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

Narcisse Batangouna, Cyr Séraphin Ngamouyih Moussata, Urbain Cyriaque Mavoungou

Faculté des Sciences et Techniques Université Marien Ngouabi Brazzaville, République du Congo.

DOI: 10.4236/jamp.2020.812203                

Abstract

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.

Keywords

A Conserved Phase-Field, Polynomial Potentiel of Order 2p - 1, Dirichlet Boundary Conditions, Maxwell-Cattaneo Law

1. Introduction

The Caginalp phase-field model

$\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\theta$ (1)

$\frac{\partial \theta }{\partial t}-\Delta \theta =-\frac{\partial u}{\partial t}$ (2)

proposed in [1] , has been extensively studied (see, e.g., [2] - [7] and [8] ). Here, u denotes the order parameter and $\theta$ the (relative) temperature.

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

$\psi \left(u\mathrm{,}\theta \right)={\displaystyle {\int }_{\Omega }}\left(\frac{1}{2}{\left|\nabla u\right|}^2+f\left(u\right)-u\theta -\frac{1}{2}{\theta }^2\right)\text{d}x\mathrm{,}$ (3)

where $\Omega$ is the domain occupied by the materiel.

We then define the enthalpy H as

$H=-\frac{\partial \psi }{\partial \theta }$ (4)

where $\partial$ denotes a variational derivative, which gives

$H=u+\theta .$ (5)

The governing equations for u and $\theta$ are then given by (see [9] )

$\frac{\partial u}{\partial t}=-\frac{\partial \psi }{\partial u},$ (6)

$\frac{\partial H}{\partial t}+divq=0,$ (7)

where q is the thermal flux vector. Assuming the classical Fourier Law

$q=-\nabla \theta ,$ (8)

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.

$\left(1+\frac{\partial }{\partial t}\right)q=-\nabla \theta ,$ (9)

In that case, it follows from (7) that

$\left(1+\frac{\partial }{\partial t}\right)\frac{\partial H}{\partial t}-\Delta \theta =0,$

hence,

$\frac{{\partial }^2\theta }{\partial t^2}+\frac{\partial \theta }{\partial t}-\Delta \theta =\frac{{\partial }^{2}u}{\partial t^2}+\frac{\partial u}{\partial t}.$ (10)

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

$q\left(t\right)=-{\displaystyle {\int }_0^{+\infty }}k\left(s\right)\nabla \theta \left(t-s\right)\text{d}s.$ (11)

for an exponentially decaying memory kernel k, namely,

$k\left(s\right)={\text{e}}^{-s}\mathrm{.}$ (12)

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

$\alpha ={\displaystyle {\int }_0^t}\theta \left(s\right)\text{d}s,\theta =\frac{\partial \alpha }{\partial t},$ (13)

and we have, integrating (10) with respect to $s\in \left[\mathrm{0,1}\right]$ .

$\frac{{\partial }^2\alpha }{\partial t^2}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (14)

where

$\alpha \left(t\mathrm{,}x\right)={\displaystyle {\int }_0^t}T\left(\tau \mathrm{,}x\right)\text{d}\tau +{\alpha }_0\left(x\right)$ (15)

is the conductive thermal displacement. Noting that $T=\frac{\partial \alpha }{\partial t}$ , we finally deduce

from (33) and (36)-(37) the following variant of the Caginalp phase-field system (see [17] ):

$\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\frac{\partial \alpha }{\partial t}$ (16)

$\frac{{\partial }^2\alpha }{\partial t^2}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (17)

In this paper, we consider the following conserved phase-field model:

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta f\left(u\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (18)

$\frac{{\partial }^2\alpha }{\partial t^2}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (19)

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 $\Omega$ , we have the conservation of mass,

$\langle u\left(t\right)\rangle =\langle u\left(0\right)\rangle ,t\ge 0$ (20)

$\langle \cdot \rangle =\frac{1}{vol\Omega }{\displaystyle {\int }_{\Omega }}\text{d}x$ (21)

denotes the spatial average. Furthermore, integrating (19) over, we obtain

$\langle \alpha \left(t\right)\rangle =\langle \alpha \left(0\right)\rangle \mathrm{,}t\ge 0$ (22)

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] ).

