Solutions to Three-Phase-Field Model for Solidification

Abstract

Our research on applied mathematics is in line with the research scope of the journal. The phase-field model is applied to simulate the material and other areas. A phase-field model describing the non-isothermal solidification of an ideal multi-component alloy system is proposed in this paper. The time and space variation of a three-phase-field function and the governing equations of the temperature field are established. The global existence of weak solutions for three-dimensional parabolic differential equations is proved by the Faedo–Galerkin method. The existence of a maximum theorem is also extensively studied.

1. Introduction

Our research on applied mathematics is in line with the research scope of the journal. A multi-component alloy system is a kind of important material, especially in technology applications and technology. Therefore, it plays an important role in the formation of mechanical properties and the microstructure of materials. The phase-field model is studied in solid materials (cf. [1]). The multi-component in the alloy combines with the appearance of multi-phase, resulting in different phase transformations and different types of solidification. The solidification of binary alloys is the basis of studying the basic principles of the solidification process. The solidification of a multi-component system can be analyzed by the solidification characteristics of a binary system. In recent years, the multi-phase field (MPF) method has attracted extensive attention. This model is a generalization of Steinbach et al. (cf. [2]), which describes isothermal phase transitions of certain kinds of alloys. The multi-phase field method has been applied to the computer simulation of microstructure evolution (cf. [3,4]) and it was applied to simulate the dendrite solidification process in Fe-Cr-Ni-Mo-C (cf. [5]). It was also applied to the computer simulation of ice-formation in sea-water (cf. [6]). The classic solutions are proved in [7], where the authors set $\theta =e-l(u+v+w)$. In this paper, we study the non-isothermal solidification of ideal multi-component and multi-phase alloy systems. We set

$$\begin{array}{ccc}& & \theta =e-\frac{l_1}{4}(\frac{1}{3}{u}^3+u^{2}v-\frac{1}{3}{v}^3-u{v}^2)-\frac{l_3}{4}(\frac{1}{3}{u}^3+u^{2}w-\frac{1}{3}{w}^3-u{w}^2)\hfill \\ & & -\frac{l_2}{4}(\frac{1}{3}{v}^3+v^{2}w-\frac{1}{3}{w}^3-v{w}^2).\hfill \end{array}$$

Thus, we allow a temperature, which is assumed to be a priori given the free energy functional $F[u,v,w,\theta ]$. This means that the model not only considers the phase transformation caused by the difference in solute, but also considers the phase transformation caused by temperature change.

We studied a system of partial differential equations simulating the evolution of three-phase boundary problems in seawater, ice, and snow, and proved that the system had a global solution in the case of a one-dimensional initial boundary value problem (cf. [8]), where we cannot especially have $h(u,v,w)$.

In this paper we prove the global solution by the Galerkin method and we conclude a maximum theorem.

The order parameter represents the solidification state of the alloy, which is equal to 0 in the solid phase and 1 in the liquid phase. The model is a generalization of the one introduced by Steinbach et al. in [2], for isothermal solidification/melting processes of certain kinds of alloys. Our model must satisfy the following partial differential equations:

$$\begin{array}{c}{u}_t-k_1(w\mathsf{\Delta }u-u\mathsf{\Delta }w)-k_3(v\mathsf{\Delta }u-u\mathsf{\Delta }v)=-a_{1}uw(w-u)\hfill \\ -a_{3}uv(v-u)+\theta (l_{1}uv+l_{3}uw),\hfill \end{array}$$

(1)

$$\begin{array}{c}{v}_t-k_2(w\mathsf{\Delta }v-v\mathsf{\Delta }w)-k_3(u\mathsf{\Delta }v-v\mathsf{\Delta }u)=-a_{2}vw(w-v)\hfill \\ -a_{3}vu(u-v)+\theta (-l_{1}uv+l_{2}vw),\hfill \end{array}$$

(2)

$$\begin{array}{c}{w}_t-k_1(u\mathsf{\Delta }w-w\mathsf{\Delta }u)-k_2(v\mathsf{\Delta }w-w\mathsf{\Delta }v)=-a_{1}wu(u-w)\hfill \\ -a_{2}wv(v-w)+\theta (-l_{3}uw-l_{2}vw),\hfill \end{array}$$

(3)

$$\begin{array}{c}{\theta }_t+(l_{1}uv+l_{3}uw)u_t+(-l_{1}uv+l_{2}vw)v_t+(-l_{2}vw-l_{3}uw)w_t=D\mathsf{\Delta }\theta .\hfill \end{array}$$

(4)

for $(t,x)\in (0,T_e)\times \mathsf{\Omega }$. The boundary conditions and initial conditions are as follows:

$$\begin{array}{c}\hfill u(t,x)=0,v(t,x)=0,w(t,x)=0,(t,x)\in [0,T_e)\times \partial \mathsf{\Omega },\end{array}$$

(5)

$$\begin{array}{c}\hfill u(0,x)=u_0,v(0,x)=v_0\left(x\right),w(0,x)=w_0\left(x\right),x\in \mathsf{\Omega }.\end{array}$$

(6)

This system satisfies the second law of thermodynamics. Here, $\mathsf{\Omega }\subset {\mathbb{R}}^3$ is a bounded open domain. The function $\theta$ is the temperature. The phase-field functions u, v, and w are the fractions of two possible solid and liquid crystal states. For 3 phases we obtain the constraint $u+v+w=1$. For physical reasons, $k_1$, $k_2$, $k_3$, $a_1$, $a_2$, $a_3$, D are positive. In the free energy

$$\begin{array}{ccc}& & F[u,v,w,\theta ]={\int }_{\mathsf{\Omega }}\{\frac{k_1}{2}{|u\nabla w-w\nabla u|}^2+\frac{k_2}{2}{|w\nabla v-v\nabla w|}^2\hfill \\ & & +\frac{k_3}{2}{|u\nabla v-v\nabla u|}^2+\widehat{\psi }(u,v,w)-\frac{1}{4}\theta (l_1(\frac{1}{3}{u}^3+u^{2}v-\frac{1}{3}{v}^3-u{v}^2)\hfill \\ & & +l_3(\frac{1}{3}{u}^3+u^{2}w-\frac{1}{3}{w}^3-u{w}^2)+l_2(\frac{1}{3}{v}^3+v^{2}w-\frac{1}{3}{w}^3-v{w}^2))\}dx,\hfill \end{array}$$

where

$$\widehat{\psi }(u,v,w)=\frac{a_1}{2}{u}^2{w}^2+\frac{a_2}{2}{v}^2{w}^2+\frac{a_3}{2}{u}^2{v}^2,$$

we choose for $\widehat{\psi }\in C^2(\mathbb{R},[0,\infty ))$, which represent the double well potential.

