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.

• Cahn-Hilliard Hyperbolic Phase-Field System
• Regular Potential
• Dirichlet Boundary Conditions

1. Introduction

G. Caginalp introduced in [1] the following phase-field system

where u is the order parameter and 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 potentials.

In [11] , Ntsokongo and Batangouna have studied the following Cahn-Hilliard phase- field system

where, u is the order parameter and 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. In the above hyperbolic system is a bounded and regular domain of with 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 which satisfies the following properties:

2. Notations

We denote by the usual L²-norm (with associated product scalar (.,.)) and set

, where denotes the minus Laplace operator with Dirichlet

boundary conditions. More generally, denote the norm of Banach space X.

Throughout this paper, the same letters and denote (generally positive) constants which may change from line to line, or even a same line.

3. A Priori Estimates

We multiply (5) by and (6) by, integrate over and add the two resulting differential equalities. We find

where

satisfies

Finaly, we conclude that,

and

for all.

Multiply (6) by and integrate over. We get.

Then.

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

4. Existence and Uniqueness of Solutions

Theorem 4.1. (Existence) We assume then the system (5) - (8) possesses at least one solution such that

,

,

and, for all.

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 such that

,

,

and for all.

Proof. Let and be two solutions of the system (5)-(8) with initial data and, respectively. We set and, then is solution of the following system

We multiply (12) by and integrate over. We find

Multiplying (13) by and integrating over, we get

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 such that

,

,

,

and, for all.

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

,

,

and, for all.

Multiply (2.1) by and integrate over. We have

we deduce the following inequality

Thanks to use, we find the following estimate

Since, then the estimate (17) implies

Multiplying (6) by and integrating over, we get

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

where

Appling the Gronwall’s lemma, we deduce that,

and

.

Multiplying (5) by and integrating ovre, we obtain

Thanks to use and the fact that, we get

Inserting the above estimate into (20), we obtain

which implies that.

Multiplying (6) by and integrating over, we find

that implies.

5. 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.

References

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