2. Setting of the Problem

We consider the following initial and boundary value problem

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta f\left(u\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (23)

$\frac{{\partial }^2\alpha }{\partial t^2}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (24)

${u|}_{\Gamma }={\Delta u|}_{\Gamma }={\alpha |}_{\Gamma }=0,\text{on}\partial \Omega ,$ (25)

${u|}_{t=0}=u_0,{\alpha |}_{t=0}={\alpha }_0,\frac{\partial \alpha }{\partial t}={\alpha }_1$ (26)

As far as the nonlinear term f is concerned, we assume that

$f\in C^{\infty }\left(R\right),f\left(0\right)=0$ (27)

Consider the following polynomial potential of order 2p − 1

$f\left(s\right)={\displaystyle \underset{i=1}{\overset{2p-1}{\sum }}}{a}_i{s}^i,p\in N^{\ast },p\ge 2;a_{2p-1}=2p{b}_{2p}\ge 0$ (28)

The function f satisfies the following properties

$\frac{1}{2}{a}_{2p-1}{s}^{2p}-c_1\le f\left(s\right)s\le \frac{3}{2}{a}_{2p-1}{s}^{2p}+c_1,$ (29)

$f^{\prime }\left(s\right)\ge \frac{1}{2}{a}_{2p-1}{s}^{2p-2}-c_2\ge -k,\forall s\in R,k\ge 0$ (30)

where

$F\left(s\right)={\displaystyle {\int }_0^s}f\left(\tau \right)\text{d}\tau$ (31)

such as

$\frac{1}{4p}{a}_{2p-1}{s}^{2p}-c_3\le F\left(s\right)\le \frac{3}{4p}{a}_{2p-1}{s}^{2p}+c_3$ (32)

Remark 2.1. We take here, for simplicity, Dirichlet Boundary Conditions. However, we can obtain the same results for Neumann Boundary Conditions, namely,

$\frac{\partial u}{\partial \nu }=\frac{\partial \Delta u}{\partial \nu }=\frac{\partial \phi }{\partial \nu }\text{on}\Gamma$ (33)

where v denotes the unit outer normal to $\Gamma$ . To do so, we rewrite, owing to (23) and (24), the equations in the form

$\frac{\partial \bar{u}}{\partial t}+{\Delta }^2\bar{u}-\Delta \left(f\left(u\right)-\langle f\left(u\right)\rangle \right)=-\Delta \frac{\partial \bar{\alpha }}{\partial t}$

$\frac{{\partial }^2\bar{\phi }}{\partial t^2}+\frac{\partial \bar{\phi }}{\partial t}-\Delta \bar{\phi }=-\frac{\partial \bar{u}}{\partial t},$

where $\bar{v}=v-\langle v\rangle$ , $\left|\langle v_0\rangle \right|\le M_1$ , $\left|\langle v_0\rangle \right|\le M_2$ , for fixed positive constants $M_1$ and $M_2$ . Then, we note that

$v\to {\left({\Vert {\left(-\Delta \right)}^{\frac{-1}{2}}v\Vert }^2+{\langle v\rangle }^2\right)}^{\frac{1}{2}}$

where, here, $-\Delta$ denotes the minus Laplace operator with Neumann boundary conditions and acting on functions with null average and where it is understood that

$\langle \cdot \rangle \mathrm{<mo>\; =\; </mo>}\frac{1}{vol\left(\Omega \right)}{\langle \cdot \mathrm{,1}\rangle }_{H^{-1}\left(\Omega \right),H^1(\; \Omega \; )}$

Furthermore

$v\mapsto {\left({\Vert \bar{v}\Vert }^2+{\langle v\rangle }^2\right)}^{\frac{1}{2}}\mathrm{,}$

$v\mapsto {\left({\Vert \nabla v\Vert }^2+{\langle v\rangle }^2\right)}^{\frac{1}{2}}\mathrm{,}$

$v\mapsto {\left({\Vert \Delta v\Vert }^2+{\langle v\rangle }^2\right)}^{\frac{1}{2}}$

are norms in $H^{-1}\left(\Omega \right)$ , $L^2\left(\Omega \right)$ , $H^1\left(\Omega \right)$ and $H^2\left(\Omega \right)$ , respectively, which are equivalent to the usual ones.

We further assume that

$\left|f\left(s\right)\right|\le \epsilon F\left(s\right)+c_{\epsilon },\forall \epsilon >0,s\in R,$ (34)

which allows to deal with term $\langle f\left(u\right)\rangle$ .

3. Notations

We denote by $\Vert \cdot \Vert$ the usual L²-norm (with associated product scalar (.,.) and set ${\Vert \cdot \Vert }_{-1}=\Vert {\left(-\Delta \right)}^{\frac{-1}{2}}\cdot \Vert$ , where $-\Delta$ denotes the minus Laplace operator with Dirichlet Boundary Conditions. More generally, ${\Vert \cdot \Vert }_X$ denote the norm of Banach space X.

Throughout this paper, the same letters $c_1\mathrm{,}{c}_2$ and $c_3$ denote (generally positive) constants which may change from line to line, or even a same line.

