\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 35, 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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/35\hfil Rosenau and Benjamin-Bona-Mahony equations] {Existence of solutions to the Rosenau and Benjamin-Bona-Mahony equation in domains with moving boundary} \author[R. K. Barreto, C. de Caldas, P. Gamboa, \& J. Limaco\hfil EJDE-2004/35\hfilneg] {Rioco K. Barreto, Cruz S. Q. de Caldas, \\ Pedro Gamboa, \& Juan Limaco} % in alphabetical order \address{Rioco K. Barreto\hfill\break Instituto de Matem\'{a}tica - UFF\\ Rua M\'{a}rio Santos Braga s/n$^o$, CEP: 24020-140, Niter\'{o}i, RJ, Brasil} \email{rikaba@vm.uff.br} \address{Cruz S. Q. de Caldas, \hfill\break Instituto de Matem\'{a}tica - UFF\\ Rua M\'{a}rio Santos Braga s/n$^o$, CEP: 24020-140, Niter\'{o}i, RJ, Brasil} \email{gmacruz@vm.uff.br} \address{Pedro Gamboa \hfill\break Instituto de Matem\'{a}tica - UFRJ\\ Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil} \email{pgamboa@dmm.im.ufrj.br} \address{Juan Limaco\hfill\break Instituto de Matem\'{a}tica - UFF\\ Rua M\'{a}rio Santos Braga s/n$^o$, CEP: 24020-140, Niter\'{o}i, RJ, Brasil} \email{juanbrj@hotmail.com} \date{} \thanks{Submitted April 28, 2003. Published March 11, 2004.} \subjclass{35M10, 35B30} \keywords{Benjamin-Bona-Mahony equation, Rosenau equation, \hfill\break\indent noncylindrical domains} \begin{abstract} In this article, we prove the existence of solutions for a hyperbolic equation known as the the Rosenau and Benjamin-Bona-Mahony equations. We study increasing, decreasing, and mixed non-cylindrical domains. Our main tools are the Galerkin method, multiplier techniques, and energy estimates. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} To investigate the dynamics of certain discrete systems, Philip Rosenau obtained the equation $u_t+(u+u^2)_x+u_{xxxxt}=0$. The study of this equation in cylindrical domains was done by Mi Ai Park \cite{r1}, who proved the existence and uniqueness of local and global solutions. The Rosenau equation could be seen as a variant of Benjamin-Bona-Mahony (BBM) equation, $u_t+(u+u^2)_x-u_{xxt}=0$, which models long waves in a non linear dispersive system. In \cite{b1}, Benjamin-Bona-Mahony proved the existence and uniqueness of global solutions for the BBM equation in cylindrical domains. In this work, we study the existence of solutions for the Rosenau and BBM equations for increasing, decreasing, and mixed noncylindrical domains. We introduce the following notation: Let $\alpha $, $\beta $, $\gamma =\beta -\alpha $, be $C^2$-functions of a real variable, such that $\alpha (t)<\beta (t)$, for all $t\geq 0$. We represent the noncylindrical domain by $$ \widehat{Q}=\{(x,t)\in \mathbb{R}^2: \alpha (t)t_1$, then the projection of $[\alpha (t_2),\beta (t_2)]$ in the subspace $t=0$ is contained in the projection of $[\alpha (t_1),\beta (t_1)]$ in the same subspace. In the third section of this article, we study the existence of solutions for \eqref{I} and \eqref{II} satisfying the hypothesis \begin{itemize} \item[(H2)] $\alpha '(t)\leq 0$ and $\beta '(t)\geq 0$ for $t\in [0,T]$. \end{itemize} Analogously hypothesis (H2) implies that $\widehat{Q}$ increases In the last section of this article, we study the \eqref{I} and \eqref{II}, satisfying the hypothesis: \begin{itemize} \item[(H3)] $\widehat{Q}=\widehat{Q_1}\cup \widehat{Q_2}$ where $\widehat{Q_1}$ is increasing and $\widehat{Q_2}$ is decreasing. \end{itemize} In the following, by $\Omega $ we represent the interval $]0,1[$, $\Omega _t$ and $\Omega _0$ denote the intervals $]\alpha (t),\beta (t)[$ and $]\alpha(0),\beta (0)[$ respectively. We denote, as usual, by $(.,.)