$\theta$ satisfies

$$\begin{array}{ccc}& & \theta =e-\frac{l_1}{4}(\frac{1}{3}{u}^3+u^{2}v-\frac{1}{3}{v}^3-u{v}^2)-\frac{l_3}{4}(\frac{1}{3}{u}^3+u^{2}w-\frac{1}{3}{w}^3-u{w}^2)\hfill \\ & & -\frac{l_2}{4}(\frac{1}{3}{v}^3+v^{2}w-\frac{1}{3}{w}^3-v{w}^2),\hfill \end{array}$$

where e is the local enthalpy and $l_1,l_2,l_3$ are the latent heat of fusion.

It is easy to see in the case of two-phase systems, $u=1-v$, or $u=1-w$, or $v=1-w$, and $\frac{\partial u}{\partial v}=-1$, or $\frac{\partial u}{\partial w}=-1$, or $\frac{\partial v}{\partial w}=-1$.

We write $Q_T:=(0,T_e)\times \mathsf{\Omega }$, where $T_e$ is a positive constant, and define

$${(\upsilon ,\phi )}_{\mathbb{Z}}={\int }_{\mathbb{Z}}\upsilon \left(y\right)\phi \left(y\right)dy,$$

for $\mathbb{Z}=\mathsf{\Omega }$. Since we must have $u+v+w=1$ and $u_t+v_t+w_t=0$, the model can be reduced to the following:

$$\begin{array}{ccc}& & u_t-\left(k_1(1-v)+k_{3}v\right)\mathsf{\Delta }u-(k_1-k_3)u\mathsf{\Delta }v=-a_{1}u(1-u-v)(1-2u-v)\hfill \\ & & -a_{3}uv(v-u)+\theta \left(l_{1}uv+l_{3}u(1-u-v)\right),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & v_t-\left(k_2(1-u)+k_{3}u\right)\mathsf{\Delta }v-(k_2-k_3)v\mathsf{\Delta }u=-a_{2}v(1-u-v)(1-u-2v)\hfill \\ & & -a_{3}vu(u-v)+\theta \left(-l_{1}uv+l_{2}v(1-u-v)\right),\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & {\theta }_t-D\mathsf{\Delta }\theta =-\left(l_{1}uv+l_{2}v(1-u-v)+2l_{3}u(1-u-v)\right)u_t\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & -\left(-l_{1}uv+2l_{2}v(1-u-v)+l_{3}u(1-u-v)\right)v_t.\hfill \end{array}$$

(10)

The boundary and initial conditions are therefore

$$\begin{array}{ccc}& & u(t,x)=0,v(t,x)=0,\theta (t,x)=0,(t,x)\in (0,T_e)\times \partial \mathsf{\Omega },\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & u(0,x)=u_0\left(x\right),v(0,x)=v_0\left(x\right),\theta (0,x)={\theta }_0\left(x\right),x\in \mathsf{\Omega }.\hfill \end{array}$$

(12)

Definition 1.

Let $(u_0,v_0,{\theta }_0)\in H_0^1\left(\mathsf{\Omega }\right)\times H_0^1\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)$. A triple $(u,v,\theta )$ with

$$\begin{array}{ccc}& & u,v\in L^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H^2\left(\mathsf{\Omega }\right)),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \theta \in L^{\infty }(0,T_e;L^2\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H_0^1\left(\mathsf{\Omega }\right)),\hfill \end{array}$$

(14)

is a weak solution to problems(7)–(10), if for all $\phi \in C_0^{\infty }((-\infty ,T_e)\times \mathsf{\Omega }))$, there hold

$$\begin{array}{ccc}\hfill 0& =& {(u,{\phi }_t)}_{Q_{T_e}}-\alpha {(\nabla u,\nabla \phi )}_{Q_{T_e}}-\beta \left(\nabla v,\nabla \left(u\phi \right)\right)-a_1{\left(u(1-u-v)(1-2u-v),\phi \right)}_{Q_{T_e}}\hfill \\ & & -a_3{(uv(v-u),\phi )}_{Q_{T_e}}+{(u_0,\phi \left(0\right))}_{\mathsf{\Omega }}+{\left(\theta \left(l_{1}uv+l_{3}u(1-u-v)\right),\phi \right)}_{Q_{T_e}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill 0& =& {(v,{\phi }_t)}_{Q_{T_e}}-\gamma {(\nabla v,\nabla \phi )}_{Q_{T_e}}-\lambda \left(\nabla u,\nabla \left(v\phi \right)\right)-a_2{\left(v(1-u-v)(1-u-2v),\phi \right)}_{Q_{T_e}}\hfill \\ & & -a_3{(uv(u-v),\phi )}_{Q_{T_e}}+{(v_0,\phi \left(0\right))}_{\mathsf{\Omega }}+{\left(\theta \left(-l_{1}uv+l_{2}v(1-u-v)\right),\phi \right)}_{Q_{T_e}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill 0& =& {(\theta ,{\phi }_t)}_{Q_{T_e}}-{(D\nabla \theta ,\nabla \phi )}_{Q_{T_e}}+{({\theta }_0,\phi \left(0\right))}_{\mathsf{\Omega }}-(l_1(\frac{1}{3}{u}^3+u^{2}v-\frac{1}{3}{v}^3-u{v}^2)\hfill \\ & & +l_3\left(\frac{1}{3}{u}^3+u^2(1-u-v)-\frac{1}{3}{(1-u-v)}^3-u{(1-u-v)}^2\right)\hfill \\ & & +l_2(\frac{1}{3}{v}^3+v^2(1-u-v)-\frac{1}{3}{(1-u-v)}^3-v{(1-u-v)}^2),{\phi }_t{)}_{Q_{T_e}}\hfill \\ & & -(l_1(\frac{1}{3}{u}_0^3+u_0^2{v}_0-\frac{1}{3}{v}_0^3-u_0{v}_0^2)\hfill \\ & & +l_3(\frac{1}{3}{u}_0^3+u_0^2(1-u_0-v_0)-\frac{1}{3}{(1-u_0-v_0)}^3-u_0{(1-u_0-v_0)}^2)\hfill \\ & & +l_2(\frac{1}{3}{v}_0^3+v_0^2(1-u_0-v_0)-\frac{1}{3}{(1-u_0-v_0)}^3-v_0{(1-u_0-v_0)}^2),\phi \left(0\right){)}_{Q_{T_e}},\hfill \end{array}$$

(17)

where $\alpha =\left(k_1(1-v)+k_{3}v\right)>0$, $\beta =k_1-k_3$, $\gamma =\left(k_2(1-u)+k_{3}u\right)>0$, $\lambda =k_2-k_3$, $w_0=1-u_0-v_0$.

