\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 28, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/28\hfil Existence of solutions] {Existence of solutions to a phase-field model with phase-dependent heat absorption} \author[G. Planas\hfil EJDE-2007/28\hfilneg] {Gabriela Planas} \address{Gabriela Planas \newline Departamento de Matem{\'a}tica \\ Instituto de Ci\^encias Matem\'aticas e de Computa\c{c}\~ao\\ Universidade de S\~ao Paulo - Campus de S\~ao Carlos \\ Caixa Postal 668, S{\~a}o Carlos, SP 13560-970, Brazil} \email{gplanas@icmc.usp.br} \thanks{Submitted November 16, 2006. Published February 12, 2007.} \subjclass[2000]{35K55, 80A22, 82B26, 35B65} \keywords{Phase transitions; parabolic system; phase-field models} \begin{abstract} We consider a phase-field model for a phase change process with phase-dependent heat absorption. This model describes the behaviour of films exposed to radiative heating, where the film can change reversibly between amorphous and crystalline states. Existence and uniqueness of solutions as well as stability are established. Moreover, a maximum principle is proved for the phase-field equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} In recent years the phase-field method has emerged as a powerful computational approach to modelling and predicting a range of phase transitions and complex growth structures occurring during solidification. This has spurred many articles using this approach and proposing several mathematical models. Phase-field models have also been developed to treat both pure materials and binary alloys; as examples of papers where mathematical analysis of such models is performed, we single out \cite{Boldrini,Caginalp,ColliHoffmann,Colli,Hoffman,Moro,Rappaz,Stefanelli}, where existence of solutions is investigated for various types of nonlinearities. We consider in this paper a phase-field model for a phase change process with phase-dependent heat absorption. Such a model was proposed by Blyuss et al. \cite{Blyuss} to model the behaviour of films exposed to radiative heating, where the film can change reversibly between amorphous and crystalline states. The models adopted so far have neglected the difference in the response of different phases to external heat sources considering only external forces which do not depend neither on the phase-field nor on the temperature. To be specific, the model we consider incorporates illumination and phase-dependent absorption in the equation for the temperature, and this influences the phase change process, as described by the phase-field equation. This model can be expressed as the following coupled system \begin{gather}\label{originalpfe} \phi_t - \epsilon^2 \Delta \phi = \phi - \phi^3 + \dfrac{(\theta_M -\theta)}{\delta}\hat{\lambda}(1-\phi^2)^2 \quad \mbox{in } \Omega \times (0,T), \\ \label{originalte} \theta_t- K \Delta \theta = \frac{\delta}{2}\phi_t+ \big[a_1 + a_{-1} + (a_1-a_{-1})\phi\big]\frac{I}{2} + b (\theta_a -\theta) \quad \mbox{in } \Omega \times (0,T), \\ \frac{\partial \phi}{\partial n} = 0 , \quad \frac{\partial \theta}{\partial n} = 0, \quad \mbox{on } \partial \Omega \times (0,T), \\ \label{dados} \phi(0) = \phi_0 , \quad \theta (0) = \theta_0 \quad \mbox{in } \Omega. \end{gather} Here $\Omega$ is an open bounded domain of $\mathbb{R}^N$, $N = 2,3$, with smooth boundary $\partial \Omega$ and $ T >0$. The order parameter (phase-field) $\phi$ is the state variable characterizing the different phases; the convection adopted is that $ \phi \in [-1,1]$, with the lower limit $ \phi =-1$ corresponding to pure melt while $ \phi =1 $ represents solid. The function $\theta$ represents the temperature of a material which melts at $ \theta = \theta_M$. The interface thickness $ \epsilon $ is a small parameter and $ \hat{\lambda}$ is a measure of the strength of coupling between the phase field and a dimensionless temperature field $ (\theta - \theta_M)/\delta$, where $\delta $ is given by $ \delta = L/C_p$, being $L>0$ the latent heat and $ C_p >0 $ the specific heat at constant pressure. For simplicity of exposition it will be assumed $\hat{\lambda} = 1 $. The constant $ K >0 $ denotes a thermal diffusivity; $ a_{\pm 1}$ are the radiative absorption coefficients for the solid and molten phases; $ I $ is the rate of incident heating; $ b $ is a thermal emission coefficient and $\theta_a$ is the ambient temperature. Our aim is to prove existence and uniqueness of solutions as well as stability. Moreover, a