$, $\|\cdot\|$ respectively the scalar product and norm in $L^2(\Omega )$. In the sequel, $w_{m,x}$ denotes $\frac{\partial w_m}{\partial x}$ , analogously $w_{m,xx}=\frac{ \partial ^2w_m}{\partial x^2}$, $w_{m,xt}=\frac{\partial ^2w_m}{\partial t\,\partial x}$ , etc. \section{Solutions on decreasing domains} In this section we study the existence and uniqueness for \eqref{I} and \eqref{II} satisfying the hypothesis (H1). Let $\gamma (t)=\beta (t)-\alpha (t)>0$, for all $t\geq 0$. Then $0<\frac{x-\alpha (t)}{\gamma (t)}<1$, for all $t\in [0,T]$. With the change of variable $u(x,t)=v(y,t)$ where $y=\frac{x-\alpha (t)}{\gamma (t)}$, for all $t\in [0,T]$, problem \eqref{I} is transformed into \begin{equation} \label{III} \begin{gathered} v_t+\frac 1\gamma (v+v^2)_y+\frac 1{\gamma ^4}v_{yyyyt}-\frac{(\alpha '+\gamma 'y)}\gamma v_y-\frac{4\gamma '}{\gamma ^5}v_{yyyy} \\ -\frac{(\alpha '+\gamma 'y)}{\gamma ^5}v_{yyyyy}=0 \quad \mbox{in }\Omega \times ]0,T[ \\ v(0,t)=v(1,t)=0\quad \mbox{in }]0,T[ \\ v_y(0,t)=v_y(1,t)=0\quad \mbox{in } ]0,T[ \\ v(y,0)=v^0(y)\quad \mbox{in }\Omega\,. \end{gathered} \end{equation} Also problem \eqref{II} is transformed into \begin{equation} \label{IV} \begin{gathered} v_t+\frac 1\gamma (v+v^2)_y-\frac 1{\gamma 2}v_{yyt}-\frac{(\alpha '+\gamma 'y)}\gamma v_y+\frac{2\gamma '}{\gamma ^3}v_{yy} \\ +\frac{(\alpha '+\gamma 'y)}{\gamma ^3}v_{yyy}=0 \quad \mbox{in }\Omega \times ]0,T[\\ v(0,t)=v(1,t)=0\quad \mbox{in }\,]0,T[ \\ v(y,0)=v^0(y)\quad \mbox{in }\Omega\,. \end{gathered} \end{equation} Under these conditions, we establish the following existence results. \begin{theorem} \label{thm2.1} For each $u^0\in H_0^2(\Omega _0)\cap H^4(\Omega _0)$, there exists a unique function $u:\widehat{Q}\to \mathbb{R}$, satisfying $u\in C^1 ([0,T];H_0^2(\Omega _t))\cap C(0,T;H^3(\Omega_t)\cap H_0^2(\Omega _t))$ and \[ \int_{\widehat{Q}}u_t \phi\,dx\,dt+\int_{\widehat{Q}}(u+u^2)_x\phi \,dx\,dt+\int_{\widehat{Q}}u_{xxt}\phi_{xx}\,dx\,dt=0, \] for all $\phi \in L^2(0,T;H_0^2(\Omega _t))$, $u(x,0)=u^0(x)$, for all $x\in \Omega _0$. \end{theorem} \begin{theorem} \label{thm2.2} For each $u^0\in H_0^1(\Omega _0)\cap H^2(\Omega _0)$, there exists a unique function $u:\widehat{Q}\to \mathbb{R}$, satisfying $u\in L^\infty (0,T;H_0^1(\Omega _t))$, $u_t\in L^\infty(0,T;H_0^1(\Omega _t))$ and \[ \int_{\widehat{Q}}u_t\phi \,dx\,dt+\int_{\widehat{Q}}(u+u^2)_x\phi \,dx\,dt+\int_{\widehat{Q}}u_{xt}\phi _x\,dx\,dt=0, \] for all $\phi \in L^2(0,T;H_0^1(\Omega _t))$, $u(x,0)=u^0(x)$, for all $x\in \Omega _0$. \end{theorem} To prove Theorem \ref{thm2.1}, we need the following lemmas. \begin{lemma} \label{lm2.3} For each $v^0\in H_0^2(\Omega )\cap H^4(\Omega )$, there exists a unique function $v:\Omega \times ]0,T[\to \mathbb{R}$, satisfying $v\in L^\infty (0,T;H_0^2(\Omega)\cap H^4(\Omega))$, $v_t\in L^\infty(0,T;H_0^2(\Omega ))$, and \begin{align*} \int_{\Omega \times ]0,T[}[v_t\psi +\frac 1\gamma (v+v^2)_y\psi +\frac 1{\gamma ^4}v_{yyt}\psi _{yy}&\\ -\frac{(\alpha '+\gamma'y)}\gamma v_y\psi -\frac{4\gamma '}{\gamma ^5}v_{yy}\psi _{yy} +(\frac{(\alpha '+\gamma 'y)}{\gamma^{5}} \psi)_y v_{yyyy}]dy\,dt&=0, \end{align*} for all $\psi \in L^2(0,T;H_0^2(\Omega))$, $v(y,0)=v^0(y)$, for all $y\in \Omega $. \end{lemma} \begin{lemma} \label{lm2.4} For each $f\in C([0,T];H^{-2}(\Omega ))$, there exists a unique function $z:\Omega \times ]0,T[\to \mathbb{R}$, satisfying $z\in C([0,T];H_0^2(\Omega ))$ and $z+\frac 1{\gamma ^4}\Delta ^2z=f$ \end{lemma} \begin{lemma} \label{lm2.5} For each $v^0\in H_0^2(\Omega )\cap H^4(\Omega )$, there exists a unique function $v:\Omega \times ]0,T[\to \mathbb{R}$, satisfying $v\in C^1([0,T];H_0^2(\Omega ))\cap C([0,T];H^3(\Omega)\cap H_0^2(\Omega ))$ and \begin{align*} \int_{\Omega \times ]0,T[} [v_t\psi +\frac 1\gamma (v+v^2)_y\psi +\frac 1{\gamma ^4} v_{yyt}\psi _{yy}&\\ -\frac{(\alpha '+\gamma 'y)}\gamma v_y\psi -\frac{4\gamma '}{\gamma ^{5}} v_{yy}\psi _{yy} +(\frac{(\alpha '+\gamma '\,y)\psi }{\gamma ^5})_y v_{yyyy}\,]\,dy\,dt&=0\,, \end{align*} for all $\psi \in L^2(0,T;H_0^2(\Omega ))$; $v(y,0)=v^0(y)$, for all $y\in \Omega $. \end{lemma} \begin{lemma} \label{lm2.6} For each $v^0\in H_0^1(\Omega )\cap H^2(\Omega )$, there exists a unique function $v:\Omega \times ]0,T[\to \mathbb{R}$, satisfying $v\in L^\infty (0,T;H_0^1(\Omega )\cap H^2(\Omega ))$, $v_t\in L^\infty(0,T;H_0^1(\Omega ))$ \\ and \begin{align*} \int_{\Omega \times ]0,T[}[v_t\psi +\frac 1\gamma (v+v^2)_y\psi -\frac 1{\gamma ^2}v_{yyt}\psi &\\ -\frac{(\alpha '+\gamma'y)}\gamma v_y\psi +\frac{2\gamma '}{\gamma ^3}v_{yy}\psi -(\frac{(\alpha '+\gamma 'y)\psi}{\gamma ^3})_y v_{yy}\, ]dy\,dt&=0, \end{align*} for all $\psi \in L^2(0,T;H_0^1(\Omega))$; $v(y,0)=v^0(y)$, for all $y\in \Omega $. \end{lemma} In this article, we prove Theorem \ref{thm2.1} and Lemmas \ref{lm2.3}, \ref{lm2.4}, \ref{lm2.5} which correspond to Rosenau Equation. However, we omit the proofs of Theorem \ref{thm2.2} and Lemma \ref{lm2.6} which correspond to Benjamin Bona-Mahony Equation; because the proofs are made in a similar way. \begin{proof}[Proof of Lemma \ref{lm2.3}] Let $(w_i)_{i\in \mathbb{N}}$ be the special basis of $H_0^2(\Omega )$, such that \begin{gather*} w_{i,yyyy} =\lambda _iw_i,\quad \mbox{in }\Omega \\ w_i(0) = w_i(1)=w_{i,y}(0)=w_{i,y}(1)=0,\quad i\in \mathbb{N}. \end{gather*} We denote by $V_m$ the subspace generated by $w_1,\dots ,w_m$. Our starting point is to construct the Galerkin approximation of the solution $v_{m}\in V_m$, which is given by the solution of the approximate equation \begin{equation} \label{V} \begin{gathered} (v_{m,t},w)+(\frac 1\gamma (v_m+v_m^2)_y,w)+\frac 1{\gamma ^4}(v_{m,yyyyt},w)- \frac{4\gamma '}{\gamma ^5}(v_{m,yyyy},w) \\ +(-\frac{(\alpha '+\gamma 'y)}{\gamma ^5}v_{m,yyyyy},w)+(-% \frac{(\alpha '+\gamma 'y)}{\gamma}v_{m,y},w)=0\quad \mbox{for all }w\in V_m \\ v_m(0)=v_m^0\to v^0\quad \mbox{in }H^4(\Omega ) \end{gathered} \end{equation} \textit{First Estimate}. Taking $w=v_m(t)$ in $($V$)_{1,}$ we have: \begin{equation} \begin{aligned} \frac