4. 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

${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}-\Delta u+f\left(u\right)=\frac{\partial \alpha }{\partial t}\mathrm{.}$ (35)

We multiply (35) by $\frac{\partial u}{\partial t}$ and have, integrating over $\Omega$ and by parts;

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2=2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$ (36)

We then multiply (24) by $\frac{\partial \alpha }{\partial t}$ to obtain

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2=-2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$ (37)

Summing (36) and (37), we find the differential inequality of the form

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x+{\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2=0$ (38)

Integrating from 0 to t with $t\in \left[\mathrm{0\; <mo>\; ;\; </mo>}T\right]$ we obtain

$\begin{array}{l}{\displaystyle {\int }_0^t}\left(\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x+{\Vert \nabla \alpha \left(s\right)\Vert }^2+{\Vert \frac{\partial \alpha \left(s\right)}{\partial t}\Vert }^2\right)\text{d}s\\ +2{\displaystyle \int }{\Vert \frac{\partial \alpha \left(s\right)}{\partial t}\Vert }^2\text{d}s+2{\displaystyle \int }{\Vert \frac{\partial u\left(s\right)}{\partial t}\Vert }_{-1}^2\text{d}s=0\end{array}$

of (35) we deduce

$F\left(u_0\right)\le \frac{3}{4p}{a}_{2p-1}{u}_0^{2p}+c_3$

which involves

$2{\displaystyle {\int }_{\Omega }}F\left(u_0\right)\text{d}x\le \frac{3}{2p}{a}_{2p-1}{\Vert u_0\Vert }_{L^{2p}}^{2p}+2c_3\left|\Omega \right|$

still of (35) we have

$\frac{3}{4p}{a}_{2p-1}{u}_0^{2p}-c_3\le F(\; u\; )$

which involves

$\frac{1}{2p}{a}_{2p-1}{\Vert u_0\Vert }_{L^{2p}}^{2p}-2c_3\left|\Omega \right|\le F(\; u\; )$

where

$E\left(t\right)+2{\displaystyle {\int }_0^t}\left({\Vert \frac{\partial \alpha \left(s\right)}{\partial t}\Vert }^2+{\Vert \frac{\partial u\left(s\right)}{\partial t}\Vert }_{-1}^2\right)\text{d}s\le C$

with

$E\left(t\right)={\Vert \nabla u\left(t\right)\Vert }^2+\frac{1}{2p}{a}_{2p-1}{\Vert u\left(t\right)\Vert }_{L^{2p}}^{2p}+{\Vert \frac{\partial \alpha \left(t\right)}{\partial t}\Vert }^2+{\Vert \nabla \alpha \left(t\right)\Vert }^2$ (39)

and $C={\Vert \nabla u_0\Vert }^2+\frac{3}{2p}{a}_{2p-1}{\Vert u_0\Vert }_{L^{2p}}^{2p}+{\Vert {\alpha }_1\Vert }^2+{\Vert \nabla {\alpha }_0\Vert }^2+C_3$ .

Finally, we conclude that $u\in L^{\infty }\left(R^{\ast }\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\alpha \in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)$ ;

$\frac{\partial u}{\partial t}\in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\frac{\partial \alpha }{\partial t}\in L^{\infty }\left(R_{+}^{\ast }\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)$ $\forall T>0$

Theorem 4.1. (Existence) We assume $\left(u_0\mathrm{,}{\alpha }_0\mathrm{,}{\alpha }_1\right)\in \left(H_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)\times H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)$ then the system (18)-(19) possesses at least one solution $\left(u\mathrm{,}\alpha \right)$ such that

$u\in L^{\infty }\left(R^{\ast }\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\alpha \in L^2(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}(\; \Omega \; )\; )$

$\frac{\partial u}{\partial t}\in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\frac{\partial \alpha }{\partial t}\in L^{\infty }\left(R_{+}^{\ast }\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\cap L^2(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2(\; \Omega \; )\; )$

$\forall T>0$

Theorem 4.2. (Uniqueness) Let the assumptions of Theorem 4.1 hold. Then, the system (18)-(19) possesses a unique solution $\left(u\mathrm{,}\alpha \right)$ such that

$u\in L^{\infty }\left(R^{\ast }\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\alpha \in L^2(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}(\; \Omega \; )\; )$

$\frac{\partial u}{\partial t}\in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)\mathrm{<mo>\; ;\; </mo>}\frac{\partial \alpha }{\partial t}\in L^{\infty }(R_{+}^{\ast }\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\cap L^2\mathrm{<mo\; stretchy="false">\; (\; </mo>0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2(\; \Omega \; )\; )$