maximum principle is established for the phase-field equation which ensures that $ \phi $ stays between $-1$ and $ 1 $ as long as the initial data $ \phi_0 $ does. We observe that this bound on the phase-field will allow us to show a stability result and, subsequently, the uniqueness of the solution. Existence of solutions will be obtained by using an auxiliary problem. The approach is to modify the problem by introducing an appropriate truncation of $ (1-\phi^2)^2$. This auxiliary problem will then be studied by using fixed point arguments. Standard notation will be used. For a given fixed $T > 0 $, we denote $Q = \Omega \times (0,T)$ and we consider the following spaces, for $ q \geq 1$, \[ W^{2,1}_q (Q) = \{ w \in L^q(Q) : D_xw,D^2_xw \in L^q(Q), w_t \in L^q(Q) \}. \] The outline of this paper is as follows. In the next section we study an auxiliary problem. The last section is devoted to prove the well-posedness of problem \eqref{originalpfe}-\eqref{dados}. First, we study the existence of solutions, secondly we establish a stability result which will give us uniqueness at the same time and, finally, a result of regularity of the solution will be obtained by applying $L^p$-theory of parabolic linear equations together with bootstrapping arguments. \section{An auxiliary problem} In this section, we introduce an auxiliary problem related to \eqref{originalpfe}-\eqref{dados} for which we will prove a result of existence of solutions by using Leray-Schauder's fixed point theorem \cite{Friedman}. Let $ \Pi $ be the function \[ \Pi(r) = \begin{cases} -1, & r < -1 \\ r, & -1 \leq r \leq 1 \\ 1, & r > 1\,. \end{cases} \] Consider the problem \begin{gather} \phi_t - \epsilon^2 \Delta \phi = \phi - \phi^3 + \dfrac{(\theta_M -\theta)}{\delta}(1-\Pi(\phi)^2)^2 \quad \mbox{in } Q, \label{regularpfe} \\ \theta_t - K \Delta \theta + b \theta = \frac{\delta}{2} \phi_t + \alpha \phi + \beta \quad \mbox{in } Q, \label{regularte} \\ \frac{\partial \phi}{\partial n} = 0 ,\quad \frac{\partial \theta} {\partial n} = 0, \quad \mbox{on } \partial \Omega \times (0,T), \\ \phi(0) = \phi_0 , \quad \theta(0) = \theta_0, \quad \mbox{in } \Omega, \label{regulardados} \end{gather} where $ \alpha = (a_1-a_{-1})\dfrac{I}{2} $ and $ \beta = (a_1 + a_{-1}) \dfrac{I}{2} + b \theta_a$. We then have the following existence result. \begin{proposition} \label{regularizado} Let $(\phi_0,\theta_0) \in H^{1+\gamma}(\Omega) \times H^{1+\gamma}(\Omega) $, $1/2 < \gamma \leq 1$, satisfying the compatibility condition $\displaystyle \frac{\partial \phi_0}{\partial n} =\frac{\partial \theta_0 }{\partial n} =0 $ a.e. on $\partial \Omega$. Then there exists $ (\phi , \theta) \in W^{2,1}_2(Q)\times W^{2,1}_2(Q) $ solution to problem \eqref{regularpfe}-\eqref{regulardados} for any fixed $T> 0$, which verifies the estimate \begin{equation} \label{regularest} \|\phi\|_{W^{2,1}_2(Q)} + \|\theta\|_{W^{2,1}_2(Q)} \leq C \bigl(\|\phi_0\|_{H^1(\Omega)} + \|\theta_0\|_{H^1(\Omega)} + 1\bigr), \end{equation} where $C$ depends on $ \Omega$, and some physical parameters. \end{proposition} \begin{proof} In order to apply Leray-Schauder's fixed point theorem we consider the following family of operators, indexed by the parameter $0 \leq \lambda \leq 1$, \[ \mathcal{T}_{\lambda} : B \to B, \] where $B$ is the Banach space \[ B = L^2(Q) \times L^2(Q) , \] and is defined as follows: given $(\hat{\phi}, \hat{\theta}) \in B$, let $\mathcal{T}_\lambda(\hat{\phi}, \hat{\theta})=(\phi, \theta)$, where $(\phi, \theta)$ is obtained by solving the problem \begin{gather} \phi_t - \epsilon^2 \Delta \phi - ( \phi - \phi^3 ) = \lambda \dfrac{(\theta_M- \hat{\theta})}{\delta}(1-\Pi(\hat{\phi})^2)^2 \quad \mbox{in } Q, \label{auxpfe} \\ \theta_t - K \Delta \theta + b \theta = \frac{\delta}{2} \phi_t + \alpha \phi + \beta \quad \mbox{in } Q, \label{auxte} \\ \frac{\partial \phi}{\partial n} = 0 ,\quad \frac{\partial \theta} {\partial n} = 0 \quad \mbox{on } \partial \Omega \times (0,T), \label{auxbound}\\ \phi(0) = \phi_0 , \quad \theta(0) = \theta_0 \quad \mbox{in } \Omega. \label{auxdados} \end{gather} Before we prove that $\mathcal{T}_\lambda$ is well defined, we observe that clearly $(\phi, \theta)$ is a solution of \eqref{regularpfe}-\eqref{regulardados} if and only if it is a fixed point of the operator $\mathcal{T}_1$. To verify that the operator $\mathcal{T}_\lambda$ is well defined, observe that since $\hat{\theta}\in L^2(Q)$ and $|(1-\Pi(\hat{\phi})^2)^2| \leq 1$, we