\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 176, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/176\hfil viscous Cahn-Hilliard equation] {Critical case for the viscous Cahn-Hilliard equation} \author[L. T. T. Bui, A. N. Dao, J. I. D\'iaz \hfil EJDE-2017/176\hfilneg] {Le Trong Thanh Bui, Anh Nguyen Dao, Jes\'us Ildefonso D\'iaz} \address{Le Trong Thanh Bui \newline Faculty of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh city, 227 Nguyen Van Cu, D. 5, Ho Chi Minh city, Vietnam} \email{bltthanh@hcmus.edu.vn} \address{Anh Nguyen Dao (corresponding author)\newline Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Vietnam} \email{daonguyenanh@tdt.edu.vn} \address{Jes\'us Ildefonso D\'iaz \newline Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain} \email{ildefonso.diaz@mat.ucm.es} \thanks{Submitted April 30, 2016. Published July 11, 2017.} \subjclass[2010]{35B25, 35K55, 35R25, 28A33, 35D99} \keywords{Forward-backward parabolic equations; singular limits; \hfill\break\indent pseudo-parabolic regularization; Cahn-Hilliard regularization; viscous Cahn-Hilliard equation} \begin{abstract} We prove the existence of solutions of the viscous Cahn-Hilliard equation in whole domain when the nonlinear term in the second order diffusion grows as $u^q$ for the critical case when $N\geq 3$. Our results improve the ones in \cite{BuiST2,DKS}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction}\label{intro} In this article, we study the initial-value problem \begin{equation} %\label{PabD} \begin{gathered} u_t=\Delta [\varphi(u)-\alpha\Delta u+\beta u_t]\quad \text{in } \mathbb{R}^N\times (0,T):=Q, \\ u(x,0)=u_0\quad \text{in }\mathbb{R}^N\times\{0\}\,, \end{gathered} \label{eP} \end{equation} where the nonlinearity $\varphi$ satisfies the following assumptions: \begin{itemize} \item[(H1)] $\varphi\in W^{1,\infty}_{loc}(\mathbb{R})$, $\varphi(0)=0$, and $\varphi(s)s\geq 0$, \hspace{0.05in} for any $s\in\mathbb{R}$. \item[(H2)] There exists $K>0$ such that \begin{equation}\label{ass.phi.1} |\varphi(u)|\leq K (|u|+|u|^{q}), \end{equation} for some $q\in (1,\infty)$ if $N=1, 2$; or $q\in \big(1,\frac{N+2}{N-2}\big]$ if $N\ge3$. \item[(H3)] There exists $s_0>0$ such that $\varphi'(s)\geq 0$, if $|s|\geq s_0$. \end{itemize} Forward-backward parabolic equations arise in a variety of applications, such as edge detection in image processing \cite{PM}, aggregation models in population dynamics \cite{Pa}, and stratified turbulent shear flow \cite{BBDPU}, theory of phase transitions \cite{BFG,BS,Mu}, control theory in \cite{Diaz}, etc. A different well-known equation of this type is the Perona-Malik equation, \begin{equation}\label{PeMa} w_t=\operatorname{div}\Big( \frac {\nabla w}{1+|\nabla w|^2}\Big)\,, \end{equation} which is parabolic if $|\nabla w|<1$ and backward parabolic if $|\nabla w|>1$. Similarly, the