d{dt}(\|v_m(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yy}(t)\|^2) +\frac{\gamma'}{\gamma ^5}\|v_{m,yy}(t)\|^2 &\\ +\frac{\gamma '}{\gamma}\|v_m(t)\|^2 +\frac{\alpha '}{\gamma ^5}v_{m,yy}^2(0)-\frac{\beta '}{\gamma ^5}v_{m,yy}^2(1) &=0 \end{aligned} \label{e1} \end{equation} Integrating this equation over $[0,t]$ and using hypothesis (H1), we obtain \begin{equation} \begin{aligned} &\|v_m(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yy}(t)\|^2\\ &\leq \|v^0\|^2+\frac1{\gamma ^4(0)}\|v_{yy}^0\|^2 +\frac{\gamma _2}{\gamma _0}\int_0^t\frac 1{\gamma ^4}\|v_{m,yy}(s)\|^2ds +\frac{\gamma _2}{\gamma _0^5}\int_0^t\|v_m(s)\|^2ds\, \end{aligned} \label{e2} \end{equation} where $\gamma _0$, $\gamma _{1}$, $\gamma _2$ are positive constants, such that $\gamma _0\leq \gamma (t)\leq \gamma _{1}$, and $$ \gamma_2=\max_{0\leq t\leq T}|\gamma '(t)|,\quad \gamma_1=\max_{0\leq t\leq T}|\alpha '(t)|. $$ This implies \begin{equation} \|v_m(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yy}(t)\|^2 \leq C_0+C_1\int_0^t[\|v_m(s)\|^2+\frac 1{\gamma ^4}\|v_{m,yy}(s)\|^2]\,ds \label{e3} \end{equation} where $C_0$, $C_{1},\dots$ denote positive constants. Applying Gronwall inequality, we have the first estimate \begin{equation} \|v_m(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yy}(t)\|^2 \leq C_{2} \label{e4} \end{equation} \emph{Second Estimate} Taking $w=v_{m,yyyy}(t)$ in the first equation of \eqref{V}, we obtain \begin{equation} \begin{aligned} &\frac 12\frac d{dt}[\|v_{m,yy}(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yyyy}(t)\|^2]\\ &\leq \frac{3\gamma _2}{2\gamma _0}\frac 1{\gamma ^4}\|v_{m,yyyy}(t)\|^2 +\frac 1\gamma \|v_{m,y}(t)\| \|v_{m,yyyy}(t)\| \\ &\quad +\frac{(\alpha _1+\gamma _2)}{\gamma _0}\|v_{m,y}(t)\|^2\|v_{m,yyyy}(t)\|. \end{aligned} \label{e5} \end{equation} From \eqref{e4} and using Schwartz's inequality and Poincare's inequality, we obtain \begin{align*} &\frac 12\frac d{dt}[\|v_{m,yy}(t)\|^2+\frac 1{\gamma^4}\|v_{m,yyyy}(t)\|^2]\\ &\leq \frac{3\gamma _2}{2\gamma _0}\frac 1{\gamma^4}\|v_{m,yyyy}(t)\|^2 +\frac 1{2\gamma ^4}\|v_{m,yyyy}(t)\|^2+\frac{\gamma _1^2}2C_3\|v_{m,yy}(t)\|^2\\ &\quad +C_3\|v_{m,yy}(t)\|^2\|v_{m,yyyy}(t)\| \\ &\leq \frac 12(1+\frac{3\gamma _2}{\gamma _0})\frac 1{\gamma^4}\|v_{m,yyyy}(t)\|^2 +\frac{\gamma _1^2}2C_3C_2 +\frac{(\alpha_1+\gamma _2)}{\gamma _0}C_3C_2 \|v_{m,yyyy}(t)\| \\ &\leq \frac 12(1+\frac{3\gamma _2}{\gamma _0})\frac 1{\gamma^4}\|v_{m,yyyy}(t)\|^2 +\frac{\gamma _1^2}2C_3C_2+\frac{\gamma _1^2}2C_4^2 +\frac 1{2\gamma ^4}\|v_{m,yyyy}(t)\|^2\,. \end{align*} Let $c_4=\frac{(\alpha_1+\gamma _2)}{\gamma _0}C_3C_2$ and let $c_5=\frac{\gamma _1^2}2C_3C_2+\frac{\gamma _1^2}2C_4^2$. Then \begin{equation} \frac d{dt}[\|v_{m,yy}(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yyyy}(t)\|^2] \leq C_5+(2+\frac{3\gamma _2}{\gamma _0})\frac 1{\gamma^4}\|v_{m,yyyy}(t)\|^2. \label{e6} \end{equation} Integrating this inequality over $[0,t]$ and applying Gronwall inequality, we obtain \begin{equation} \|v_{m,yy}(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yyyy}(t)\|^2\leq C_{6} \label{e7} \end{equation} \emph{Third Estimate} Taking $w=v_{m,t}(t)$ in (V)$_1$, we have \begin{equation} \begin{aligned} &\|v_{m,t}(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yyt}(t)\|^2\\ &=-\frac 1\gamma((v_m(t)+v_m^2(t))_y, v_{m,t}(t)) +\frac{4\gamma '}{\gamma ^5}(v_{m,yyyy}(t),v_{m,t}(t))\\ &\quad -(\frac{(\alpha '+\gamma 'y)}{\gamma ^5}v_{m,yyyy}(t),v_{m,yt}(t)) +(\frac{(\alpha '+\gamma 'y)}{\gamma ^5}v_{m,y}(t),v_{m,t}(t)). \end{aligned}\label{e8} \end{equation} >From \eqref{e4}, \eqref{e7}, and \eqref{e8}, we obtain \begin{equation} \|v_{m,t}(t)\|^2+\frac 1{\gamma ^4}\|v_{m,yyt}(t)\|^2\leq C_{7}. \label{e9} \end{equation} These three estimates permit to pass to the limit in the approximate equation and we obtain a weak solution $v$ in the sense of Lemma \ref{lm2.3}. The uniqueness of solution and the verification of initial data are showed by the standard arguments. \end{proof} \begin{proof}[Proof of Lemma \ref{lm2.4}] To prove the existence we consider two stages: First stage $f\in C([0,T];H_0^2(\Omega ))$. Let $(w_i)_{i\in \mathbb{N}}$ be the special basis of $H_0^2(\Omega )$ used in the proof of Lemma \ref{lm2.3}. Consider the sequence $(f_n)$, such that $f_n(t)=\sum_{i=1}^n(f(t),w_i)w_i$. It is clear that $f_n\to f$ strongly in $C([0,T];H_0^2(\Omega ))$. The approximated solution $z_m(t)$ to $z+\frac 1{\gamma ^4}\Delta ^2z=f$ is $z_m(t)=\sum_{i=1}^mg_{im}(t)w_i$, where $g_{im}$ are solutions of the approximated system \begin{equation} (z_m(t),w_i)+\frac 1{\gamma ^4}(\Delta z_m(t),\Delta w_i) =(f_m(t),w_i)\,,\quad i=1,\dots m \label{e10} \end{equation} \textit{A priori estimate}. Let us prove that $(z_m)$ is a Cauchy sequence in $C([0,T];H_0^2(\Omega ))$. In fact, let $m$ and $n$ be positive integer such that $m>n$ and $g_{in}(t)=0$ for $n\leq i\leq m$. Then $z_m$ and $z_n$ are solutions of \eqref{e10} in $V_{m}$. Consider the Cauchy difference $z_m-z_n$. We have from \eqref{e10} that, for $i=1,\dots , m$, \begin{equation} (z_m(t)-z_n(t),w_i)+\frac 1{\gamma ^4}(\Delta (z_m(t)-z_n(t)),\Delta w_i)=(f_m(t)-f_n(t),w_i)\,, \label{e11} \end{equation} Taking $w_i=z_m(t)-z_n(t)$ in \eqref{e11}, using the Cauchy-Schwarz inequality and the equivalent norms, we obtain $$ |z_m-z_n|_{C([0,T];H_0^2(\Omega ))}\leq c|f_m-f_n|_{C([0,T];H_0^2(\Omega))} $$ Then $z_n\to z$ strongly in $C([0,T];H_0^2(\Omega ))$. Therefore, taking limit in \eqref{e10}, we obtain $z+\frac 1{\gamma ^4}\Delta ^2 z=f$ in $C([0,T];H^{-2}(\Omega ))$. \noindent Second stage $f\in C([0,T];H^{-2}(\Omega ))$. By density, there exists a sequence $(f_n)$, $f_n\in C([0,T];H_0^2(\Omega ))$, such that $f_n\to f$ strongly in $C([0,T];H^{-2}(\Omega ))$. Using the first stage we have that there exist a sequence $(z_n)$, $z_n\in C([0,T];H_0^2(\Omega ))$ such that \begin{equation} z_n+\frac 1{\gamma ^4}\Delta ^2z_n=f_n\quad \text{in } C([0,T];H^{-2}(\Omega)). \label{e12} \end{equation} Consider the Cauchy difference $z_m-z_n$, $m>n$. We obtain \begin{equation} z_{m-}z_n+\frac 1{\gamma ^4}\Delta ^2(z_m-z_n)=f_m-f_n\quad \text{in } C([0,T];H^{-2}(\Omega )).