$\forall T>0$

Let $\left(u^{\left(1\right)}\mathrm{,}{\alpha }^{\left(1\right)}\mathrm{,}\frac{\partial {\alpha }^{\left(1\right)}}{\partial t}\right)$ and $\left(u^{\left(2\right)}\mathrm{,}{\alpha }^{\left(2\right)}\mathrm{,}\frac{\partial {\alpha }^{\left(2\right)}}{\partial t}\right)$ be two solutions (23)-(25) with initial data $\left(u_0^{\left(1\right)}\mathrm{,}{\alpha }_0^{\left(1\right)}\mathrm{,}{\alpha }_1^{\left(1\right)}\right)$ and $\left(u_0^{\left(2\right)}\mathrm{,}{\alpha }_0^{\left(2\right)}\mathrm{,}{\alpha }_1^{\left(2\right)}\right)$ , respectively. We set

$\left(u\mathrm{,}\alpha \mathrm{,}\frac{\partial \alpha }{\partial t}\right)=\left(u^{\left(1\right)}\mathrm{,}{\alpha }^{\left(1\right)}\mathrm{,}\frac{\partial {\alpha }^{\left(1\right)}}{\partial t}\right)-\left(u^{\left(2\right)}\mathrm{,}{\alpha }^{\left(2\right)}\mathrm{,}\frac{\partial {\alpha }^{\left(2\right)}}{\partial t}\right)$

and

$\left(u_0,{\alpha }_0,{\alpha }_1\right)=\left(u_0^{\left(1\right)},{\alpha }_0^{\left(1\right)},{\alpha }_1^{\left(1\right)}\right)-(u_0^{\left(2\right)},{\alpha }_0^{\left(2\right)},{\alpha }_1^{(\; 2\; )\; )}$

Then, $\left(u\mathrm{,}\alpha \right)$ satisfies

$\frac{\partial u}{\partial t}+{\Delta }^{2}u-\Delta \left(f\left(u^{\left(1\right)}\right)-f\left(u^{\left(2\right)}\right)\right)=-\Delta \frac{\partial \alpha }{\partial t}$ (40)

$\frac{{\partial }^2\alpha }{\partial t^2}+\frac{\partial \alpha }{\partial t}-\Delta \alpha =-\frac{\partial u}{\partial t}$ (41)

${u|}_{\Gamma }={\Delta u|}_{\Gamma }={\alpha |}_{\Gamma }=0,\text{on}\partial \Omega ,$ (42)

${u|}_{t=0}=u_0,{\alpha |}_{t=0}={\alpha }_0,\frac{\partial \alpha }{\partial t}={\alpha }_1$ (43)

We multiply (40) by ${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}$ , we have

${\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+\left(\frac{\partial u}{\partial t}\mathrm{,}-\Delta u\right)+\left(-\Delta \left(f\left(u^{\left(1\right)}\right)-f\left(u^{\left(2\right)}\right)\right)\mathrm{,}{\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}\right)=\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$

$\frac{\text{d}}{\text{d}t}{\Vert \nabla u\Vert }^2+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2=-2\left(f\left(u^{\left(1\right)}\right)-f\left(u^{\left(2\right)}\right)\mathrm{,}\frac{\partial u}{\partial t}\right)+2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)\mathrm{.}$ (44)

We multiply by (41) by $\frac{\partial \alpha }{\partial t}$ , we have

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2=-2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$ (45)

Now summing (44) and (45) we obtain

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert \nabla u\Vert }^2+{\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\\ =-2\left(f\left(u^{\left(1\right)}\right)-f\left(u^{\left(2\right)}\right)\mathrm{,}\frac{\partial u}{\partial t}\right)\end{array}$ (46)

We know that

$f\left(u^1\right)-f\left(u^2\right)={\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}{a}_k\left(u^{\left(1\right)k}\right)-{\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}{a}_k\left(u^{\left(2\right)k}\right)={\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}{a}_k\left(u^{\left(1\right)k}-u^{\left(2\right)k}\right)$

which involves

$\begin{array}{l}\left|f\left(u^1\right)-f\left(u^2\right)\right|\le {\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}\left|a_k\right|\left|u^{\left(1\right)k}-u^{\left(2\right)k}\right|\\ \le {\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}\left|a_k\right|\left|u^{\left(1\right)}-u^{\left(2\right)}\right|{\left|u^{\left(1\right)}\right|}^{k-1}+{\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}{\left|u^{\left(1\right)}\right|}^{k-1-j}{\left|u^{\left(2\right)}\right|}^j+{\left|u^{\left(2\right)}\right|}^{k-1}.\end{array}$