equation \begin{equation}\label{PeMai} u_t=\Delta \Big( \frac {u}{1+u^2}\Big) \end{equation} is parabolic if $|u|<1$ and backward parabolic if $|u|>1$. Observe that in one space dimension the above equations are formally related setting $u=w_x$. A different well-known equation of application in theory of phase transitions is \begin{equation}\label{I1} u_t=\Delta \varphi (u) \end{equation} where the famous choice of nonlinearity $\varphi(u)=u^3-u$. Clearly, forward-backward parabolic equations lead to ill-posed problems. Often a higher order term is added to the right-hand side to regularize the equation. Two main classes of additional terms are encountered in the mathematical literature, which, e.g. in case of equation \eqref{PeMai}, \eqref{I1}, reduce to: (i) $\epsilon\Delta[\psi(u)]_t$, with $\psi'>0$, leading to third-order pseudo-parabolic equations ($\epsilon>0$ being a small parameter; for example, see \cite{BBDU,BuiST1,EP,MTT,NP,P1,P2,S,ST1,ST2}); (ii) $-\epsilon\Delta^2 u$, leading to fourth-order Cahn-Hilliard type equations (for example, see \cite{BBMN,BFG,P3,Sl} and references therein). Remarkably, when $\psi(u)=u$ either of the above regularizations can be regarded as a particular case of the viscous Cahn-Hilliard equation, \begin{equation}\label{viCH} \nu u_t=\Delta [\varphi(u)-\alpha\Delta u+\beta u_t] \quad (\alpha, \beta, \nu>0)\,, \end{equation} choosing either $\alpha=\epsilon$ or $\beta=\epsilon$; here $\varphi(u)= u^3-u$ or $\varphi(u)= \frac {u}{1+u^2}$ for equation \eqref{I1}, whereas in general it involves a non-monotonic function. Equation \eqref{viCH} has been derived by several authors using different physical considerations (in particular, see \cite{G,JF,N}). It is worth mentioning the wide literature concerning both the relationship between the viscous Cahn-Hilliard equation and {\it phase field models}, and generalized versions of the equation suggested in \cite{G} (and references therein). Besides, the existence results were obtained under suitable nonlinearity $\varphi$ in bounded smooth domain of $\mathbb{R}^N$ (see \cite{BuiST2,CD,ES}). Moreover, in the latter reference authors give us the rigorous proof of convergence to solutions of either the Cahn-Hilliard equation, or of the Allen-Cahn equation, or of the Sobolev equation, depending on the choice of the parameter $\alpha, \beta$. Recently, in \cite{DKS} the authors gave the analysis of equation \eqref{viCH} in $\mathbb{R}^N$ under some assumptions on the growth of nonlinearity $\varphi$ satisfying (H2), but not including the critical case $q=\frac{N+2}{N-2}$. In light of the above considerations, by using some sharp a priori estimates for a suitable auxiliary approximation problem, we will prove the existence of solutions of problem \eqref{eP} for a class of nonlinear functions $\varphi$ satisfying the growth condition (H2) including the critical case $q=\frac{N+2}{N-2}$. Thus, our existence results enhance a part of the ones of Dlotko, et al.