Existence and Uniqueness of Solution for Caginalp Hyperbolic Phase-Field System with a Polynomial Potential

We prove the existence and the uniqueness of solutions for Caginalp hyperbolic phase-field system with initial conditions, Dirichlet boundary homogeneous conditions and a regular potential of order 2p − 1, in bounded domain.


Introduction
We are interested on the study of the following Caginalp hyperbolic phase-field system in a smooth and bounded domain with homogenous Dirichlet conditions u| ∂Ω = α| ∂Ω = 0 (3) and initial conditions u| t=0 = u 0 , ∂u ∂t where ϵ > 0 is a relaxation parameter, u = u(x, t) the order parameter and α = α(t, x) are the unknown functions, f is a regular potential.
Consider the following polynomial potential of order 2p − 1 The function f satisfies the following properties We denote by ∥.∥ the usual L 2 -norm (with associated product scalar (., .)),△ denotes the Laplace operator with Dirichlet boundary conditions.Throughout this paper, C i (i = 1, ..., n) denote positive constants which may change from line to line, or even the same line.

Method
To prove our main results, we have to use classical methods of functional analysis applied in the theory of Partial Differential Equations.

Results
In this paper, we first prove the existence and the uniqueness of solutions theorems.The two first results being proven in a larger space, we will seek the solution with more regularity.

A priori Estimates
We multiply (1) by ∂u ∂t and (2) by γ ∂α ∂t where γ > 0, integrate over Ω.This gives d dt and d dt Sum the two resulting differential inequalities with 1 − 2γ > 0. Due to the equation ( 7), we get where Using (7) we have 2 Inserting the above estimate in (10), we obtain a differential inequality of the form where the strictly positive constant k is independent of ϵ.
Applying Gronwall's lemma, we obtain for all t ∈ (0, T ) which implies, in view of (7) Then we deduce that The main result of this paper is the proof of the existence and the uniqueness theorems.

Existence and Uniqueness of Solutions
The proof is based on a priori estimates obtained in the previous section and on a standard Galerkin scheme.
Theorem 2. (Uniqueness) If the assumptions of the theorem (1) hold.Then, the system ( 1)-( 2) has a unique solution with the above regularity.
Therefore, one gets In this proof, we consider the two cases n = 1 and n = 2 or 3.
Applying the Hölder inequality to (19), we obtain the following inequality ) Then, we have and Finally, for n = 1, 2 or 3, we obtain We deduce the continuous dependence of the solution with respect to the initial conditions, hence the uniqueness of the solution of problem ( 1)-( 7), is proved.
The existence and the uniqueness of the solution to problem (1)-( 7) being proven in a larger space, we now establish the solution with more regularity.

Discussion
In this paper, we have confirmed the existence and the uniqueness of the solution for hyperbolic Caginalp phase-field system with all regular potential.We next study the existence of global and exponential attractors.We can also complete this work by studying this system with any other types of boundary conditions.