Based on Young’s inequality, we have

${\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}{\left|u^{\left(1\right)}\right|}^{k-1-j}{\left|u^{\left(2\right)}\right|}^j\le {\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}\left(\frac{k-j-1}{k-1}{\left|u^{\left(1\right)}\right|}^{k-1}+\frac{j}{k-1}{\left|u^{\left(2\right)}\right|}^{k-1}\right)$

with $p=\frac{k-1}{k-j-1}$ and $q=\frac{k-1}{j}$ such as $\frac{1}{p}+\frac{1}{q}=1$ . So

${\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}{\left|u^{\left(1\right)}\right|}^{k-1-j}{\left|u^{\left(2\right)}\right|}^j\le \frac{1}{k-1}{\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}\left(k-1\right){\left|u^{\left(1\right)}\right|}^{k-1}+\frac{1}{k-1}{\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}j\left({\left|u^{\left(2\right)}\right|}^{k-1}-{\left|u^{\left(1\right)}\right|}^{k-1}\right).$

As

${\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}j=\frac{\left(k-2\right)\left(k-1\right)}{2}$

then

$\begin{array}{c}{\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}{\left|u^{\left(1\right)}\right|}^{k-1-j}{\left|u^{\left(2\right)}\right|}^j\le \left(k-2\right){\left|u^{\left(1\right)}\right|}^{k-1}+\frac{k-2}{2}{\left|u^{\left(2\right)}\right|}^{k-1}-\frac{k-2}{2}{\left|u^{\left(1\right)}\right|}^{k-1}\\ \le \frac{k-2}{2}\left({\left|u^{\left(1\right)}\right|}^{k-1}+{\left|u^{\left(2\right)}\right|}^{k-1}\right).\end{array}$

We know that

$\forall k\in N$ ; $k-2\le k$ then $\frac{k-2}{2}\le \frac{k}{2}\le k$

${\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}{\left|u^{\left(1\right)}\right|}^{k-1-j}{\left|u^{\left(2\right)}\right|}^j\le k\left({\left|u^{\left(1\right)}\right|}^{k-1}+{\left|u^{\left(2\right)}\right|}^{k-1}\right)$

which gives

$\begin{array}{c}\left|f\left(u^1\right)-f\left(u^2\right)\right|\le {\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}\left|a_k\right|\left|u^{\left(1\right)}-u^{\left(2\right)}\right|\left(\left(k+1\right){\left|u^{\left(1\right)}\right|}^{k-1}+\left(k+1\right){\left|u^{\left(2\right)}\right|}^{k-2}\right)\\ \le \left|u\right|{\displaystyle \underset{j=1}{\overset{k-2}{\sum }}}\left(k+1\right)\left|a_k\right|\left({\left|u^{\left(1\right)}\right|}^{k-1}+{\left|u^{\left(2\right)}\right|}^{k-1}\right)\end{array}$

$\exists k>0$ such as

$\left(k+1\right)\left|a_k\right|\le k$ ; $\forall k\in \mathrm{1,2,}\cdots \mathrm{,2}p-1$

so

$\left|f\left(u^1\right)-f\left(u^2\right)\right|\le \left|u\right|k{\displaystyle \underset{k=1}{\overset{k-2}{\sum }}}\left({\left|u^{\left(1\right)}\right|}^{k-1}+{\left|u^{\left(2\right)}\right|}^{k-1}\right).$

Based on Young’s inequality, we have $\forall k\ge 2$

${\left|u^{\left(1\right)}\right|}^{k-1}\le \frac{k-1}{2p-2}{\left({\left|u^{\left(1\right)}\right|}^{k-1}\right)}^{\frac{2p-2}{k-1}}+\frac{2p-k-1}{2p-2}$

and

${\left|u^{\left(2\right)}\right|}^{k-1}\le \frac{k-1}{2p-2}{\left({\left|u^{\left(2\right)}\right|}^{k-1}\right)}^{\frac{2p-2}{k-1}}+\frac{2p-k-1}{2p-2}$

that involve

$\begin{array}{c}\left|f\left(u^1\right)-f\left(u^2\right)\right|\le \left|u\right|\frac{k}{2p-2}{\displaystyle \underset{k=1}{\overset{2p-1}{\sum }}}\left(\left(k-1\right)\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(2\right)}\right|}^{2p-2}\right)+2\left(\frac{2p-k-1}{2p-2}\right)\right)\\ \le c\left|u\right|\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(2\right)}\right|}^{2p-2}+1\right)\mathrm{.}\end{array}$

We finally

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le c{\displaystyle {\int }_{\Omega }}\left|u\right|\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(2\right)}\right|}^{2p-2}+1\right)\left|\frac{\partial u}{\partial t}\right|\text{d}x\mathrm{.}$ (47)

The second member of (45) is increased in $R^n$ for $n=1,2,3$ .