\ \cite{DKS}. Our existence theorem is as follows: \begin{theorem}\label{th.exi.regular1} Let $u_0\in H^1(\mathbb{R}^N)$, and $q=\frac{N+2}{N-2}$. Let $\varphi$ satisfy {\rm (H1)--(H3)}. Then, there exists a weak solution of problem \eqref{eP}. \end{theorem} \begin{remark} \rm Note that we do not assume the boundedness on $\varphi'$, see \cite[(1.1)]{BuiST2}. Thus, our results also improve the ones of Bui, et al.\ \cite{BuiST2}. \end{remark} Before proving Theorem \ref{th.exi.regular1}, we give a definition of weak solutions of \eqref{eP}. \begin{definition}\label{def.sol.completa} \rm Let $\alpha, \beta>0$, and let $u_0\in H^1(\mathbb{R}^N)$. By a weak solution of problem \eqref{eP} we mean any function $u\in C([0,T];H^2(\mathbb{R}^N))\cap C^1([0,T];L^2(\mathbb{R}^N))$ such that $\varphi(u)\in C([0,T];L^2(\mathbb{R}^N))$, and \begin{equation}\label{eq.strong} \begin{gathered} u_t=\Delta v \quad \text{in }Q\\ u=u_0\quad \text{in }\mathbb{R}^N\times\{0\}\, \end{gathered} \end{equation} in the sense of distribution. Here $v\in C([0,T];H^2(\mathbb{R}^N)\cap H^1(\mathbb{R}^N))$ and for every $t\in[0,T]$ the function $v(\cdot,t)$ is the unique solution of the elliptic problem \begin{equation}\label{eq.elli} \begin{gathered} -\beta \Delta v(\cdot,t) + v(\cdot,t) =\varphi(u)(\cdot,t)-\alpha\Delta u(\cdot,t)\quad \text{in }\mathbb{R}^N,\\ \lim_{|x| \to \infty}v(x,t)=0. \end{gathered} \end{equation} The function $v$ is called a {\em chemical potential}. \end{definition} \section{Proof of Theorem \ref{th.exi.regular1}} We first mention the description of our method. We start by considering the existence of weak solutions of the viscous Cahn-Hilliard problem with Dirichlet boundary conditions in the ball $B_n$, which has center at the origin and radius $n \geq 1$: \begin{equation}\label{ePn} \begin{gathered} u_t=\Delta [\varphi_n(u)-\alpha\Delta u+\beta u_t] \quad \text{in } B_n\times (0,T)=:Q_n \\ u= \Delta u=0\quad \text{on } \partial B_n\times (0,T) \\ u=u_{0n}=u_0 \phi_n \quad \text{in }B_n\times\{0\}\,, \end{gathered} \end{equation} where $\phi_n(x)=\phi(x/n)$, and $\phi\in\mathcal{C}^\infty(\mathbb{R}^N)$ such that $\phi(x)=1$ if $|x|<1/2$, and $\phi(x)=0$ if $|x|>1$. And $\varphi_n$ is just a truncated function of $\varphi$ as in \cite{BuiST2}: \[ \varphi_n(u)=\begin{cases} \varphi(u), & \text{if } |u|\leq n,\\ \varphi(n)+ (u-n), & \text{if } u> n,\\ \varphi(-n)+ (u+n), & \text{if } u< n. \end{cases} \] Secondly, we establish a priori estimates for those solutions of problem \eqref{ePn} being independent of $n$. Finally, we shall pass to the limit as $n \to \infty$ (in a suitable way) to get a desired result. It is not difficult to verify that $\varphi_n$ is a globally Lipschitz function, and $\varphi_n(u)u\geq 0$. A