infer from \cite[Theorem 2.1]{Hoffman} that there is a unique solution $\phi$ of equation \eqref{auxpfe} with $\phi \in W^{2,1}_2(Q) $ satisfying the first of the boundary conditions \eqref{auxbound}. Since $\phi $ and $ \phi_t \in L^2(Q), $ according to $L^p$-theory of parabolic equations \cite[Theorem 9.1]{Lady} there is a unique solution $\theta $ of equation \eqref{auxte} with $\theta \in W^{2,1}_2(Q)$ satisfying the second of the boundary conditions \eqref{auxbound}. Therefore, for each $\lambda \in [0,1]$, the mapping $\mathcal{T}_{\lambda} $ is well defined from $B$ into $B$. To prove continuity of $\mathcal{T}_{\lambda}$, let $ (\hat{\phi}_n, \hat{\theta}_n) \in B$ strongly converging to $(\hat{\phi}, \hat{\theta}) \in B$; for each $n$, let $(\phi_n, \theta_n)$ the corresponding solution of problem \begin{gather} {\phi_n}_t - \epsilon^2 \Delta \phi_n - ( \phi_n - \phi_n^3 ) = \lambda \dfrac{(\theta_M-\hat{\theta}_n )}{\delta}(1-\Pi(\hat{\phi}_n)^2)^2 \quad \mbox{in } Q, \label{npfe} \\ {\theta_n}_t - K \Delta \theta_n + b \theta_n = \frac{\delta}{2} {\phi_n}_t + \alpha \phi_n + \beta \quad \mbox{in } Q, \label{nte} \\ \frac{\partial \phi_n}{\partial n} = 0 ,\quad \frac{\partial \theta_n} {\partial n} = 0 \quad \mbox{on } \partial \Omega \times (0,T), \\ \phi_n(0) = \phi_0 , \quad \theta_n(0) = \theta_0 \quad \mbox{in }\Omega. \label{ndados} \end{gather} Next, we show that the sequence $(\phi_n, \theta_n)$ converges strongly to $(\phi, \theta) = \mathcal{T}_{\lambda}( \hat{\phi}, \hat{\theta}) $ in $B$. For that purpose, we will obtain estimates, uniformly with respect to $n$, for $(\phi_n, \theta_n)$. We denote by $ C_i $ any positive constant independent of $n$. We multiply \eqref{npfe} successively by $\phi_n$, ${\phi_n}_t$ and $-\Delta \phi_n$, and integrate over $ \Omega \times (0,t)$. After integration by parts and the use of H\"{o}lder and Young inequalities, we obtain the following three estimates \begin{gather} \begin{aligned} \frac{1}{2} \int_{\Omega} | \phi_n|^2 dx + & \int_0^t \int_{\Omega}\left(\epsilon^2 | \nabla \phi_n |^2 + |\phi_n|^4 \right)\,dx\,ds \\ &\leq C_1 + C_2 \int_0^t \int_{\Omega} \left( | \hat{\theta}_n |^2 + | \phi_n|^2 \right) \,dx\,ds, \end{aligned}\label{a} \\ \begin{aligned} \frac{1}{2} \int_0^t \int_{\Omega}| {\phi_n}_t |^2 \,dx\,ds + & \int_{\Omega} \left( \frac{\epsilon^2}{2} | \nabla \phi_n|^2 + \frac{|\phi_n|^4}{4} - \frac{|\phi_n|^2}{2} \right)dx \\ &\leq C_1 + C_2 \int_0^t \int_{\Omega} | \hat{\theta}_n |^2 \,dx\,ds , \end{aligned}\label{b} \\ \begin{aligned} \frac{1}{2}\int_{\Omega}| \nabla \phi_n |^2 dx + & \frac{\epsilon^2}{2} \int_0^t \int_{\Omega} | \Delta \phi_n |^2 \,dx\,ds \\ &\leq C_1 + C_2 \int_0^t \int_{\Omega} \left( | \nabla \phi_n |^2 + | \hat{\theta}_n |^2 \right) \,dx\,ds. \end{aligned}\label{c} \end{gather} By multiplying \eqref{b} by $ \frac{1}{2} $ and adding the result to \eqref{a} we find \[ \int_{\Omega} \left( | \phi_n |^2 + | \nabla \phi_n |^2 + |\phi_n|^4 \right) dx \leq C_1 + C_2 \int_0^t \int_{\Omega} \left( | \hat{\theta}_n |^2 + | \phi_n|^2 \right) \,dx\,ds. \] Since $ \| \hat{\theta}_n \|_{L^2(Q)} $ is bounded independent of $n$, by using Gronwall's lemma we deduce that \begin{equation} \label{e} \| \phi_n \|_{L^{\infty}(0,T; H^1(\Omega))} \leq C_1. \end{equation} Then, thanks to estimates (\ref{a})-(\ref{c}) we arrive at \begin{equation} \label{f} \| \phi_n \|_{L^2(0,T; H^2(\Omega))} + \| {\phi_n}_t \|_{L^2(Q)} \leq C_1. \end{equation} Next, from $L^p$-theory of parabolic equations applied to equation \eqref{nte} we have \begin{equation} \label{h} \| \theta_n \|_{W^{2,1}_2(Q)} \leq C_1 ( \|\theta_0\|_{H^1(\Omega)} + \|{\phi_n}_t \|_{L^2(Q)} + \|\phi_n \|_{L^2(Q)}+1). \end{equation} We now infer from \eqref{e},\eqref{f} and \eqref{h} that the sequence $(\phi_n, \theta_n) $ is bounded in $ W^{2,1}_2(Q) $ and in \[ W = \left \{ v \in L^{\infty}(0,T; H^1(\Omega)),\, v_t \in L^2(0,T; L^2(\Omega)) \right\}. \] Since $W^{2,1}_2(Q)$ is compactly embedded in $L^2(0,T;H^1(\Omega))$ and $W$ in $ C([0,T];L^2(\Omega))$ \cite[Corollary 4]{Simon}, it follows that there exist \[ \phi, \, \theta \in L^2(0,T; H^2(\Omega)) \cap L^{\infty}(0,T; H^1(\Omega)) \quad \mbox{with } \phi_t,\, \theta_t \in L^2(Q), \] and a subsequence of $ (\phi_n, \theta_n) $ (which we still denote by $ (\phi_n, \theta_n)$), such that, as $ n \to + \infty$, \begin{equation} \label{conv} \begin{gathered} (\phi_n, \theta_n) \to (\phi, \theta) \quad \mbox{in } \bigl(L^2(0,T;H^1(\Omega)) \cap C([0,T];L^2(\Omega))\bigr)^2 \quad \mbox{strongly, }\\ (\phi_n, \theta_n) \rightharpoonup (\phi,\, \theta) \quad \mbox{in } \bigl(W^{2,1}_2(Q)\bigr)^2 \quad \mbox{weakly.