If n = 1; $u^i\in H_0^1\left(\Omega \right)\subset H^1\left(\Omega \right)=W^{1,2}\left(\Omega \right)$ for $i=1,2$ .

Thanks to the continuous injection $H^1\left(\Omega \right)\subset C\left(\bar{\Omega }\right)$ , then is $C>0$ , by applying Holder’s inegality, we get

${\displaystyle {\int }_{\Omega }}\left|u\right|\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(1\right)}\right|}^{2p-2}+1\right)\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C\Vert u\Vert \Vert \frac{\partial u}{\partial t}\Vert \mathrm{,}$

which involves using the compact injection $H^1\left(\Omega \right)\subset L^2\left(\Omega \right)$ , we have

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{H^1}\Vert \frac{\partial u}{\partial t}\Vert$ (48)

If n = 2 then $H^1\left(\Omega \right)\subset L^q\left(\Omega \right)$ , $\forall q\in \left[\mathrm{1,}\infty \right[$ .

Based on Holder’s inequality, we have

${\displaystyle {\int }_{\Omega }}\left|u\right|\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(1\right)}\right|}^{2p-2}+1\right)\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{L^3}\Vert \frac{\partial u}{\partial t}\Vert \mathrm{.}$

Finally

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{H^1}\Vert \frac{\partial u}{\partial t}\Vert$

If n = 3, then $H^1\left(\Omega \right)\subset L^q\left(\Omega \right)$ with $q\in \left[\mathrm{1,6}\right]$

In this case, we also

${\displaystyle {\int }_{\Omega }}\left|u\right|\left({\left|u^{\left(1\right)}\right|}^{2p-2}+{\left|u^{\left(1\right)}\right|}^{2p-2}+1\right)\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{L^6}\Vert \frac{\partial u}{\partial t}\Vert \mathrm{.}$

So

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{H^1}\Vert \frac{\partial u}{\partial t}\Vert \mathrm{.}$

We notice that in $R^n$ for $n=1,2,3$ , we have

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{H^1}\Vert \frac{\partial u}{\partial t}\Vert \mathrm{.}$

Using Young’s inequality, we have

${\displaystyle {\int }_{\Omega }}\left|f\left(u^1\right)-f\left(u^2\right)\right|\left|\frac{\partial u}{\partial t}\right|\text{d}x\le C{\Vert u\Vert }_{H^1}^2+{\Vert \frac{\partial u}{\partial t}\Vert }^2$ (49)

Inserting (49) into (46), we find

$\frac{\text{d}}{\text{d}t}{E}_2+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\le c^{\prime }{\Vert u\Vert }_{H^1}^2+{\Vert \frac{\partial u}{\partial t}\Vert }^2$

and recalling the interpolation inequality ${\Vert \frac{\partial u}{\partial t}\Vert }^2\le c{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}\Vert \nabla \frac{\partial u}{\partial t}\Vert$

with $E_2={\Vert \nabla u\Vert }^2+{\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2$

Finally

$\frac{\text{d}}{\text{d}t}{E}_2+c^{″}{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\le C{E}_2\mathrm{,}C>0$ (50)

Theorem 4.3. (Second theorem of the solution’s existence) The existence and uniqueness of the solution (23)-(25) problem being proven, now we seek the solution of (23)-(25) with more regularity.

Assume $\begin{array}{l}\left(u_0\mathrm{,}{\alpha }_0\mathrm{,}{\alpha }_1\right)\in H^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\\ \times \left(u_0\mathrm{,}{\alpha }_0\mathrm{,}{\alpha }_1\right)\in H^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\times H_0^1\left(\Omega \right)\end{array}$ , then the (23)-(24) system admits a unique $\left(u\mathrm{,}\alpha \right)$ solution such as

$u\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\right)\mathrm{,}\alpha \in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\right),$

$\frac{\partial \alpha }{\partial t}\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap H_0^1\left(\Omega \right)\right)\mathrm{,}$

and

$\frac{\partial u}{\partial t}\in L^2(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}(\; \Omega \; )\; )$

Theorems of existence (23) and uniqueness (24) being proven then $u\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)$ , $\alpha \in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\right)$ , $\frac{\partial \alpha }{\partial t}\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)$ and $\frac{\partial u}{\partial t}\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)$ , $\forall T>0$ .