Theorem 1.

For all $(u_0,v_0,{\theta }_0)\in H_0^1\left(\mathsf{\Omega }\right)\times H_0^1\left(\mathsf{\Omega }\right)\times L^2\left(\mathsf{\Omega }\right)$, there exists a unique weak solution $(u,v,\theta )$ of problems(7)–(12), which, in addition to(13)–(14), satisfies

$$\begin{array}{c}\hfill u_t\in L^2\left(Q_{T_e}\right),u\in L^4\left(Q_{T_e}\right),v_t\in L^2\left(Q_{T_e}\right),v\in L^4\left(Q_{T_e}\right),{\theta }_t\in L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right)).\end{array}$$

(18)

Notation. In the following sections, we use the letter $C>0$ to indicate that it will change with the line. The $L^2\left(\mathsf{\Omega }\right)$-norm is denoted by $\parallel ·\parallel$.

The outline of this paper is as follows. In Section 2, we will prove the existence of solutions to the initial boundary-value problem of the nonlinear Equations (7)–(12) by the Faedo–Galerkin method.

In Section 3 we shall show that a maximum theorem holds.

2. Existence of Solutions

In this section, we construct the approximate solutions by the Galerkin method and derive the a priori estimates; then, we propose to send $m\to \infty$ and to show that a subsequence of our solutions $u_m,v_m,{\theta }_m$ converges to a weak solution of problems (7)–(12).

2.1. Construction of Approximate Solutions

Let ${\left\{{\omega }_k\right\}}_{k=1}^{\infty }$ be a basis in $H_0^1\left(\mathsf{\Omega }\right)$ and ${\omega }_k$ be a solution to the eigen-problem

$$\left\{\begin{array}{c}-\mathsf{\Delta }{\omega }_k={\lambda }_k{\omega }_k,in\mathsf{\Omega }\hfill \\ {\omega }_k=0,k=1,\cdots ,on\partial \mathsf{\Omega }.\hfill \end{array}\right.$$

For a positive integer m, we will look for approximate solutions $u_m,v_m,e_m$ of the form

$$\begin{array}{c}\hfill u_m\left(t\right)=\sum _{k=1}^m{d}_m^k\left(t\right){\omega }_k,v_m\left(t\right)=\sum _{k=1}^m{g}_m^k\left(t\right){\omega }_k,e_m\left(t\right)=\sum _{k=1}^m{h}_m^k\left(t\right){\omega }_k,(k=1,2,...,m),\end{array}$$

(19)

where we select the coefficients $d_m^k\left(t\right),g_m^k\left(t\right),h_m^k\left(t\right)$ so that

$$\begin{array}{c}\hfill d_m^k\left(0\right)=(u_0,{\omega }_k)={\delta }_m^k,g_m^k\left(0\right)=(v_0,{\omega }_k)={\eta }_m^k,h_m^k\left(0\right)=({\theta }_0,{\omega }_k)={\zeta }_m^k,\end{array}$$

(20)

and

$$\begin{array}{ccc}& & (u_{mt},{\omega }_j)-\alpha (\mathsf{\Delta }{u}_m,{\omega }_j)-\beta (u_m\mathsf{\Delta }{v}_m,{\omega }_j)=\hfill \\ & & -a_1\left(u_m(1-u_m-v_m)(1-2u_m-v_m),{\omega }_j\right)-a_3\left(u_m{v}_m(v_m-u_m),{\omega }_j\right)\hfill \\ & & +((e_m-\frac{l_1}{4}(\frac{1}{3}{u}_m^3+u_m^2{v}_m-\frac{1}{3}{v}_m^3-u_m{v}_m^2)\hfill \\ & & -\frac{l_3}{4}(\frac{1}{3}{u}_m^3+u_m^2{w}_m-\frac{1}{3}{w}_m^3-u_m{w}_m^2)\hfill \\ & & -\frac{l_2}{4}(\frac{1}{3}{v}_m^3+v_m^2{w}_m-\frac{1}{3}{w}_m^3-v_m{w}_m^2))\left(l_1{u}_m{v}_m+l_3{u}_m{w}_m\right),{\omega }_j),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & (v_{mt},{\omega }_j)-\gamma (\mathsf{\Delta }{v}_m,{\omega }_j)-\lambda (v_m\mathsf{\Delta }{u}_m,{\omega }_j)=\hfill \\ & & -a_2(v_m(1-u_m-v_m)(1-u_m-2v_m),{\omega }_j)-a_3(v_m{u}_m(u_m-v_m),{\omega }_j)\hfill \\ & & +((e_m-\frac{l_1}{4}(\frac{1}{3}{u}_m^3+u_m^2{v}_m-\frac{1}{3}{v}_m^3-u_m{v}_m^2)\hfill \\ & & -\frac{l_3}{4}(\frac{1}{3}{u}_m^3+u_m^2{w}_m-\frac{1}{3}{w}_m^3-u_m{w}_m^2)\hfill \\ & & -\frac{l_2}{4}(\frac{1}{3}{v}_m^3+v_m^2{w}_m-\frac{1}{3}{w}_m^3-v_m{w}_m^2))\left(-l_1{u}_m{v}_m+l_2{v}_m{w}_m\right),{\omega }_j),\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& & (e_{mt},{\omega }_j)=D\mathsf{\Delta }({\theta }_m-\frac{l_1}{4}(\frac{1}{3}{u}_m^3+u_m^2{v}_m-\frac{1}{3}{v}_m^3-u_m{v}_m^2)\hfill \\ & & -\frac{l_3}{4}(\frac{1}{3}{u}_m^3+u_m^2{w}_m-\frac{1}{3}{w}_m^3-u_m{w}_m^2)\hfill \\ & & -\frac{l_2}{4}(\frac{1}{3}{v}_m^3+v_m^2{w}_m-\frac{1}{3}{w}_m^3-v_m{w}_m^2),{\omega }_j),(j=1,\dots ,m).\hfill \end{array}$$

(23)

where $w_m=1-u_m-v_m$. (22) and (24) comprise a system of nonlinear ordinary differential equations, and the nonlinear term is locally Lipschitz continuous.