} \end{gathered} \end{equation} It now remains to pass to the limit as $n$ tends to $ +\infty$ in (\ref{npfe})-(\ref{ndados}). Since the embedding of $ W^{2,1}_2(Q)$ into $ L^9(Q) $ is compact \cite{Lions}, we infer that $ \phi_n^3 $ converges to $ \phi^3 $ in $L^2(Q). $ Moreover, since $(1-\Pi(\cdot)^2)^2 $ is a bounded Lipschitz continuous function and $\hat{\phi_n}$ converges to $\hat{\phi} $ in $ L^2(Q)$, we have that $ (1-\Pi(\hat{\phi_n})^2)^2$ converges to $ (1-\Pi(\hat{\phi})^2)^2$ in $ L^p(Q) $ for any $ p \in [1,\infty)$. We then pass to the limit in \eqref{npfe} and get \eqref{auxpfe}. From convergence \eqref{conv}, it is easy to pass to the limit in \eqref{nte} and conclude that \eqref{auxte} holds almost everywhere. Therefore $\mathcal{T}_{\lambda}$ is continuous for all $ 0 \leq \lambda \leq 1$. At the same time, $\mathcal{T}_{\lambda}$ is bounded in $ W^{2,1}_2(Q)\times W^{2,1}_2(Q) $ but, the embedding of this space in $B$ is compact. Hence, $\mathcal{T}_{\lambda}$ is a compact operator for each $\lambda \in [0,1]$. To prove that for $(\hat{\phi}, \hat{\theta})$ in a bounded set of $B$, $T_{\lambda} $ is uniformly continuous with respect to $\lambda$, let $ 0 \leq \lambda_1, \lambda_2 \leq 1 $ and $(\phi_i, \theta_i)\, (i=1,2)$ be the corresponding solutions of \eqref{auxpfe}-\eqref{auxdados}. We observe that $\phi= \phi_1 - \phi_2 $ and $ \theta = \theta_1 - \theta_2 $ satisfy the problem \begin{gather} \begin{aligned} \phi_t - \epsilon^2 \Delta \phi = & \phi (1- (\phi_1^2 + \phi_1 \phi_2 + \phi_2^2) ) \\ & + (\lambda_1 - \lambda_2) \Bigl( \dfrac{(\theta_M- \hat{\theta})}{\delta}(1-\Pi(\hat{\phi})^2)^2 \Bigr) \mbox{ in } Q, \end{aligned}\label{lpfe} \\ \theta_t - K \Delta \theta + b \theta = \frac{\delta}{2} \phi_t + \alpha \phi \quad \mbox{in } Q, \label{lte} \\ \frac{\partial \phi}{\partial n} = 0 ,\quad \frac{\partial \theta} {\partial n} = 0 \quad \mbox{on } \partial \Omega \times (0,T), \\ \phi(0) = 0 , \quad \theta(0) = 0 \quad \mbox{in } \Omega. \label{ldados} \end{gather} We remark that $d:= \phi_1^2 + \phi_1 \phi_2 + \phi_2^2 \geq 0$. Now, multiply equation \eqref{lpfe} by $\phi$ and integrate over $\Omega \times(0,t);$ after integration by parts and the use of H\"older and Young inequalities we obtain \begin{align*} \int_{\Omega} | \phi |^2 dx + & \int_0^t \int_{\Omega} | \nabla \phi |^2 \,dx\,ds \\ & \leq C_1 \int_0^t \int_{\Omega} | \phi |^2 \,dx\,ds + C_2| \lambda_1 - \lambda_2 |^2 \int_0^t \int_{\Omega} (| \hat{\theta} |^2 +1 )\,dx\,ds . \end{align*} By applying Gronwall's lemma we arrive at \begin{equation} \label{o} \| \phi \|^2_{L^{\infty} (0,T; L^2(\Omega))} + \| \phi \|^2_{L^2(0,T; H^1(\Omega))} \leq C_1 \, | \lambda_1 - \lambda_2 |^2. \end{equation} Multiplying \eqref{lpfe} by $ \phi_t $ and using H\"older inequality, we get \begin{align*} \int_0^t \int_{\Omega} | \phi_t |^2 \,dx\,ds + & \frac{\epsilon^2}{2} \int_{\Omega} | \nabla \phi |^2 dx\\ & \leq C_1 \int_0^t \int_{\Omega} | \phi |^2 \,dx\,ds + \frac{1}{2} \int_0^t \int_{\Omega} | \phi_t |^2 \,dx\,ds \\ &\quad + C_2 \Big( \int_0^t \int_{\Omega} | \phi |^{10/3} \,dx\,ds \Big)^{3/5} \Big(\int_0^t \int_{\Omega} | d |^5 \,dx\,ds \Big)^{2/5} \\ &\quad + C_3| \lambda_1 - \lambda_2 |^2 \int_0^t \int_{\Omega} ( | \hat{\theta} |^2 + 1 ) \,dx\,ds . \end{align*} Since $ W^{2,1}_2(Q) \hookrightarrow L^{10}(Q) $, the following interpolation inequality holds \[ \| \phi \|^2_{L^{10/3}(Q)} \leq \eta \, \| \phi \|^2_{W^{2,1}_2(Q)} + \tilde{C} \, \| \phi \|^2_{L^2(Q)} \quad \mbox{for all } \eta > 0. \] Moreover, since $\| d \|_{L^5(Q)} \leq C$, with $C$ depending on $\| \phi_i \|_{L^{10}(Q)},\, i =1,2$, by rearranging the terms in the last inequality, we obtain \begin{equation} \label{r} \begin{aligned} \int_0^t \int_{\Omega} | \phi_t |^2 \,dx\,ds + \int_{\Omega} | \nabla \phi |^2 dx \leq & C_1 \int_0^t \int_{\Omega} | \phi |^2 \,dx\,ds + C_2 \eta \| \phi \|^2_{W^{2,1}_2(Q)} \\ & + C_3 | \lambda_1 - \lambda_2 |^2 \int_0^t \int_{\Omega} ( | \hat{\theta} |^2 + 1 ) \,dx\,ds . \end{aligned} \end{equation} Multiplying \eqref{lpfe} by $- \Delta \phi$, we infer in a similar way that \begin{equation} \label{s} \begin{aligned} \int_{\Omega} | \nabla \phi |^2 dx + &\int_0^t \int_{\Omega} | \Delta \phi |^2 dx ds\\ &\leq C_1 \int_0^t \int_{\Omega}\left( | \phi |^2 + | \nabla \phi |^2 \right) \,dx\,ds + C_2 \eta \| \phi \|^2_{W^{2,1}_2(Q)} \\ &\quad + C_3 | \lambda_1 - \lambda_2 |^2 \int_0^t \int_{\Omega} ( | \hat{\theta} |^2 + 1 ) \,dx\,ds . \end{aligned} \end{equation} By taking $\eta > 0 $ small enough and considering (\ref{o}), we conclude from (\ref{r}) and (\ref{s}) that \begin{equation} \label{t} \| \phi \|^2_{W^{2,1}_2(Q)} + \| \phi \|^2_{L^{\infty}(0,T;H^1(\Omega))} \leq C_1 |\lambda_1 - \lambda_2 |^2. \end{equation} Next, by multiplying \eqref{lte} by $\theta$, integrating over $\Omega\times (0,t)$ and using H\"older inequality we have \[ \int_{\Omega} | \theta |^2 dx + \int_0^t \int_{\Omega} | \nabla \theta |^2 \,dx\,ds \leq C_1 \int_0^t\int_{\Omega}( | \phi_t |^2 + | \phi |^2+ | \theta|^2 ) \,dx\,ds . \] Thus, by using Gronwall's lemma and \eqref{t}, we infer that \begin{equation} \label{u} \| \theta \|^2_{L^{\infty}(0,T; L^2(\Omega))} \leq C_1 \, | \lambda_1 - \lambda_2 |^2. \end{equation} It follows from \eqref{t} and \eqref{u} that $\mathcal{T}_{\lambda} $ is uniformly continuous with respect to $\lambda$ on bounded sets of $B$. Now we estimate the set of all fixed points of $\mathcal{T}_{\lambda}$. Let $ (\phi, \theta) \in B $ be such a fixed point, i.e. a solution of the problem \begin{gather} \phi_t - \epsilon^2 \Delta \phi -( \phi - \phi^3) = \lambda \dfrac{(\theta_M -\theta)}{\delta}(1-\Pi(\phi)^2)^2 \quad \mbox{in } Q, \label{fixpfe} \\ \theta_t - K \Delta \theta + b \theta = \frac{\delta}{2} \phi_t + \alpha \phi + \beta \quad \mbox{in } Q, \label{fixte} \\ \frac{\partial \phi}{\partial n} = 0 ,\quad \frac{\partial \theta}{\partial n }= 0 \quad \hfill \mbox{on } \partial \Omega \times (0,T), \\ \phi(0) = \phi_0 , \quad \theta(0) = \theta_0 \quad \mbox{in } \Omega. \label{fixdados} \end{gather} First, we multiply equation \eqref{fixpfe} successively by $\phi$, $\phi_t $ and $ -\Delta \phi$, and integrate over $\Omega $. After integration by parts, using H\"older and Young inequalities we obtain \begin{gather} \frac{1}{2} \frac{d}{dt} \int_{\Omega} | \phi |^2 dx + \int_{\Omega} \left( \epsilon^2 | \nabla \phi |^2 + |\phi|^4 \right) dx \leq C_1 + C_2 \int_{\Omega} ( | \theta |^2 + |\phi |^2) dx, \label{w} \\ \frac{1}{2}\int_{\Omega} | \phi_t |^2 dx + \frac{d}{dt} \int_{\Omega} \left( \frac{\epsilon^2}{2} | \nabla \phi |^2 + \frac{1}{4} |\phi|^4 - \frac{1}{2} | \phi |^2 \right) dx \leq C_1 + C_2 \int_{\Omega} | \theta |^2 dx, \label{x} \\ \frac{1}{2} \frac{d}{dt} \int_{\Omega} | \nabla \phi |^2 dx + \int_{\Omega} \frac{\epsilon^2}{2} | \Delta \phi |^2 dx \leq C_1 + C_2 \int_{\Omega} ( | \theta |^2 + |\nabla \phi |^2 ) dx. \label{y} \end{gather} Next, by multiplying \eqref{fixte} with $ \theta $, arguments similar to the previous ones lead to the following estimate \begin{equation} \frac{1}{2}\frac{d}{dt} \int_{\Omega} | \theta |^2 dx + K \int_{\Omega} | \nabla \theta |^2 dx \leq \frac{1}{8} \int_{\Omega} | \phi_t |^2 dx + C_1 \int_{\Omega} (| \theta |^2 + |\phi|^2)dx \label{z}. \end{equation} Now, multiply \eqref{x} by $ \frac{1}{2} $ and add the result to \eqref{w}, \eqref{y} and \eqref{z} to obtain \begin{equation} \begin{aligned} \frac{d}{dt} \int_{\Omega} \Big(\frac{1}{4} | \phi |^2 + \big( \frac{\epsilon^2}{4} + & \frac{1}{2} \big) | \nabla \phi |^2 + \frac{1}{8} |\phi|^4 + \frac{1}{2} | \theta |^2 \Big) dx \\ + \int_{\Omega} \Big( \epsilon^2 | \nabla \phi |^2 + & |\phi|^4 + \frac{1}{8} | \phi_t |^2 + \frac{\epsilon^2}{2} | \Delta \phi |^2 + K | \nabla \theta |^2 \Big) dx \\ & \leq C_1 + C_2 \int_{\Omega} \left( | \theta |^2 + |\phi |^2 + | \nabla \phi |^2 \right) dx. \label{bb} \end{aligned} \end{equation} Integrating with respect $t$ and using Gronwall's lemma we find \[ \| \phi \|_{L^{\infty}(0,T;H^1(\Omega))} + \| \theta \|_{L^{\infty}(0,T;L^2(\Omega))} \leq C_1 , \] where $C_1$ is independent of $\lambda$. Therefore, all fixed points of $\mathcal{T}_{\lambda}$ in $B$ are bounded independently of $\lambda \in [0,1]$. Finally, observe that the equation $x - \mathcal{T}_0(x) = 0$ is equivalent to say that problem \eqref{auxpfe}-\eqref{auxdados} for $\lambda = 0$ has a unique solution. This is concluded reasoning exactly as in the beginning of this proof, when we proved that $\mathcal{T}_\lambda$ was well defined. Therefore, we can apply Leray-Schauder's fixed point theorem, and so there is at least one fixed point $(\phi, \theta) \in B \cap W^{2,1}_2(Q) \times W^{2,1}_2(Q) $ of the operator $\mathcal{T}_1$, i.e., $(\phi, \theta ) = \mathcal{T}_1(\phi,\theta)$. This corresponds to a solution of problem \eqref{regularpfe}-\eqref{regulardados}. To prove estimate \eqref{regularest}, observe that from \eqref{bb} it follows \begin{equation} \label{cc} \|\phi\|_{W^{2,1}_2(Q)} + \|\theta\|_{L^2(0,T;H^1(\Omega))\cap L^\infty(0,T;L^2(\Omega))} \leq C \big( \|\phi_0\|_{H^1(\Omega)} + \|\theta_0\|_{L^2(\Omega)} + 1 \big). \end{equation} To obtain an estimate for $ \|\theta\|_{W^{2,1}_2(Q)}$, we apply $L^p$-theory of parabolic equations \[ \|\theta \|_{W^{2,1}_2(Q)} \leq C \bigl ( \|\theta_0\|_{H^1(\Omega)} + \|\phi_t \|_{L^2(Q)} + \|\phi\|_{L^2(Q)} + 1 \bigr).