well-posed result for problem \eqref{ePn} is proved in \cite[Theorem 2.1]{BuiST2}. Thus, there exists a unique weak solution $u_n$ of problem \eqref{ePn} in $B_n\times(0, T)$. Remind that \[ v_n=\varphi_n(u_n)-\alpha\Delta u_n+\beta u_{n t}. \] Then, multiplying both sides of this equation with $\partial_t u_n$ and integrating over $B_n \times (0,t)$ yields \begin{align*} &\int_{B_n}\Phi_n(u_n)(x,t)\,dx +\frac{\alpha }{2} \int_{B_n}|\nabla u_n|^2(x,t)\,dx +\beta\int_0^t\!\!\!\int_{B_n} u_{nt}^2\,dx\,ds\\ & = \int_0^t\!\!\!\int_{B_n} v_n \partial_t u_n \,dx\,ds + \int_{B_n}\Phi_n(u_{0n})(x)\,dx + \frac{\alpha}{2}\int_{B_n}|\nabla u_{0n}|^2\,dx, \quad \text{for } t\in(0,T), \end{align*} with $\Phi_n(u)=\int_{0}^{u}\varphi_n(s)ds$. Note that $\Delta v_n= u_{nt}$. Then, we obtain \begin{equation}\label{stima.fond} \begin{split} &\int_{\Omega}\Phi_n(u_n)(x,t)\,dx +\frac{\alpha }{2} \int_{\Omega}|\nabla u_n|^2(x,t)\,dx +\beta\int_0^t\!\!\!\int_{\Omega} u_{nt}^2\,dx\,ds\\ &+\int_0^t\!\!\!\int_{\Omega}|\nabla v_n|^2\,dx\,ds\\ &=\int_{\Omega}\Phi_n(u_{0n})(x)\,dx +\frac{\alpha}{2}\int_{\Omega}|\nabla u_{0n}|^2\,dx \end{split} \end{equation} Since $\varphi_n(s)s\geq 0$ and assumption (H2), we have \begin{equation*} 0\leq \Phi_n(u)=\int_{0}^{u}\varphi_n(s) ds\leq \frac{K}{2} u^2+\frac{K(N-2)}{2N}u^{\frac{2N}{N-2}},\end{equation*} so there is a positive constant $C=C(K, N)$ such that \begin{equation}\label{2} \int_{B_n} \Phi_n(u_{0n}) dx \leq C\Big(\int_{B_n} u^2_0 dx+ \int_{B_n} u^{\frac{2N}{N-2}}_0 dx\Big). \end{equation} From Sobolev's embedding theorem, we obtain \begin{equation}\label{1} \|u_{0n}\|_{L^\frac{2N}{N-2}(B_n)}\leq C(N) \|\nabla u_{0n}\|_{L^2(B_n)}. \end{equation} A combination of \eqref{1} and \eqref{2} implies that $\int_{B_n} \Phi(u_{0n}) \,dx$ is bounded by a constant depending only on $\|u_0\|_{H^1(\mathbb{R}^N)}$. Therefore, there is a positive constant $C=C(N, \|u_0\|_{H^1(\mathbb{R}^N)})$ such that \begin{equation}\label{stima.fond1} \begin{split} &\int_{B_n}\Phi_n(u_n)(x,t)\,dx+\frac{\alpha }{2} \int_{B_n}|\nabla u_n|^2(x,t)\,dx\\ &+ \beta\int_0^t\!\!\!\int_{B_n} u_{nt}^2\,dx\,ds + \int_0^t\!\!\!\int_{B_n}|\nabla v_n|^2\,dx\,ds \leq C. \end{split} \end{equation} Next, using $u_n$ as a test function to the first equation of \eqref{ePn} yields \begin{align*} &\frac{1}{2} \int_{B_n}u_n^2(x,t)\,dx +\frac{\beta }{2} \int_{B_n}|\nabla u_n|^2(x,t)\,dx +\alpha\int_0^t\!\!\!\int_{B_n} (\Delta u_n)^2\,dx\,ds\\ &+\int_0^t\int_{B_n} \varphi'_n(u) |\nabla u_n|^2 dx\,ds \\ &\leq \int_{B_n}u_{0n}^2(x,t)\,dx +\frac{\beta }{2} \int_{B_n}|\nabla u_{0n}|^2(x,t)\,dx\, \end{align*} Using (H3) yields \begin{align*} &\frac{1}{2} \int_{B_n}u_n^2(x,t)\,dx +\frac{\beta }{2} \int_{B_n}|\nabla u_n|^2(x,t)\,dx + \alpha\int_0^t\!\!\!