Assuming $u_m,v_m,e_m$ have the structure (19), we note that

$$\begin{array}{c}\hfill (u_m\left(t\right),{\omega }_j)=(\sum _{k=1}^m{d}_m^{k^{\prime }}\left(t\right){\omega }_k,{\omega }_j)={\int }_{\mathsf{\Omega }}\sum _{k=1}^m{d}_m^{k^{\prime }}\left(t\right){\omega }_k{\omega }_{j}dx=\sum _{k=1}^m{d}_m^{k^{\prime }}\left(t\right){\int }_{\mathsf{\Omega }}{\omega }_k{\omega }_{j}dx,\end{array}$$

(24)

$$\begin{array}{c}\hfill (v_m\left(t\right),{\omega }_j)=(\sum _{k=1}^m{g}_m^{k^{\prime }}\left(t\right){\omega }_k,{\omega }_j)={\int }_{\mathsf{\Omega }}\sum _{k=1}^m{g}_m^{k^{\prime }}\left(t\right){\omega }_k{\omega }_{j}dx=\sum _{k=1}^m{g}_m^{k^{\prime }}\left(t\right){\int }_{\mathsf{\Omega }}{\omega }_k{\omega }_{j}dx,\end{array}$$

(25)

$$\begin{array}{c}\hfill (e_m\left(t\right),{\omega }_j)=(\sum _{k=1}^m{h}_m^{k^{\prime }}\left(t\right){\omega }_k,{\omega }_j)={\int }_{\mathsf{\Omega }}\sum _{k=1}^m{h}_m^{k^{\prime }}\left(t\right){\omega }_k{\omega }_{j}dx=\sum _{k=1}^m{h}_m^{k^{\prime }}\left(t\right){\int }_{\mathsf{\Omega }}{\omega }_k{\omega }_{j}dx,\end{array}$$

(26)

We deal with other terms in the same way.

We introduce the vectors

$$D_m=D_m\left(t\right)=\left(\begin{array}{c}{d}_m^1\left(t\right)\\ ⋮\\ d_m^m\left(t\right)\end{array}\right),G_m=G_m\left(t\right)=\left(\begin{array}{c}{g}_m^1\left(t\right)\\ ⋮\\ g_m^m\left(t\right)\end{array}\right),H_m=H_m\left(t\right)=\left(\begin{array}{c}{h}_m^1\left(t\right)\\ ⋮\\ h_m^m\left(t\right)\end{array}\right)$$

and $F_1(D_m^T,G_m^T,H_m^T),F_2(D_m^T,G_m^T,H_m^T),F_3(D_m^T,G_m^T,H_m^T)$ denote the nonlinear terms. Thus, we obtain a system of ordinal differential equations.

$$\begin{array}{ccc}& & B{D}_m^{\prime }+F_1(D_m^T,G_m^T,H_m^T)=0,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill & & B{G}_m^{\prime }+F_2(D_m^T,G_m^T,H_m^T)=0,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill & & B{H}_m^{\prime }+F_3(D_m^T,G_m^T,H_m^T)=0,\hfill \end{array}$$

(29)

where

$$B=\left(b_{ij}\right)=\left[\begin{array}{cccc}({\omega }_1,{\omega }_1)& \dots & ({\omega }_1,{\omega }_m)& \\ \dots & \dots & \dots \\ ({\omega }_m,{\omega }_1)& \dots & ({\omega }_m,{\omega }_m)\end{array}\right]=\int \left[\begin{array}{c}{\omega }_1\\ ⋮\\ {\omega }_m\end{array}\right]\left[\begin{array}{c}{\omega }_1\dots {\omega }_m\end{array}\right]dx.$$

(30)

Then, if we choose a vector $\overrightarrow{X}=(x_1,\dots ,x_m)$ that is not equal to zero, there holds for ${(\overrightarrow{X})}^{T}B\overrightarrow{X}={\int }_{\mathsf{\Omega }}{(x_1{\omega }_1+\dots +x_m{\omega }_m)}^{2}dx>0$; otherwise, invoking that ${\omega }_1,\dots ,{\omega }_m$ are linearly independent, we obtain $\overrightarrow{X}=0$. B is a positive-definite matrix.

For the initial data, we make a smooth approximation

$$\begin{array}{ccc}& & u_{0m}=\sum _{k=1}^m{\delta }_m^k{\omega }_k\to u_0,stronglyin{H}_0^1\left(\mathsf{\Omega }\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& & v_{0m}=\sum _{k=1}^m{\eta }_m^k{\omega }_k\to v_0,stronglyin{H}_0^1\left(\mathsf{\Omega }\right),\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & e_m=\sum _{k=1}^m{\zeta }_m^k{\omega }_k\to e_0,stronglyin{H}_0^1\left(\mathsf{\Omega }\right).\hfill \end{array}$$

(33)

According to the existence theorem of local solutions to ordinary differential equations, there exist the solutions for a.e. $0\le t\le t_m$. We extend $t_m=T_e$. Then, we obtain the global solutions. However, we need to show that the a priori estimates hold.

2.2. A Priori Estimates

Theorem 2.

There exists a constant C, depending on $\mathsf{\Omega },T_e$, such that

$$\begin{array}{ccc}& & \parallel u_m{\parallel }_{H_0^1\left(\mathsf{\Omega }\right)}^2+\parallel v_m{\parallel }_{H_0^1\left(\mathsf{\Omega }\right)}^2+\parallel e_m{\parallel }^2+{\int }_0^{T_e}\parallel u_m{\parallel }_{L^4\left(\mathsf{\Omega }\right)}^{4}d\tau +{\int }_0^{T_e}{\parallel v_m\parallel }_{L^4\left(\mathsf{\Omega }\right)}^{4}d\tau \hfill \\ & & +{\int }_0^{T_e}(\parallel u_m{\parallel }_{H^2\left(\mathsf{\Omega }\right)}^2+\parallel v_m{\parallel }_{H^2\left(\mathsf{\Omega }\right)}^2+\parallel e_m{\parallel }_{H_0^1\left(\mathsf{\Omega }\right)}^2)d\tau \le C.\hfill \end{array}$$

(34)

for $m=1,2,\dots$ where ${\parallel u}_m{\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)},\parallel v_m{\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)},{\parallel w_m\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}$ are suitably small.

Proof. 