\] Using \eqref{cc} we deduce the desired estimate. The proof of Proposition \ref{regularizado} is thus complete. \end{proof} \section{Existence and uniqueness} In this section, we prove the well-posedness of problem \eqref{originalpfe}-\eqref{dados}. We begin with the following existence result. \begin{theorem} \label{existencia} Let be given functions satisfying: $\phi_0,\, \theta_0 \in H^{1+ \gamma}(\Omega)$ with $1/2 < \gamma \leq 1$, the compatibility condition $\displaystyle \frac{\partial \phi_0}{\partial n} = \frac{\partial \theta_0}{\partial n} =0$ a.e. on $\partial \Omega $ and such that $-1 \leq \phi_0 \leq 1 $ a.e. in $ \Omega$. Then there exists $ (\phi, \theta) \in W^{2,1}_2(Q)\times W^{2,1}_2(Q) $ solution to problem \eqref{originalpfe}-\eqref{dados} which satisfies \[ -1 \leq \phi \leq 1 \quad \mbox{for all } t \in [0,T] \mbox{ and a.e. in } \Omega. \] In addition to that the following estimate \begin{equation} \label{originalest} \|\phi\|_{W^{2,1}_2(Q)} + \|\theta\|_{W^{2,1}_2(Q)} \leq C \bigl(\|\phi_0\|_{H^1(\Omega)} + \|\theta_0\|_{H^1(\Omega)} + 1\bigr) \end{equation} holds with $C$ depending on $ \Omega, \, T $ and the physical parameters. \end{theorem} \begin{proof} Observe that it suffices to show that a solution $(\phi,\theta) \in W^{2,1}_2(Q)\times W^{2,1}_2(Q) $ to auxiliary problem \eqref{regularpfe}-\eqref{regulardados} with $-1 \leq \phi_0 \leq 1 $ a.e. in $ \Omega$ satisfies $ -1 \leq \phi \leq 1 $. In fact, if $ -1 \leq \phi \leq 1 $ by definition of the operator $ \Pi$ we have that $ \Pi(\phi) = \phi $ and, subsequently, $ (\phi,\theta) $ will be a solution of the original problem \eqref{originalpfe}-\eqref{dados}. First, we prove that if $ \phi_0 \leq 1 $ a.e. in $ \Omega $ then $ \phi(t) \leq 1 $ for all $ t \in [0,T] $ and a.e. in $ \Omega$. Let us consider the positive part of $ (\phi -1) $ namely $ (\phi -1)^+ = \max (\phi-1,0)$. According to \cite{Gilbarg}, we have that $ \nabla (\phi-1)^+ = \nabla \phi $ if $ \phi -1 \geq 0 $ and $ \nabla (\phi-1)^+ = 0 $ otherwise. Similarly, we have $ (\phi -1)^+_t = \phi_t $ if $ \phi -1 \geq 0 $ and $ (\phi-1)^+_t = 0 $ otherwise. Multiplying equation \eqref{regularpfe} by $(\phi -1)^+ $ and integrating over $ \Omega \times (0,t) $, for any $ 0 \leq t \leq T$, we obtain \begin{align*} &\| (\phi-1)^+ (t)\|_{L^2(\Omega)}^2 + \epsilon^2 \int_0^t \|\nabla (\phi-1)^+ \|^2_{L^2(\Omega)} ds \\ & = \| (\phi_0-1)^+ \|_{L^2(\Omega)}^2 + \int_0^t \int_\Omega \Bigl( \phi - \phi^3 + \dfrac{(\theta_M -\theta)}{\delta}(1-\Pi(\phi)^2)^2 \Bigr)(\phi-1)^+ \,dx\,ds. \end{align*} Since $ \phi_0 \leq 1 $ one has that $ \| (\phi_0-1)^+ \|_{L^2(\Omega)} = 0$. Moreover, if $ \phi < 1 $ the last integral vanishes. Now, observe that if $ \phi \geq 1 $ we have that $ (\phi - \phi^3)(\phi-1)^+ = \phi (1-\phi^2) (\phi-1)^+ \leq 0 $ and $\Pi(\phi) =1 $. Thus $ (1-\Pi(\phi)^2)^2 = 0$ and so we can conclude that \[ \| (\phi-1)^+ (t)\|_{L^2(\Omega)}^2 \leq 0, \quad \mbox{for all } 0 \leq t \leq T. \] Therefore, $ (\phi-1)^+ (t) = 0 $ for all $ 0 \leq t \leq T $ and a.e. in $ \Omega$, which implies that $ \phi(t) \leq 1 $ for all $ 0 \leq t \leq T $ and a.e. in $ \Omega$. Next, we prove that if $ \phi_0 \geq -1 $ a.e. in $ \Omega $ then $ \phi(t) \geq -1 $ for all $ t \in [0,T] $ and a.e. in $ \Omega$. For this we consider the negative part of $ (\phi +1) $ namely $ (\phi +1)^- = \max (-(\phi+1),0). $ By multiplying equation \eqref{regularpfe} by $ -(\phi +1)^-$ we obtain \begin{align*} & \| (\phi+1)^- (t)\|_{L^2(\Omega)}^2 + \epsilon^2 \int_0^t \|\nabla (\phi+1)^- \|^2_{L^2(\Omega)} ds \\ &= \| (\phi_0+1)^- \|_{L^2(\Omega)}^2 + \int_0^t \int_\Omega \bigl( \phi - \phi^3 + \dfrac{(\theta_M -\theta)}{\delta}(1-\Pi(\phi)^2)^2 \bigr)\bigl(-(\phi+1)^-\bigr) \,dx\,ds. \end{align*} Similarly as before, since $ \phi_0 \geq -1 $ we have that $ \|(\phi_0+1)^- \|_{L^2(\Omega)} = 