\int_{B_n} (\Delta u_n)^2\,dx\,ds\\ &\leq \int_{B_n\times(0,t)\cap \{|u_n|\leq s_0\}} -\varphi'_n(u) |\nabla u_n|^2 dx\,ds + \int_{B_n}u_{0n}^2(x,t)\,dx \\ &\quad +\frac{\beta }{2} \int_{B_n}|\nabla u_{0n}|^2(x,t)\,dx\, \end{align*} By (H1), there is a positive constant $C_0$ such that $|\varphi'_n(s)|0$ not depending on $n$ such that \begin{equation}\label{9} \|u_n(\cdot,t)\|^2_{L^{\frac{2N}{N-4}}(B_n)}\leq C \|\Delta u_n(\cdot,t)\|^2_{L^2(B_n)}. \end{equation} Now, it follows from the interpolation theorem that \[ \|u_n(\cdot,t)\|_{L^{\frac{2(N+2)}{N-2}}(B_n)} \leq \|u_n(\cdot,t)\|^{\theta}_{L^{\frac{2N}{N-4}}(B_n)} \|u_n(\cdot,t)\|^{1- \theta}_{L^{\frac{2N}{N-2}}(B_n)}, \] with $\theta = \frac{N-2}{N+2}$. By \eqref{5}, from the last inequality, we obtain \begin{equation}\label{10} \|u_n(\cdot,t)\|^{\frac{2(N+2)}{N-2}}_{L^{\frac{2(N+2)}{N-2}}(B_n)} \leq C\|u_n(\cdot,t)\|^{2}_{L^{\frac{2N}{N-4}}(B_n)} \end{equation} A combination of \eqref{10}, \eqref{9}, and \eqref{3} yields \begin{equation}\label{11} \int_0^T \|u_n(\cdot,t)\|^{\frac{2(N+2)}{N-2}}_ {L^{\frac{2(N+2)}{N-2}}(B_n)} dt\leq C \int_0^T \|\Delta u_n(\cdot,t)\|^{2}_{L^2(B_n)} dt \leq C(T,N,u_0). \end{equation} Therefore, we obtain the above claim. It remains to pass to the limit as $n\to \infty$ in the equation satisfied by $u_n$. Thanks to the uniform estimates in \eqref{stima.fond}, \eqref{stima.fond1}, \eqref{3}, and \eqref{11}, we can mimic the proof of \cite[Theorem 2.4]{BuiST2} to get \begin{gather}\label{conv_u_n_k} u_n \stackrel{*}\rightharpoonup u \quad\text{in }L^\infty((0,T);H^1(\mathbb{R}^N))\,,\\ \label{conv_u_n_k_time} u_{nt} \rightharpoonup u_t \quad\text{in }L^2(Q)\,,\\ \label{conv_Delta_u_n_k} \Delta u_{n} \rightharpoonup \Delta u \quad\text{in }L^2(Q), \\ \label{12} u_n \to u \quad\text{a.e. in } Q, \end{gather} up to a subsequence. Next, we prove that $\varphi_n(u_n)$ converges weakly to $\varphi(u)$ in $L^2(Q)$. In fact, we observe that $\varphi_n(u_n)\to \varphi(u)$ as $n\to\infty$ a.e. in $Q$ by \eqref{12}. Moreover, the sequence $\{\varphi_n(u_n)\}_{n\geq 1}$ is uniformly bounded in $L^2(B_n\times(0,T))$ for any $n\geq 1$. Thus, there is a subsequence (still denoted by $\{\varphi_n(u_n)\}_{n\geq 1}$) such that $\varphi_n(u_n)$ converges weakly to $\varphi(u)$ in $L^2(Q)$, see \cite[Theorem 13.44]{HeStr}. Now, it suffices to show that $u$ is a weak solution of \eqref{eP}. We write the equation satisfied by $u_n$ in the weak sense: For any $\psi\in\mathcal{C}^1([0,T]; \mathcal{C}^2_c(\mathbb{R}^N))$ such that $\psi(.,T)=0$, we have \begin{align*} &\int_{Q(\psi)} -u_n\psi_t \,dx\,ds -\int_{\operatorname{supp}(\psi)} u_{0n}(x)\psi(x,0) \,dx \\ & = \int_{Q(\psi)}\left( \varphi(u_n) \Delta\psi -\alpha\Delta u_n \Delta\psi +\beta u_n \Delta\psi_t\right) dx\,ds, \end{align*} for any $n\geq 1$ such that $\operatorname{supp}(\psi)\subset B_n$, and $Q(\psi)=\operatorname{supp}(\psi)\times(0,T)$. 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