\label{e13} \end{equation} Composing \eqref{e13} with $z_m - z_n \in C([0,T];H_0^{2}(\Omega ))$. and integrating in $\Omega $, we have $$ |z_m-z_n|_{C([0,T];H_0^2(\Omega ))}\leq c|f_m-f_n|_{C([0,T];H^{-2}(\Omega ))}; $$ therefore, $z_n\to z$ strongly in $C([0,T];H_0^2(\Omega ))$ and taking limit in \eqref{e12} we have $z+\frac 1{\gamma ^4}\Delta ^2z=f$ in $C([0,T];H^{-2}(\Omega ))$. The uniqueness of the solutions is showed by the standard arguments. \end{proof} \begin{proof}[Proof of Lemma \ref{lm2.5}] From Lemma \ref{lm2.4} we can define the operator $\mathcal{B}(t)=(I+\frac 1{\gamma ^4}\Delta ^2)^{-1}$ from $C([0,T];H^{-2}(\Omega))$ to $C([0,T];H_0^2(\Omega ))$ by $\mathcal{B}(t) f=z$ with $f\in \break C([0,T];H^{-2}(\Omega ))$ where $z$ is a solution of $z+\frac1{\gamma ^4}\Delta ^2z=f$. Note that $\mathcal{B}(t)$ is linear and continuous. By Lemma 2.3, $v\in L^2(0,T;H^4(\Omega )\cap H_0^2(\Omega ))$ and $v_t\in L^2(0,T;H_0^2(\Omega ))$. From Lions-Magenes, Theorems 3.1 and 9.6, chapter I \cite{l2}, we conclude that \begin{equation} v\in C([0,T];H_0^2(\Omega )\cap H^3(\Omega ))\label{e14} \end{equation} On the other hand, from the transformed problem \eqref{III}, we obtain \begin{equation} (I+\frac 1{\gamma ^4}\Delta ^2) v_t=f\,, \label{e15} \end{equation} where, $$ f=-\frac 1\gamma (v+v^2)_y\,+\frac{(\alpha '+\gamma' y)} \gamma v_y+\frac{4\gamma '}{\gamma ^{5}} v_{yyyy} +(\frac{(\alpha '+\gamma 'y)}{\gamma ^5}) v_{yyyyy} $$ From \eqref{e14}, we conclude that $f\in C([0,T];H^{-2}(\Omega ))$. Then from \eqref{e15} we have that $v_t=\mathcal{B}(t)f$, where $v_t\in C([0,T];H_0^2(\Omega ))$ and we get the required result. \end{proof} The proof of Theorem \ref{thm2.1} follows immediately from Lemma \ref{lm2.5} and the Change of Variable Theorem. Therefore, we omit it. Observe that Theorem \ref{thm2.1} in a cylindrical domain, has the regularity $u\in C^{1}([0,T];H_0^2(\Omega ))\cap C([0,T];H^4(\Omega )\cap H_0^2(\Omega ))$. In fact, as we consider an additional estimate with $w_i=u_{m,txxxx}$, in the Galerkin approximation, that allows us to obtain the regularity $u_t\in L^2(0,T;H^4(\Omega )\cap H_0^2(\Omega ))$. However, in our noncylindrical domain, this is not possible since the transformed problem (III), contains a term $v_{yyyyy}$ that does not allow us to use the estimate with $w_i=v_{m,tyyyy}$ in the Galerkin approximation. \section{Solutions on increasing domains} In this section we study the existence of solution for the systems \eqref{I} and \eqref{II} satisfying the hypothesis (H2). We use the Penalization Method given by Lions \cite{l2}. Let $Q=]a,b[\times ]0,T[$ be the cylinder such that $\widehat{Q} \subset Q$. We define the function $M:Q\to \mathbb{R}$, by $$ M(x,t)=\begin{cases} 1 &\mbox{in }Q\setminus \widehat{Q} \\ 0 &\mbox{in }\widehat{Q} \end{cases} $$ To show the existence result we will use the following Lemma. \begin{lemma} \label{lm3.1} If $u,u_t\in L^2(0,T;L^2(a,b))$, then \[ \int_0^t (M u(s),u_t(s))ds\geq \frac12\|M(t) u(t)\|_{L^2(a,b)}^2 -\frac 12\|M(0) u(0)\|_{L^2(a,b)}^2\,. \] \end{lemma} \begin{proof} We have \begin{align*} \int_0^t(Mu(s),u_t(s))ds &=\frac 12\int_0^t\int_a^b\,M(u^2(s))_t\,d\xi\,ds\\ &=\frac 12\int_{[0,t]\times [a,b]} M(u^2(s))_t\,d\xi \,ds. \end{align*} From Fubini's Theorem and recalling the definition of $M$, it follows that \begin{align*} &\int_0^t(Mu(s),u_t(s))ds\\ &=\frac 12\int_a^{\alpha(t)}\int_0^t [u^2(s)]_t\,ds\,d\xi +\frac 12\int_{\alpha(t)}^{\alpha (0)}\int_0^{\alpha ^{-1}(x)} [u^2(s)]_t\,ds\,d\xi \\ &\quad +\frac 12\int_{\beta (0)}^{\beta (t)}\int_0^{\beta ^{-1}(x)} [u^2(s)]_t\,ds\,d\xi +\frac 12\int_{\beta (t)}^b\int_0^t [u^2(s)]_t\,ds\,d\xi \\ &=\frac 12\int_a^{\alpha (t)} [u^2(t,\xi )-u^2(0,\xi )]\,d\xi +\frac 12\int_{\alpha (t)}^{\alpha (0)}[u^2(\alpha ^{-1}(x),0)-u^2(0,\xi)]\,d\xi \\ &\quad +\frac 12\int_{\beta (0)}^{\beta (t)} [u^2(\beta^{-1}(x),0)-u^2(0,\xi ) ]\,d\xi +\frac 12\int_{\beta(t)}^b [u^2(t,\xi )-u^2(0,\xi )]\,d\xi \\ &\geq \frac 12[\int_a^{\alpha (t)} u^2(t,\xi )\,d\xi +\int_{\beta (t)}^b u^2(t,\xi )\,d\xi -(\int_a^{\alpha (t)}u^2(0,\xi )\,d\xi \\ &\quad +\int_{\alpha (t)}^{\alpha (0)}u^2(0,\xi )\,d\xi +\int_{\beta (0)}^{\beta (t)} u^2(0,\xi )\,d\xi +\int_{\beta (t)}^b u^2(0,\xi)\,d\xi)] \\ &=\frac 12\int_a^bM(t,\xi )\, u^2(t,\xi )\,d\xi -\frac 12\int_a^bM(0,\xi ) u^2(0,\xi )\,d\xi \\ &=\frac 12\|M(t) u(t)\|_{L^2(a,b)}^2-\frac 12\|M(0) u(0)\|_{L^2(a,b)}^2 \end{align*} which completes the proof. \end{proof} The existence of solution for \eqref{I} and \eqref{II}, satisfying the hypothesis (H2), is established in the next theorems. \begin{theorem} \label{thm3.2} For each $u^0\in H_0^2(\Omega _0)$, there exists a function $u:\widehat{Q}\to \mathbb{R}$, satisfying $u\in L^\infty (0,T;H_0^2(\Omega _t))$, $u_t\in L^\infty(0,T;H_0^2(\Omega _t))$ and \begin{equation} \int_{\widehat{Q}}u_t\phi \,dx\,dt +\int_{\widehat{Q}}(u+u^2)_x \phi \,dx\,dt +\int_{\widehat{Q}}u_{xxt}\phi _{xx}\,dx\,dt=0 \label{e16} \end{equation} for all $\phi \in L^2(0,T;H_0^2(\Omega _t))$; $u(x,0)=u^0(x)$ \end{theorem} \begin{theorem}\label{thm3.3} For each $u^0\in H_0^1(\Omega_0)$, there exists a function $u:\widehat{Q}\to \mathbb{R}$, satisfying $u\in L^\infty (0,T;H_0^1(\Omega _t))$, $u_t\in L^\infty(0,T;H_0^1(\Omega _t))$ and \[ \int_{\widehat{Q}}u_t\phi \,dx\,dt +\int_{\widehat{Q}}(u+u^2)_x\phi \,dx\,dt +\int_{\widehat{Q}}u_{xt}\phi _x\,dx\,dt=0, \] for all $\phi \in L^2(0,T;H_0^1(\Omega _t))$; $u(x,0)=u^0(x)$ \end{theorem} \begin{proof}[Proof of Theorem \ref{thm3.2}] To prove this result we use the penalization method. For each $\epsilon >0$ we consider the problem \begin{equation} \label{VI} \begin{gathered} u_{\epsilon , t}+(u_\epsilon +u_\epsilon ^2)_x+u_{\epsilon , xxxxt}+\frac 1\epsilon Mu_{\epsilon , t}-\frac 1\epsilon (Mu_{\epsilon , xt})_x=0 \quad \mbox{in }Q \\ u_\epsilon (a,t)=u_\epsilon (b,t)=u_{\epsilon , x}(a,t)=u_{\epsilon , x}(b,t)=0 \quad \mbox{in }]0,T[ \\ u_\epsilon (x,0)=\widetilde{u}^0(x) \quad \mbox{in }]a,b[ \end{gathered} \end{equation} Let $\{w_i\}_{i\in N}$ be a basis of $H_0^2(a,b)$, such that $w_1=\widetilde{u}_0$. We denote by $V_m=[w_1,\dots ,w_m]$ the subspace of $H_0^2(a,b)$, generated by $\widetilde{u}_0,w_2,\dots ,w_m$. We seek $u_{\epsilon m}(t)$ in $V_m$ solution to the approximate problem \begin{equation} \label{VII} \begin{gathered} (u_{\epsilon m,t},w)+((u_{\epsilon m}+u_{\epsilon m} ^2)_x,w)+(u_{\epsilon m,xxxxt},w) \\ +\frac 1\epsilon (Mu_{\epsilon m,t},w)-\frac 1\epsilon ((Mu_{\epsilon m,xt})_x,w)=0 \quad \mbox{for all }w\in V_m \\ u_{\epsilon m}(0)=u^0(x)\to \widetilde{u}^0 \quad \mbox{in } H_0^2(a,b) \end{gathered} \end{equation} \textit{First Estimate.