Multiplying (22)–(24) by $d_m^k\left(t\right),g_m^k\left(t\right)$, and $h_m^k\left(t\right)$, respectively, summing up over $j=1,\dots ,m$, integrating over $\mathsf{\Omega }$, and adding and recalling (19), we find

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}(\parallel u_m{\parallel }^2+\parallel v_m{\parallel }^2+\frac{1}{D}\parallel e_m{\parallel }^2)+\frac{1}{2\gamma }(\gamma \alpha -{\lambda }^2){\parallel \nabla u_m\parallel }^2\hfill \\ & & +\frac{1}{2\alpha }(\gamma \alpha -{\beta }^2)\parallel \nabla v_m{\parallel }^2+\frac{1}{2}{\parallel \nabla e_m\parallel }^2\hfill \\ & & +\frac{1}{64}(128a_1-33|3a_1-a_3|-|3a_2-a_3|-64ϵ)\parallel u_m{\parallel }_{L^4\left(\mathsf{\Omega }\right)}^4\hfill \\ & & +\frac{1}{64}(128a_2-33|3a_2-a_3|-|3a_1-a_3|-64ϵ)\parallel v_m{\parallel }_{L^4\left(\mathsf{\Omega }\right)}^4\hfill \\ & & +\frac{1}{32}(32a_1+64a_3+32a_2-15|3a_1-a_3|-15|3a_2-a_3\left|\right)\parallel u_m{v}_m{\parallel }^2\hfill \\ & & \le C(\parallel u_m{\parallel }^2+\parallel v{\parallel }^2+\frac{1}{D}\parallel e_m{\parallel }^2)+C\left(\mathsf{\Omega }\right),\hfill \end{array}$$

(35)

where $\alpha \gamma -{\lambda }^2>0,\alpha \gamma -{\beta }^2,128a_1-33|3a_1-a_3|-|3a_2-a_3|-64ϵ>0$, $128a_2-33|3a_2-a_3|-|3a_1-a_3|-64ϵ>0,32a_1+64a_3+32a_2-15|3a_1-a_3|-15|3a_2-a_3|>0$.

By using the Gronwall inequality, we obtain

$$\begin{array}{ccc}& & (\parallel u_m{\parallel }^2+\parallel v_m{\parallel }^2+\frac{1}{D}\parallel e_m{\parallel }^2)+\frac{1}{2\gamma }(\gamma \alpha -{\lambda }^2){\int }_0^{T_e}{\parallel \nabla u_m\parallel }^{2}d\tau \hfill \\ & & +\frac{1}{2\alpha }(\gamma \alpha -{\beta }^2){\int }_0^{T_e}\parallel \nabla v_m{\parallel }^{2}d\tau +\frac{1}{2}{\int }_0^{T_e}{\parallel \nabla e_m\parallel }^{2}d\tau \hfill \\ & & +\frac{1}{64}(128a_1-33|3a_1-a_3|-|3a_2-a_3|-64ϵ){\int }_0^{T_e}{\parallel u_m\parallel }_{L^4\left(\mathsf{\Omega }\right)}^{4}d\tau \hfill \\ & & +\frac{1}{64}(128a_2-33|3a_2-a_3|-|3a_1-a_3|-64ϵ){\int }_0^{T_e}{\parallel v_m\parallel }_{L^4\left(\mathsf{\Omega }\right)}^{4}d\tau \hfill \\ & & +\frac{1}{32}(32a_1+64a_3+32a_2-15|3a_1-a_3|-15|3a_2-a_3\left|\right){\int }_0^{T_e}{\parallel u_m{v}_m\parallel }^{2}d\tau \hfill \\ & & \le C\left(T_e\right).\hfill \end{array}$$

(36)

Multiplying (22) and (23) by ${\lambda }_j{d}_m^k\left(t\right)$ and ${\lambda }_j{g}_m^k\left(t\right)$, respectively, summing up over $j=1,\dots ,m$, then formally integrating over $\mathsf{\Omega }$ and adding them, we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}(\parallel \nabla u_m{\parallel }^2+\parallel \nabla v_m{\parallel }^2)+\frac{1}{2\gamma }(\alpha \gamma -{\lambda }^2-\gamma ϵ){\parallel \mathsf{\Delta }{u}_m\parallel }^2\hfill \\ & & +\frac{1}{2\alpha }(\gamma \alpha -{\beta }^2-\alpha ϵ){\parallel \mathsf{\Delta }{v}_m\parallel }^2\hfill \\ & & +\frac{1}{2}(6a_1-|3a_1-a_3|-|5a_1-3a_3-a_2{|}^2)\parallel u_m\nabla u_m{\parallel }^2\hfill \\ & & +\frac{1}{2}(2a_1+2a_3-|3a_2-a_3|-|5a_1-3a_3-a_2{|}^2)\parallel v_m\nabla u_m{\parallel }^2\hfill \\ & & +\frac{1}{2}(6a_2-|3a_2-a_3|-|5a_2-3a_3-a_1{|}^2)\parallel v_m\nabla v_m{\parallel }^2\hfill \\ & & +\frac{1}{2}(2a_2+2a_3-|3a_1-a_3|-|5a_2-3a_3-a_1{|}^2)\parallel u_m\nabla v_m{\parallel }^2\hfill \\ & & \le C(\parallel \nabla u_m{\parallel }^2+\parallel \nabla v_m{\parallel }^2+\parallel \nabla e_m{\parallel }^2+1),\hfill \end{array}$$

(37)

where $\alpha \gamma -{\lambda }^2-\gamma ϵ>0$, $\gamma \alpha -{\beta }^2-\alpha ϵ>0$, $6a_1-|3a_1-a_3|-|5a_1-3a_3-a_2{|}^2>0$, $2a_1+2a_3-|3a_2-a_3|-|5a_1-3a_3-a_2{|}^2>0$, $2a_2-|3a_2-a_3|-|5a_2-3a_3-a_1{|}^2>0$, $2a_2+2a_3-|3a_1-a_3|-|5a_2-3a_3-a_1{|}^2>0$.

By using the Gronwall inequality, we obtain

$$\begin{array}{ccc}& & \parallel \nabla u_m{\parallel }^2+\parallel \nabla v_m{\parallel }^2+\frac{1}{2\gamma }(\alpha \gamma -{\lambda }^2-\gamma ϵ){\int }_0^{T_e}{\parallel \mathsf{\Delta }{u}_m\parallel }^{2}d\tau \hfill \\ & & +\frac{1}{2\alpha }(\gamma \alpha -{\beta }^2-\alpha ϵ){\int }_0^{T_e}{\parallel \mathsf{\Delta }{v}_m\parallel }^{2}d\tau \le C\left(T_e\right).\hfill \end{array}$$

(38)

It follows from (37) and (38) that

$$\begin{array}{c}\hfill {\parallel u}_{mt}{\parallel }_{L^2\left(Q_{T_e}\right)+L^{\frac{4}{3}}\left(Q_{T_e}\right)}+\parallel v_{mt}{\parallel }_{L^2\left(Q_{T_e}\right)+L^{\frac{4}{3}}\left(Q_{T_e}\right)}+{\parallel e_{mt}\parallel }_{L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))}\le C\left(T_e\right).\end{array}$$

(39)

□

2.3. Existence of Weak Solutions

Next we pass to limits as $m\to \infty$, to build the weak solutions to the initial boundary-value problems (7)–(12).

