have  Applying Gronwall's lemma, we obtain for all ( ) We deduce the continuous dependence of the solution relative to the initial conditions, hence the uniqueness of the solution.
The existence and uniqueness of the solution of problem ( 5)-( 8) being proven in a larger space, we will seek the solution with more regularity.

Conclusion
We have just shown the theorems of existence and uniqueness of the solutions for perturbed Cahn-Hilliard hyperbolic phase-field system with regular potentials.
, where −∆ denotes the minus Laplace operator with Dirichlet boundary conditions.More generally, .X denote the norm of Banach space X.Throughout this paper, the same letters 1 2 , c c and 3 c denote (generally positive) constants which may change from line to line, or even a same line.
s and the fact that 