On the Caginalp for a Conserve Phase-Field with a Polynomial Potentiel of Order 2 p − 1

Our aim in this paper is to study on the Caginalp for a conserved phase-field with a polynomial potentiel of order 2p − 1. In this part, one treats the conservative version of the problem of generalized phase field. We consider a regular potential, more precisely a polynomial term of the order 2p − 1 with edge conditions of Dirichlet type. Existence and uniqueness are analyzed. More precisely, we precisely, we prove the existence and uniqueness of solutions.

Furthermore, all physical constants have been set equal to one. This system models, e.g., melting-solidification phenomena in certain classes of materials.
The Caginalp system can be derived as follows. We first consider the (total) free energy ( ) ( ) 2 where Ω is the domain occupied by the materiel. We then define the enthalpy H as where ∂ denotes a variational derivative, which gives .
The governing equations for u and θ are then given by (see [9]) where q is the thermal flux vector. Assuming the classical Fourier Law we find (1) and (2). Now, a drawback of the Fourier Law is the so-called "paradox of heat conduction", namely, it predicts that thermal signals propagate with infinite speed, which, in particular, violates causality (see, e.g. [10] and [11]). One possible modification, in order to correct this unrealistic feature, is the Maxwell-Cattaneo Law. 1 , In that case, it follows from (7) that This model can also be derived by considering, as in [12] (see also [13]- [20]), the Caginalp phase-field model with the so-called Gurtin-Pipkin Law for an exponentially decaying memory kernel k, namely, Indeed, differentiating (11) with respect to t and integrating by parts, we recover the Maxwell-Cattaneo Law (9). Now, in view of the mathematical treatment of the problem, it is more convenient to introduce the new variable is the conductive thermal displacement. Noting that T t α ∂ = ∂ , we finally deduce from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see [17]): In this paper, we consider the following conserved phase-field model: These equations are known as the conserved phase-field model (see [21]- [30]) based on type II heat conduction and with two temperatures (see [3] and [4]), conservative in the sense that, when endowed with Neumann boundary conditions, the spatial average of the order parameter is a conserved quantity. Indeed, in that case, integrating (18) over the spatial domain Ω , we have the conservation of mass, 1 dx vol Ω ⋅ = Ω ∫ (21) denotes the spatial average. Furthermore, integrating (19) over, we obtain Our aim in this paper is to study the existence and uniqueness of solution of (17)-(39). We consider here only one type of boundary condition, namely, Dirichlet (see [31] [32] [33]).

Setting of the Problem
We consider the following initial and boundary value problem As far as the nonlinear term f is concerned, we assume that Consider the following polynomial potential of order 2p − 1 ( ) The function f satisfies the following properties ( ) Remark 2.1. We take here, for simplicity, Dirichlet Boundary Conditions. However, we can obtain the same results for Neumann Boundary Conditions, namely, where v denotes the unit outer normal to Γ . To do so, we rewrite, owing to (23) and (24), the equations in the form M . Then, we note that ( ) where, here, −∆ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that H Ω , respectively, which are equivalent to the usual ones.
We further assume that which allows to deal with term

Notations
We denote by ⋅ the usual L 2 -norm (with associated product scalar (.,.) and set ( )

A Priori Estimates
The estimates derived in this subsection will be formal, but they can easily be justified within a Galerkin scheme. We rewrite (23) in the equivalent form We multiply (35) by u t ∂ ∂ and have, integrating over Ω and by parts; We then multiply (24) by t Summing (36) and (37), we find the differential inequality of the form ( ) We multiply (40) by ( ) We multiply by (41) by t α ∂ ∂ , we have Now summing (44) and (45) We know that and recalling the interpolation inequality    We multiply (24)

Conclusion
In this work we have studied the existence and uniqueness of the solution of a conservative-type Caginalp system with Dirichlet-type boundary conditions. Finally we have also succeeded in this work to establish the existence theorems of the solution of this system with low regularity and more regularity. As a perspective, we plan to study this problem in a bounded or unbounded domain with different types of potentials and Neumann-type conditions.