Theorem 3

(Aubin–Lions). Let $B_0$ be a normed linear space embedded compactly into another normed linear space B, which is continuously embedded into a Hausdorff locally convex space $B_1$, and $1\le p<+\infty$. If $v,v_i\in L^p(0,t;B_0),i\in \mathbb{N}$, the sequence ${\left\{v_i\right\}}_{i\in \mathbb{N}}$ converges weakly to v in $L^p(0,t;B_0)$, and ${\left\{\frac{\partial v_i}{\partial t}\right\}}_{i\in \mathbb{N}}$ is bounded in $L^1(0,t;B_1)$, then $v_i$ converges to v strongly in $L^p(0,t;B)$.

Lemma 1.

Let $(0,t)\times \mathsf{\Omega }$ be an open set in ${\mathbb{R}}^{+}\times {\mathbb{R}}^n$. Suppose functions $g_n,g$ are in $L^q((0,t)\times \mathsf{\Omega }$ for any given $1<q<\infty$, which satisfy

$$\begin{array}{ccc}\hfill {\parallel g}_n{\parallel }_{L^q((0,t)\times \mathsf{\Omega })}\le C,& g_n\to g& a.e.in(0,t)\times \mathsf{\Omega }.\hfill \end{array}$$

Then $g_n$ converges to g weakly in $L^q((0,t)\times \mathsf{\Omega })$.

Theorem 3 is a general version of the Aubin–Lions lemma valid under the weak assumption ${\partial }_t{v}_i\in L^1(0,t;B_1)$. This version, which we need here, is proved in [9]. A proof of Lemma 1 can be found in [9].

Lemma 2.

Problems (7)–(12) have at least one weak solution $(u,v,e)$ in the sense of Definition 1.1. Each of the weak solutions satisfies $e_t\in L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))$ and $(u_t,v_t)\in L^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right)$.

Proof of Lemma 2.

According to the energy estimates (34), we see sequences ${\left\{u_m\right\}}_{m=1}^{\infty }$, ${\left\{v_m\right\}}_{m=1}^{\infty },{\left\{e_m\right\}}_{m=1}^{\infty }$ are bounded in $L^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H^2\left(\mathsf{\Omega }\right))$,

$L^{\infty }(0,T_e;L^2\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H_0^1\left(\mathsf{\Omega }\right))$, respectively, and $({\left\{u_{mt}\right\}}_{m=1}^{\infty },{\left\{v_{mt}\right\}}_{m=1}^{\infty }),{\left\{e_{mt}\right\}}_{m=1}^{\infty }$ are bounded in $L^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right),L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))$, respectively.

Consequently there exist subsequences ${\left\{u_{m_l}\right\}}_{l=1}^{\infty }\subset {\left\{u_m\right\}}_{m=1}^{\infty }$, ${\left\{v_{m_l}\right\}}_{l=1}^{\infty }\subset {\left\{v_m\right\}}_{m=1}^{\infty }$, ${\left\{e_{m_l}\right\}}_{l=1}^{\infty }\subset {\left\{e_m\right\}}_{m=1}^{\infty }$ and functions $(u,v)\in L^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H^2\left(\mathsf{\Omega }\right))$,

$e\in L^2(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\cap L^{\infty }(0,T_e;L^2\left(\mathsf{\Omega }\right))$, with $(u_t,v_t)\in L^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right)$,

$e_t\in L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))$, such that

$$\left\{\begin{array}{c}{u}_{m_l}\rightharpoonup u,weaklyin{L}^2(0,T_e;H^2\left(\mathsf{\Omega }\right))\hfill \\ u_{m_l}\stackrel{*}{\rightharpoonup }u,weaklyin{L}^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\hfill \\ u_{m_{l}t}\rightharpoonup u_t,weaklyin{L}^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right)\hfill \\ v_{m_l}\rightharpoonup v,weaklyin{L}^2(0,T_e;H^2\left(\mathsf{\Omega }\right))\hfill \\ v_{m_l}\stackrel{*}{\rightharpoonup }v,weaklyin{L}^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\hfill \\ v_{m_{l}t}\rightharpoonup v_t,weaklyin{L}^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right).\hfill \\ e_{m_l}\stackrel{*}{\rightharpoonup }e,weaklyin{L}^{\infty }(0,T_e;L^2\left(\mathsf{\Omega }\right))\hfill \\ e_{m_l}\rightharpoonup e,weaklyin{L}^2(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\hfill \\ e_{m_{l}t}\rightharpoonup e_t,weaklyin{L}^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))\hfill \end{array}\right.$$

There exist functions $\xi ,\varphi$ such that

$$\begin{array}{ccc}& & u_m^3\rightharpoonup \xi ,weaklyin{L}^{\frac{4}{3}}\left(Q_{T_e}\right)\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& & v_m^3\rightharpoonup \varphi ,weaklyin{L}^{\frac{4}{3}}\left(Q_{T_e}\right).\hfill \end{array}$$

(41)

It remains to show that $\xi =u^3,\varphi =v^3$. To this end, we show first that $u_m\to u,v_m\to v$ in $L^2(0,T_e;H^1\left(\mathsf{\Omega }\right))$ by applying Theorem 3. To apply this theorem, $u_t,v_t$ have estimates $u_t\in L^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right)\subset L^{\frac{4}{3}}\left(Q_{T_e}\right)\subset L^1(0,T_e;L^{\frac{4}{3}}\left(\mathsf{\Omega }\right))$, $v_t\in L^2(0,T_e;L^2\left(\mathsf{\Omega }\right))+L^{\frac{4}{3}}\left(Q_{T_e}\right)\subset L^{\frac{4}{3}}\left(Q_{T_e}\right)\subset L^1(0,T_e;L^{\frac{4}{3}}\left(\mathsf{\Omega }\right))$, $e_t\in L^2(0,T_e;H^{-1}\left(\mathsf{\Omega }\right))$.