We multiply (23) by ${\left(-\Delta \right)}^{-1}\frac{\partial u}{\partial t}$ and have, integrating over $\Omega$ , we have

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2=2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$ (51)

Multiplying (24) by $\frac{\partial \alpha }{\partial t}$ , we have

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2=-2\left(\frac{\partial u}{\partial t}\mathrm{,}\frac{\partial \alpha }{\partial t}\right)$ (52)

Now summing (51) and (52) we obtain

$\frac{\text{d}}{\text{d}t}\left({\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x+{\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2\right)+2{\Vert \frac{\partial u}{\partial t}\Vert }_{-1}^2+2{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2=0$ (53)

where

$E_3={\Vert \nabla u\Vert }^2+2{\displaystyle {\int }_{\Omega }}F\left(u\right)\text{d}x+{\Vert \nabla \alpha \Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2$

finally

${\Vert \nabla u\left(t\right)\Vert }^2+c{\Vert u\left(t\right)\Vert }_{L^{2p}}^{2p}+{\Vert \nabla \alpha \left(t\right)\Vert }^2+{\Vert \frac{\partial \alpha }{\partial t}\Vert }^2+2{\displaystyle {\int }_0^t}\left({\Vert \frac{\partial \alpha \left(s\right)}{\partial t}\Vert }^2+{\Vert \frac{\partial u\left(s\right)}{\partial t}\Vert }_{-1}^2\right)\text{d}s\le c_1\mathrm{.}$

We infer that

$u\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^2\left(\Omega \right)\cap L^{2p}\left(\Omega \right)\right)$ , $\alpha \in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}_0^1\left(\Omega \right)\right)$ ,

$\frac{\partial \alpha }{\partial t}\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)$ and $\frac{\partial u}{\partial t}\in L^{\infty }\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}^{-1}\left(\Omega \right)\right)$ .

We multiply (24) by $\frac{{\partial }^2\alpha }{\partial t^2}$ , we have

$\frac{\text{d}}{\text{d}t}\left({\Vert \frac{\partial \alpha }{\partial t}\Vert }^2+{\Vert \nabla \alpha \Vert }^2\right)+{\Vert \frac{{\partial }^2\alpha }{\partial t^2}\Vert }^2\le {\Vert \frac{\partial u}{\partial t}\Vert }^2\mathrm{.}$

We infer from this that $\frac{{\partial }^2\alpha }{\partial t^2}\in L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)$ .