0$. Moreover, if $ \phi \geq - 1 $ the last integral vanishes. Now, observe that if $ \phi < - 1 $ we have that $ (\phi - \phi^3)\bigl(-(\phi+1)^-\bigr) = \phi (1-\phi)(1+\phi)\bigl( -(\phi+1)^-\bigr) \leq 0 $ and $ \Pi(\phi) =-1 $. Thus $ (1-\Pi(\phi)^2)^2 = 0$ and we deduce \[ \| (\phi+1)^- (t)\|_{L^2(\Omega)}^2 \leq 0, \quad \mbox{for all } 0 \leq t \leq T. \] Therefore, $ (\phi+1)^- (t) = 0 $ for all $ 0 \leq t \leq T $ and a.e. in $ \Omega$, which implies that $ \phi(t) \geq -1 $ for all $ 0 \leq t \leq T $ and a.e. in $ \Omega$. The proof is then complete. \end{proof} We will prove stability of the solutions which will give us uniqueness at the same time. We will denote by $C$ a positive constant that may change from one relation to another. \begin{theorem} \label{stability} Let be given functions satisfying: $\phi_0^i,\, \theta_0^i \in H^{1+ \gamma}(\Omega)$ with $1/2 < \gamma \leq 1$, $ \dfrac{\partial \phi_0^i}{\partial n} = \dfrac{\partial \theta_0^i}{\partial n} =0$ a.e. on $\partial \Omega $ and such that $-1 \leq \phi_0^i \leq 1 $ a.e. in $ \Omega, \, i = 1,2$. Let $ (\phi_i, \theta_i) $ be the corresponding solutions to problem \eqref{originalpfe}-\eqref{dados}. Then the following stability estimate holds \[ \|\phi_1 - \phi_2\|_{W^{2,1}_2(Q)} + \|\theta_1 - \theta_2\|_{W^{2,1}_2(Q)} \leq C \bigl(\|\phi_0^1 - \phi_0^2\|_{H^1(\Omega)} + \|\theta_0^1- \theta_0^2\|_{H^1(\Omega)} \bigr), \] where $C$ depends on $ \|\phi_i\|_{W^{2,1}_2(Q)}$ and $ \| \theta_i\|_{W^{2,1}_2(Q)}$. \end{theorem} \begin{proof} We observe that $\phi= \phi_1 - \phi_2 $ and $\theta = \theta_1 - \theta_2 $ verify the following problem \begin{gather} \begin{aligned} \phi_t - \epsilon^2 \Delta \phi = & \phi (1- (\phi_1^2 + \phi_1 \phi_2 + \phi_2^2) ) \\ & + \dfrac{(\theta_M- \theta_1)}{\delta}(1-\phi_1^2)^2 - \dfrac{(\theta_M- \theta_2)}{\delta}(1-\phi_2^2)^2 \quad \mbox{in } Q, \end{aligned}\label{estapfe} \\ \theta_t - K \Delta \theta + b \theta = \frac{\delta}{2} \phi_t + \alpha \phi \quad \mbox{in } Q, \label{estate} \\ \frac{\partial \phi}{\partial n} = 0 ,\quad \frac{\partial \theta} {\partial n} = 0 \quad \mbox{on } \partial \Omega \times (0,T), \\ \phi(0) = \phi_0^1 - \phi_0^2 = \phi_0 ,\quad \theta(0) = \theta_0^1- \theta_0^2 = \theta_0 \quad \mbox{in } \Omega. \label{estadados} \end{gather} Now, using the identity $ (1-\phi_1^2)^2 - (1-\phi_2^2)^2 = \phi (\phi_1+\phi_2) (\phi_1^2 + \phi_2^2 -2) $ equation \eqref{estapfe} can be written as \begin{align*} \phi_t - \epsilon^2 \Delta \phi &= \phi \bigl(1- (\phi_1^2 + \phi_1 \phi_2 + \phi_2^2) \bigr) + \dfrac{\theta_M}{\delta}\phi (\phi_1+\phi_2) (\phi_1^2 + \phi_2^2 -2) \\ &\quad + \dfrac{1}{\delta}\theta_1 \phi (\phi_1+\phi_2) (\phi_1^2 + \phi_2^2 -2) + \dfrac{1}{\delta} \theta (1-\phi_2^2)^2. \end{align*} Since $ |\phi_i| \leq 1, $ from $ L^p$-theory of parabolic equations we have \[ \|\phi\|_{W^{2,1}_2(Q)} \leq C \bigl( \|\phi_0 \|_{H^1(\Omega)} + \|\phi \|_{L^2(Q)} + \|\theta \|_{L^2(Q)} + \|\theta_1 \phi \|_{L^2(Q)} \bigr) \] and \begin{align*} \|\theta\|_{W^{2,1}_2(Q)} & \leq C \bigl( \|\theta_0 \|_{H^1(\Omega)} + \|\phi_t \|_{L^2(Q)} + \|\phi \|_{L^2(Q)} \bigr) \\ & \leq C \bigl( \|\theta_0 \|_{H^1(\Omega)} + \|\phi_0 \|_{H^1(\Omega)} + \|\phi \|_{L^2(Q)} + \|\theta \|_{L^2(Q)} + \|\theta_1 \phi \|_{L^2(Q)} \bigr) . \end{align*} The $L^2$-norm of $ \theta_1 \phi $ can be bounded by using H\"older inequality and the Sobolev embedding \begin{align*} \|\theta_1 \phi \|_{L^2(Q)} & \leq \Bigl(\int_0^T \|\theta_1\|_{L^4(\Omega)}^2 \|\phi\|_{L^4(\Omega)}^2dt\Bigr)^{1/2}\\ & \leq C \|\theta_1\|_{L^\infty(0,T;H^1(\Omega))} \|\phi\|_{L^2(0,T;H^1(\Omega))}. \end{align*} Thus, we conclude that \begin{equation} \label{ambas} \|\phi\|_{W^{2,1}_2(Q)} + \|\theta\|_{W^{2,1}_2(Q)} \leq C \bigl( \|\phi_0 \|_{H^1(\Omega)} + \|\theta_0 \|_{H^1(\Omega)} + \|\theta \|_{L^2(Q)} + \|\phi\|_{L^2(0,T;H^1(\Omega))} \bigr). \end{equation} To obtain estimates for $ \phi $ and $ \theta$ we return to equations \eqref{estapfe}-\eqref{estate} and use standard techniques. We first deduce that \[ \frac{1}{2} \frac{d}{dt}\|\phi\|_{L^2(\Omega)}^2 + \epsilon^2\|\nabla \phi \|_{L^2(\Omega)}^2 \leq C \bigl(\|\phi \|_{L^2(\Omega)}^2 + \|\theta \|_{L^2(\Omega)}^2 + \int_\Omega | \theta_1|\, |\phi|^2 dx\bigr), \] where we used that $ |\phi_i | \leq 1$. The last term can be bounded by using H\"older and Young inequalities \begin{align*} \int_\Omega | \theta_1|\, |\phi|^2 dx & \leq \|\theta_1\|_{L^4(\Omega)} \|\phi\|_{L^4(\Omega)} \|\phi\|_{L^2(\Omega)} \\ &\leq C\|\theta_1\|_{L^\infty(0,T;H^1(\Omega))}^2 \|\phi\|_{L^2(\Omega)}^2 + \frac{\epsilon^2}{2}\|\nabla \phi\|_{L^2(\Omega)}^2. \end{align*} By rearranging terms we arrive at \[ \frac{d}{dt}\|\phi\|_{L^2(\Omega)}^2 + \epsilon^2\|\nabla \phi \|_{L^2(\Omega)}^2 \leq C \bigl(\|\phi \|_{L^2(\Omega)}^2 + \|\theta \|_{L^2(\Omega)}^2 \bigr) . \] Next, by multiplying equation \eqref{estate} by $ \theta $, we obtain, for any $ \eta > 0 $, \[ \frac{1}{2} \frac{d}{dt} \|\theta\|^2_{L^2(\Omega)} + K \|\nabla \theta\|^2_{L^2(\Omega)} \leq \eta \|\phi_t \|_{L^2(\Omega)}^2 + C \bigl( \|\phi \|_{L^2(\Omega)}^2 + \|\theta \|_{L^2(\Omega)}^2 \bigr). \] By integrating in time we deduce from the above relations that \begin{align*} &\|\phi\|_{L^2(\Omega)}^2 + \|\theta\|^2_{L^2(\Omega)} + \int_0^t \bigl( \|\nabla \phi \|_{L^2(\Omega)}^2 + \|\nabla \theta\|^2_{L^2(\Omega)} \bigr) ds \\ &\leq C \bigl( \|\phi_0 \|_{L^2(\Omega)}^2 + \|\theta_0 \|_{L^2(\Omega)}^2 + \int_0^t (\|\phi \|_{L^2(\Omega)}^2 + \|\theta \|_{L^2(\Omega)}^2)ds \bigr)+ \eta \|\phi_t \|_{L^2(Q)}^2 \end{align*} Taking $\eta $ small enough and using \eqref{ambas} yields \begin{align*} &\|\phi\|_{L^2(\Omega)}^2 + \|\theta\|^2_{L^2(\Omega)} + \int_0^t \bigl( \|\nabla \phi \|_{L^2(\Omega)}^2 + \|\nabla \theta\|^2_{L^2(\Omega)} \bigr) ds \\ &\leq C \bigl( \|\phi_0 \|_{H^1(\Omega)}^2 + \|\theta_0 \|_{H^1(\Omega)}^2 + \int_0^t( \|\phi \|_{L^2(\Omega)}^2 + \|\theta \|_{L^2(\Omega)}^2 )ds \bigr). \end{align*} Gronwall's lemma implies \begin{align*} &\|\phi\|_{L^2(\Omega)}^2 + \|\theta\|^2_{L^2(\Omega)} + \int_0^t \bigl( \|\nabla \phi \|_{L^2(\Omega)}^2 + \|\nabla \theta\|^2_{L^2(\Omega)} \bigr) ds \\ &\leq C \bigl( \|\phi_0 \|_{H^1(\Omega)}^2 + \|\theta_0 \|_{H^1(\Omega)}^2 \bigr). \end{align*} By plugging this in \eqref{ambas} we obtain the desired stability result. \end{proof} \begin{corollary} \label{unicidade} Let assumptions in theorem \ref{existencia} be fulfilled. Then there exists a unique solution $ (\phi, \theta) \in W^{2,1}_2(Q) \times W^{2,1}_2(Q)$ to problem \eqref{originalpfe}-\eqref{dados}. \end{corollary} \begin{remark} \label{rmk5} \rm The results stated in Theorems \ref{existencia} and \ref{stability} still hold, exactly with the same proofs, for initial conditions $\phi_0$ and $ \theta_0$ in any functional space including $H^1(\Omega)$ and for which it makes sense to require that $ \dfrac{\partial \phi_0}{\partial n} = \dfrac{\partial \theta_0}{\partial n}= 0$ a.e. on $\partial \Omega$ in order to apply $L^p$-theory of the parabolic linear equations. Moreover, a weaker version of theorems hold, with a natural weaker formulation of (\ref{originalpfe})-(\ref{dados}), for initial conditions $\phi_0$ and $ \theta_0$ just in $H^1(\Omega)$. For the proof, it is enough to take sequences in $ H^{1+ \gamma}(\Omega)$ with $1/2 < \gamma \leq 1$ satisfying the compatibility condition and converging to $\phi_0$ and $ \theta_0$ in $H^1(\Omega)$, and then to consider a sequence of approximate problems with these initial conditions. Since the sequence of approximate solutions will satisfy estimate \eqref{originalest}, it will be possible to pass to the limit and recover a solution of the original problem. \end{remark} We will prove a regularity result under the additional assumption that the initial data are smooth enough by using $L^p$-theory of the parabolic linear equations together with bootstrapping arguments. \begin{theorem} Let $ p \geq 2$. Let be given functions satisfying: $\phi_0,\, \theta_0 \in W^{2-\frac{2}{p}}_p(\Omega) \cap H^{1+ \gamma}(\Omega)$ with $1/2 <\gamma \leq 1$, $ \dfrac{\partial \phi_0}{\partial n} = \dfrac{\partial \theta_0}{\partial n} =0$ a.e. on $\partial \Omega $ and such that $-1 \leq \phi_0 \leq 1 $ a.e. in $ \Omega$. Then the unique solution to problem \eqref{originalpfe}-\eqref{dados} satisfies \[ (\phi, \theta) \in W^{2,1}_p(Q)\times W^{2,1}_p(Q). \] \end{theorem} \begin{proof} According to theorem \ref{existencia} and corollary \ref{unicidade} there exists a unique solution $ (\phi,\theta) \in W^{2,1}_2(Q) \times W^{2,1}_2(Q)$ to problem \eqref{originalpfe}-\eqref{dados}. Since $ |\phi| \leq 1 $ and $ W^{2,1}_2(Q) \hookrightarrow L^{10}(Q)$ from $L^p$-theory of parabolic equations applied to the phase-field equation we have that $ \phi \in W^{2,1}_{10}(Q)$ and, subsequently, from the temperature equation we conclude $ \theta \in W^{2,1}_{10}(Q)$. 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