Applying Theorem 3 with $B_0=H^2\left(\mathsf{\Omega }\right),B=H^1\left(\mathsf{\Omega }\right),B_1=L^{\frac{4}{3}}\left(\mathsf{\Omega }\right)$, and $p=2$ we obtain

$$\begin{array}{ccc}& & u_m\to u,stronglyin{L}^2(0,T_e;H^1\left(\mathsf{\Omega }\right))\hfill \\ & & v_m\to v,stronglyin{L}^2(0,T_e;H^1\left(\mathsf{\Omega }\right)).\hfill \end{array}$$

Therefore, we choose the other sequences from these sequences and denote them in the same way. They converge almost everywhere in $Q_{T_e}$. This implies the convergence $u_m^3\to u^3,v_m^3\to v^3$ almost everywhere in $Q_{T_e}$. Using the embedding $H^1⋐L^4\left(\mathsf{\Omega }\right)$ and applying Lemma 1, we obtain $\xi =u^3,\varphi =v^3$. Then, we obtain the others in the same way. Equation (15) follows from these relations if we show that

$$\begin{array}{}\mathrm{(42)}& & & {(u_{0m},\phi \left(0\right))}_{\mathsf{\Omega }}\to (u_0,\phi \left(0\right)),\hfill \mathrm{(43)}& & & {(u_m,{\phi }_t)}_{Q_{T_e}}\to {(u,{\phi }_t)}_{Q_{T_e}},\hfill \mathrm{(44)}& & & {(\nabla u_m,\nabla \phi )}_{Q_{T_e}}\to {(\nabla u,\nabla \phi )}_{Q_{T_e}},\hfill \mathrm{(45)}& & & {(\nabla v_m,\nabla \left(u_m\phi \right))}_{Q_{T_e}}\to {(\nabla v_m,\nabla \left(u\phi \right))}_{Q_{T_e}},\hfill \mathrm{(46)}& & & {\left(u_m{v}_m(v_m-u_m),\phi \right)}_{Q_{T_e}}{\to (uv(v-u),\phi )}_{Q_{T_e}},\hfill \mathrm{(47)}& & & {({\theta }_m{u}_m{v}_m,\phi )}_{Q_{T_e}}\to {(\theta uv,\phi )}_{Q_{T_e}}.\hfill \end{array}$$

for $m\to \infty$. Now, the relation (42) follows from (31), the relation (43) is a consequence of $u_{mt}\in L^{\frac{4}{3}}\left(Q_{T_e}\right)$, the relation (44) is consequence of $u_m\in L^2(0,T_e;H^2\left(\mathsf{\Omega }\right))$, the relation (45) is a consequence of $u_m\to u,\nabla u_m\to \nabla u$, $\nabla v_m\to \nabla v$ in $L^2\left(Q_{T_e}\right)$, the relation (46) is a consequence of $u_m{v}_m\in L^2\left(Q_{T_e}\right)$ and $u_m\to u,v_m\to v$ in $L^2\left(Q_{T_e}\right)$, and the relation (47) is obtained from $u_m{v}_m\in L^2\left(Q_{T_e}\right)$ and ${\theta }_m\to \theta$ in $L^2\left(Q_{T_e}\right)$. We obtain other terms in the same way as above. □

2.4. Uniqueness

In this subsection we show uniqueness of the solution of $(u,v,e)$ that was obtained in Section 2.2 and Section 2.3.

Theorem 4.

Let $(u_{0i},v_{0i})\in H_0^1\left(\mathsf{\Omega }\right),e_{0i}\in L^2\left(\mathsf{\Omega }\right)$, $i=1,2$ be given functions. Let $(u_i,v_i,e_i)$ be weak solutions of problems(7)–(12)with $(u_i,v_i)\in L^{\infty }(0,T_e;H_0^1\left(\mathsf{\Omega }\right))\cap L^2(0,T_e;H^2\left(\mathsf{\Omega }\right))$, and $e_i\in L^2(0,T:H_0^1\left(\mathsf{\Omega }\right))\cap L^{\infty }(0,T_e;L^2\left(\mathsf{\Omega }\right))$, $i=1,2$. Then

$$\begin{array}{ccc}& & \widetilde{\parallel u}{\parallel }^2+\parallel \tilde{v}{\parallel }^2+\parallel \tilde{e}{\parallel }^2\le C(1+T_e{e}^{C_1{T}_e})(\parallel {\tilde{u}}_0{\parallel }^2+\parallel {\tilde{v}}_0{\parallel }^2+\parallel {\tilde{e}}_0{\parallel }^2).\hfill \end{array}$$

(48)

where $\tilde{u}=u_1-u_2,\tilde{v}=v_1-v_2$ and $\tilde{e}=e_1-e_2$. The constant C is independent of $u_i,v_i,e_i,u_{0i},v_{0i},e_{0i}$.

Proof. 

The result equation of the difference of (7) for $u_1$ and $u_2$ is multiplied by $\tilde{u}$. We obtain the others in the same way. Then, $\tilde{u},\tilde{v}$, and $\tilde{e}$ satisfy the inequality

$$\begin{array}{ccc}& & \frac{1}{2}(\parallel \tilde{u}{\parallel }^2+\parallel \tilde{v}{\parallel }^2+\frac{1}{D}\parallel \tilde{e}{\parallel }^2)+\frac{1}{2\gamma }(\gamma \alpha -{\lambda }^2){\parallel \nabla \tilde{u}\parallel }^2\hfill \\ & & +\frac{1}{2\alpha }(\gamma \alpha -{\beta }^2)\parallel \nabla \tilde{v}{\parallel }^2+\frac{1}{4}{\parallel \nabla \tilde{e}\parallel }^2\hfill \\ & & \le C\left\{\right(\parallel {\tilde{u}}_0{\parallel }^2+\parallel {\tilde{v}}_0{\parallel }^2+\parallel {\tilde{e}}_0{\parallel }^2)+{\int }_0^t{\int }_{\mathsf{\Omega }}(\parallel \tilde{u}{\parallel }^2+\parallel \tilde{v}{\parallel }^2+\parallel \tilde{e}{\parallel }^2\left)\right\}.\hfill \end{array}$$

(49)

We use the Gronwall inequality, which yields

$$\begin{array}{ccc}& & \parallel \overline{u}{\parallel }^2+\parallel \overline{v}{\parallel }^2+\parallel \overline{e}{\parallel }^2\le C(1+T_e{e}^{C_1{T}_e})(\parallel {\overline{u}}_0{\parallel }^2+\parallel {\overline{v}}_0{\parallel }^2+\parallel {\overline{e}}_0{\parallel }^2).\hfill \end{array}$$

(50)

In particular, for ${\overline{u}}_0={\overline{v}}_0={\overline{e}}_0=0$, we obtain the uniqueness of the solution.

3. A Maximum Theorem

If we want the model to have physical meaning, we must prove that the order parameters $u,v,w$ still change in the interval $[0,1]$ during its evolution. We will study a maximum theorem.