5. 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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Altundas, Y.B. and Caginalp, G. (2005) Velocity Selection in 3D Dendrites: Phase Field Computations and Microgravity Experiments. Nonlinear Analysis, 62, 467-481. https://doi.org/10.1016/j.na.2005.02.122
[2]	Babin, A.V. and Vishik, M.I. (1992) Attractors of Evolution Equations. Vol. 25 of Studies in Mathematics and Its Applications, North-Holland Publishing Co., Amsterdam.
[3]	Batangouna, N. and Pierre, M. (2018) Convergence of Exponential Attractors for a Time Splitting Approximation of the Caginalp Phase-Field System. Communications on Pure & Applied Analysis, 17, 1-19. https://doi.org/10.3934/cpaa.2018001
[4]	Bai, F., Elliott, C.M., Gardiner, A., et al. (1995) The Viscous Cahn-Hilliard Equation. I. Computations. Nonlinearity, 8, 131-160. https://doi.org/10.1088/0951-7715/8/2/002
[5]	Bates, P.W. and Zheng, S.M. (1992) Inertial Manifolds and Inertial Sets for the Phase-Field Equations. Journal of Dynamics and Differential Equations, 4, 375-398. https://doi.org/10.1007/BF01049391
[6]	Brezis, H. (1973) Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam, London, American Elsevier Publishing Co. Inc., New York.
[7]	Brochet, D., Hilhorst, D. and Chen, X. (1993) Finite-Dimensional Exponential Attractor for the Phase Field Model. Applicable Analysis, 49, 197-212. https://doi.org/10.1080/00036819108840173
[8]	Brokate, M. and Sprekels, J. (1996) Hysteresis and Phase Transitions. Vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-4048-8
[9]	Landau, L.D. and Lifshitz, E.M. (1980) Statistical Physics I. 3rd Edition, Butterworth-Heinemann, Oxford.
[10]	Caginalp, G. and Socolovsky, E.A. (1989 Efficient Computation of a Sharp Interface by Spreading via Phase Field Methods. Applied Mathematics Letters, 2, 117-120. https://doi.org/10.1016/0893-9659(89)90002-5
[11]	Caginalp, G. (1986) An Analysis of a Phase Field Model of a Free Boundary. Archive for Rational Mechanics and Analysis, 92, 205-245. https://doi.org/10.1007/BF00254827
[12]	Chen, P.J., Gurtin, M.E. and Williams, W.O. (1968) A Note on Non-Simple Heat Conduction. Journal of Applied Physics (ZAMP), 19, 969-970. https://doi.org/10.1007/BF01602278
[13]	Chepyzhov, V.V. and Vishik, M.I. (2002) Attractors for Equations of Mathematical Physics. Vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence. https://doi.org/10.1090/coll/049
[14]	Chill, R., Fašangová, E. and Prüss, J. (2006) Convergence to Steady State of Solutions of the Cahn-Hilliard and Caginalp Equations with Dynamic Boundary Conditions. Mathematische Nachrichten, 279, 1448-1462. https://doi.org/10.1002/mana.200410431
[15]	Dupaix, C., Hilhorst, D. and Kostin, I.N. (1999) The Viscous Cahn-Hilliard Equation as a Limit of the Phase Field Model: Lower Semicontinuity of the Attractor. Journal of Dynamics and Differential Equations, 11, 333-353. https://doi.org/10.1023/A:1021985631123
[16]	Doumbe, B. (2013) Etude de modeles de champ de phase de type Caginalp. PhD Thesis, Université de Poiters.
[17]	Efendiev, M., Miranville, A. and Zelik, S. (2000) Exponential Attractors for a Nonlinear Reaction-Diffusion System in R3. Comptes Rendus de l’Académie des Sciences—Series I—Mathematics, 330, 713-718. https://doi.org/10.1016/S0764-4442(00)00259-7
[18]	Efendiev, M., Miranville, A. and Zelik, S. (2004) Exponential Attractors for a Singularly Perturbed Cahn-Hilliard System. Mathematische Nachrichten, 272, 11-31. https://doi.org/10.1002/mana.200310186
[19]	Elliott, C.M. and Stuart, A.M. (1993) The Global Dynamics of Discrete Semilinear Parabolic Equations. SIAM Journal on Numerical Analysis, 30, 1622-1663. https://doi.org/10.1137/0730084
[20]	Miranville, A. and Quintanilla, R. (2009) Some Generalizations of the Caginalp Phase-Field System. Applicable Analysis, 88, 877-894. https://doi.org/10.1080/00036810903042182
[21]	Miranville, A. (2014) Some Mathematical Models in Phase Transition. Discrete & Continuous Dynamical Systems Series S, 7, 271-306. https://doi.org/10.3934/dcdss.2014.7.271
[22]	Miranville, A. and Quintanilla, R. (2016) A Caginalp Phase-Field System Based on Type III Heat Conduction with Two Temperatures. Quarterly of Applied Mathematics, 74, 375-398. https://doi.org/10.1090/qam/1430
[23]	Penrose, O. and Fife, P.C. (1990) Thermodynamically Consistent Models of Phase-Field Type for the Kinetics of Phase Transitions. Journal of Physics D, 43, 44-62. https://doi.org/10.1016/0167-2789(90)90015-H
[24]	Quintanilla, R. (2009) A Well-Posed Problem for the Three-Dual-Phase-Lag Heat Conduction. Journal of Thermal Stresses, 32, 1270-1278. https://doi.org/10.1080/01495730903310599
[25]	Temam, R. (1997) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Vol. 68 of Applied Mathematical Sciences, 2nd Edition, Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-0645-3
[26]	Mavoungou, U.C. (2016) Existence and Uniqueness of Solution for Caginalp Hyperbolic Phase-Field System with a Singular Potential.
[27]	Fakih, H. (2015) A Cahn Hilliard Equation with a Proliferation Term for Biological and Chemical Applications. Asymptotic Analysis, 94, 71-104. https://doi.org/10.3233/ASY-151306
[28]	Ntsokongo, A.J. and Batangouna, N. (2016) Existence and Uniqueness of Solutions for a Conserved Phase-Field Type Model. AIMS Mathematics, 1, 144-155. https://doi.org/10.3934/Math.2016.2.144
[29]	Raugel, G. (2002) Global Attractors in Partial Differential Equations. In: Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 885-982. https://doi.org/10.1016/S1874-575X(02)80038-8
[30]	Stuart, A.M. and Humphries, A.R. (1996) Dynamical Systems and Numerical Analysis. Vol. 2 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge.
[31]	Temam, R. (1969) Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Archive for Rational Mechanics and Analysis, 33, 377-385. https://doi.org/10.1007/BF00247696
[32]	Zhang, Z. (2005) Asymptotic Behavior of Solutions to the Phase-Field Equations with Neumann Boundary Conditions. Communications on Pure & Applied Analysis, 4, 683-693. https://doi.org/10.3934/cpaa.2005.4.683
[33]	Zhu, C. (2015) Attractor of a Semi-Discrete Benjamin-Bona-Mahony Equation on R1. Annales Polonici Mathematici, 115, 219-234. https://doi.org/10.4064/ap115-3-2