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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.

G. Caginalp introduced in [

where u is the order parameter and

These Cahn-Hilliard phase-fiel system are known as the conserved phase-field system (see [

In [

where

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

The hyperbolic system has been extensively studied for Dirichlet boundary conditions and regular or singular potentials (see [

In this paper we prove the existence and the uniqueness of solutions of (5)-(8). We consider the regular potential

We denote by ^{2}-norm (with associated product scalar (.,.)) and set

boundary conditions. More generally,

Throughout this paper, the same letters

We multiply (5) by

where

satisfies

Finaly, we conclude that

and

for all

Multiply (6) by

Then

In this study, we have three main results; existence theorem, uniqueness theorem and existence theorem with more regularity.

Theorem 4.1. (Existence) We assume

and

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

and

Proof. Let

We multiply (12) by

Multiplying (13) by

Now summing (14) and (15) we obtain

where

Lagrange theorem gives a estimates

which implies

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.

Theorem 4.3. Assume

then the system (5)-(8) possesses a unique solution

and

Proof. Following theorems 4.1 and 4.2, the system (5)-(8) possesses the unique solution

and

Multiply (2.1) by

we deduce the following inequality

Thanks to use

Since

Multiplying (6) by

Now summing (18) and (19), we obtain

where

Appling the Gronwall’s lemma, we deduce that

and

Multiplying (5) by

Thanks to use

Inserting the above estimate into (20), we obtain

which implies that

Multiplying (6) by

that implies

We have just shown the theorems of existence and uniqueness of the solutions for perturbed Cahn-Hilliard hyperbolic phase-field system with regular potentials.

De Dieu Mangoubi, J., Moukoko, D., Moukamba, F. and Langa, F.D.R. (2016) Existence and Uniqueness of Solution for Cahn-Hilliard Hyperbolic Phase-Field System with Dirichlet Boundary Condition and Regular Potentials. Applied Mathematics, 7, 1919-1926. http://dx.doi.org/10.4236/am.2016.716157