Theorem 5.

Let $(u,v,w,\theta )$ be the solution of problems(7)–(12)and $u_0\in [0,1],v_0\in [0,1],w_0\in [0,1]$ for each $x\in \mathsf{\Omega }$. Then such solution satisfies $u\in [0,1],v\in [0,1],w\in [0,1]$ and $u+v+w=1$ for each $x\in \mathsf{\Omega }$ and $t\in {\mathbb{R}}^{+}$.

Proof. 

We only analyze Equation (7); Equation (8) is treated in a similar way. To prove that $u(t,x)\ge 0$, let us define

$$u^{-}=\left\{\begin{array}{cc}-u,\hfill & \mathrm{if}u<0,\hfill \\ 0,\hfill & \mathrm{if}u\ge 0,\hfill \end{array}\right.$$

so that $u^{-}\ge 0$ and satisfies boundary and initial conditions

$$\begin{array}{ccc}\hfill u^{-}(t,x)& =& 0,t\ge 0,x\in \partial \mathsf{\Omega },\hfill \\ \hfill u^{-}(0,x)& =& 0,x\in \mathsf{\Omega }.\hfill \end{array}$$

We multiply (7) by $-u^{-}$ and integrate in $\mathsf{\Omega }$ to obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}{\int }_{\mathsf{\Omega }}{\left(u^{-}\right)}^{2}dx+\frac{\alpha }{2}{\int }_{\mathsf{\Omega }}{(\nabla u^{-})}^{2}dx\le C{\int }_{\mathsf{\Omega }}{(\nabla v)}^2{\left(u^{-}\right)}^{2}dx+C{\int }_{\mathsf{\Omega }}\theta {\left(u^{-}\right)}^{2}dx\hfill \\ & & -a_1{\int }_{\mathsf{\Omega }}{\left(u^{-}\right)}^2(1-u^{-}-v)(1-2u^{-}-v)dx-a_3{\int }_{\mathsf{\Omega }}{\left(u^{-}\right)}^{2}v(v-u^{-})dx.\hfill \end{array}$$

(51)

Using Gronwall’ s lemma, we obtain that ${\left(u^{-}\right)}^2=0$ a.e. in $\mathsf{\Omega }$ for all $t\in (0,T_e)$, and thus $u\ge 0$ a.e. in $Q_{T_e}$. We obtain in a similar way as earlier that $v\ge 0$, $w\ge 0$ a.e. in $Q_{T_e}$. Using a similar argument to prove that $u(t,x)\le 1,v(t,x)\le 1,w(t,x)\le 1$, we define

$$u^{+}=\left\{\begin{array}{cc}u-1,\hfill & \mathrm{if}u>1,\hfill \\ 0,\hfill & \mathrm{if}u\le 1,\hfill \end{array}\right.$$

$$v^{+}=\left\{\begin{array}{cc}v-1,\hfill & \mathrm{if}v>1,\hfill \\ 0,\hfill & \mathrm{if}v\le 1,\hfill \end{array}\right.$$

$$w^{+}=\left\{\begin{array}{cc}w-1,\hfill & \mathrm{if}w>1,\hfill \\ 0,\hfill & \mathrm{if}w\le 1,\hfill \end{array}\right.$$

so that $u^{+}(t,x)\ge 0$ and satisfies boundary and initial conditions

$$\begin{array}{c}\hfill u^{+}(t,x)=0,v^{+}(t,x)=0,w^{+}(t,x)=0,t\ge 0,x\in \partial \mathsf{\Omega },\\ \hfill u^{+}(0,x)=0,v^{+}(t,x)=0,w^{+}(t,x)=0,x\in \mathsf{\Omega }.\end{array}$$

Multiplying (1)–(3) by $u^{+},v^{+}$, and $w^{+}$, respectively, integrating in $\mathsf{\Omega }$, and adding the resulting equations, we have

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}{\int }_{\mathsf{\Omega }}{u}^{+}(1+u^{+})dx+\frac{1}{2}\frac{d}{dt}{\int }_{\mathsf{\Omega }}{v}^{+}(1+v^{+})dx+\frac{1}{2}\frac{d}{dt}{\int }_{\mathsf{\Omega }}{w}^{+}(1+w^{+})dx\hfill \\ & & +\alpha {\int }_{\mathsf{\Omega }}{(\nabla u^{+})}^{2}dx+\gamma {\int }_{\mathsf{\Omega }}{(\nabla v^{+})}^{2}dx+\left(k_{1}u+k_2(1-u)\right){\int }_{\mathsf{\Omega }}{(\nabla w^{+})}^{2}dx\hfill \\ & & \le \left|\beta \right|{\int }_{\mathsf{\Omega }}\mathsf{\Delta }v{u}^{+}(1+u^{+})dx+C{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u{v}^{+}(1+v^{+})dx+C{\int }_{\mathsf{\Omega }}\mathsf{\Delta }u{w}^{+}(1+w^{+})dx\hfill \\ & & -a_1{\int }_{\mathsf{\Omega }}{u}^{+}(1+u^{+})(-1-u^{+}-v^{+})(-2-2u^{+}-v^{+})dx+C{\int }_{\mathsf{\Omega }}\theta u^{+}(1+u^{+})dx\hfill \\ & & -a_3({\int }_{\mathsf{\Omega }}{u}^{+}(1+u^{+})(1+v^{+})(v^{+}-u^{+})dx+C{\int }_{\mathsf{\Omega }}\theta v^{+}(1+v^{+})dx\hfill \\ & & +{\int }_{\mathsf{\Omega }}{v}^{+}(1+v^{+})(1+u^{+})(u^{+}-v^{+})dx)+C{\int }_{\mathsf{\Omega }}{w}^{+}(1+w^{+})\theta dx\hfill \\ & & -a_2{\int }_{\mathsf{\Omega }}{v}^{+}(1+v^{+})(-1-u^{+}-v^{+})(-2-2v^{+}-u^{+})dx\hfill \\ & & -a_1{\int }_{\mathsf{\Omega }}{w}^{+}(1+w^{+})(u^{+}-w^{+})dx-a_2{\int }_{\mathsf{\Omega }}{w}^{+}(1+w^{+})(1+v^{+})(v^{+}-w^{+})dx.\hfill \end{array}$$

(52)

By using Gronwall’s lemma, we obtain that $u^{+}=0$, $v^{+}=0$, $w^{+}=0$ a.e. in $\mathsf{\Omega }$ for all $t\in (0,T_e)$, and thus $u\le 1$, $v\le 1$, $w\le 1$ a.e. in $Q_{T_e}$. □