} Taking $w=u_{\epsilon m}$ in \eqref{VII} and applying Lemma \ref{lm3.1}, we obtain \begin{equation} \||u_{\epsilon m}(t)\||^2+\||u_{\epsilon m,xx}(t)\||^2 +\frac 1\epsilon \||M(t) u_{\epsilon m}(t)\||^2 +\frac 1\epsilon\||M(t) u_{\epsilon m,x}(t)\||^2\leq c_8, \label{e17} \end{equation} where $\||\cdot \||$ denotes the norm in $L^2(a,b)$. \smallskip \noindent\textit{Second Estimate.} Taking $w=u_{\epsilon m,t}(t)$ in \eqref{VII} and using \eqref{e17} we have \begin{equation} \begin{aligned} &\||u_{\epsilon m,t}(t)\||^2+\||u_{\epsilon m,xxt}(t)\||^2+\frac 1\epsilon (M(t) u_{\epsilon m,t}(t),u_{\epsilon m,t}(t))\\ &\frac 1\epsilon (M(t) u_{\epsilon m,xt}(t),u_{\epsilon m,xt}(t))\\ &\leq c_{9}+\frac 12\||u_{\epsilon m,t}(t)\||^2 \end{aligned} \label{e18} \end{equation} From where we obtain \[ \||u_{\epsilon m,t}(s)\||^2+\||u_{\epsilon m,xxt}(s)\||^2 +\frac1{\epsilon }\||M(t) u_{\epsilon m,t}(t)\||^2 +\frac 1{\epsilon}\||M(t) u_{\epsilon m,xt}(t)\||^2 \leq c_9 %\label{e19} \] From the estimates above, we pass to the limit in the approximate equation, and we obtain that $u_\epsilon $ is solution of the penalized problem \begin{equation} \begin{aligned} \int_0^T\int_a^bu_{\epsilon , t} v\,dx\,dt +\int_0^T\int_a^b(u_\epsilon +u_\epsilon^2)_x v\,dx\,dt +\int_0^T\int_a^bu_{\epsilon ,xxt} v_{xx}\,dx\,dt&\\ +\frac 1\epsilon \int_0^T\int_a^bMu_{\epsilon , t} v\,dx\,dt +\frac 1\epsilon \int_0^T\int_a^bMu_{\epsilon , xt} v_x\,dx\,dt&=0 \end{aligned} \label{e20} \end{equation} for all $v\in L^2(0,T;H_0^2(a,b))$. From \eqref{e17}, \eqref{e18} and the Banach-Steinhauss Theorem, we pass to the limit as $\epsilon \to 0$ in \eqref{e20} and we obtain \eqref{e16}. \noindent \textit{Regularity.} From the first estimate, we have \[ \frac 1\epsilon \int_0^t(M u_{\epsilon m , t}(s),u_{\epsilon m}(s))\,ds \leq c \] On the other hand, from Lemma \ref{lm3.1} we obtain \[ \frac 1\epsilon \int_0^t(M u_{\epsilon m , t}(s),u_{\epsilon m}(s))\,ds\geq \frac 1{2\epsilon }\||M(t) u_{\epsilon m}(t)\||^2. \] Then $\||M(t) u_{\epsilon m}(t)\||^2\leq 2c\epsilon $. Thus $\int_0^T\int_a^bM(t)\, u_{\epsilon m}^2(t)\,dx\,dt\leq 2c\epsilon T$ or \[ \int_0^T\int_a^b|M(t) u_\epsilon (t)|^2\,dx\,dt\leq \liminf \int_0^T\int_a^b|M(t) u_\epsilon (t)|^2\,dx\,dt\leq 2c\epsilon T \] Then $Mu_\epsilon \to 0$ in $L^2(0,T;L^2(a,b))$. On the other hand, $Mu_\epsilon \to M u$ in $L^2(0,T;L^2(a,b))$ and $Mu_{\epsilon m}\to M u_\epsilon$ in $L^2(0,T;L^2(a,b))$. So, we conclude that $Mu=0$ a.e. in $Q$ or $u=0$ in $Q\setminus \widehat{Q}$. Analogously, applying the Lemma \ref{lm3.1} to $u_{\epsilon m , xt}$ instead of $u_{\epsilon m ,t}$, we obtain: $u_{x}=0$ in $Q\setminus\widehat{Q}$. Since $u\in L^\infty (0,T;H_0^1(a,b))$ and $u_t\in L^\infty(0,T;H_0^1(a,b))$, then $u\in C([0,T];H_0^1(a,b))$. Therefore, $u(t)\in H_0^1(a,b)$ for all $t$ and $u=0$ in $]a,b[\setminus ]\alpha (t),\beta (t)[$. From where $u(t)\in H_0^1(\alpha (t),\beta (t))$, for all $t$. Thus $u\in L^\infty(0,T;H_0^1(\Omega _t))$. %\eqref{e21} Analogously, $u_x\in L^\infty(0,T;H_0^1(\Omega _t))$. %\eqref{e22} From these two statements, we have that $u\in L^\infty(0,T;H_0^2(\Omega _t))$. From the second estimate, \[ \int_0^T\int_a^b|M(t) u_{\epsilon m ,t}(t)|^2\,dx\,dt+\int_0^T\int_a^b|M(t) u_{\epsilon m , xt}(t)|^2\,dx\,dt\leq 2c\epsilon T. \] By similar arguments, we obtain that $u_t\in L^\infty(0,T;H_0^2(\Omega _t))$, which prove the regularity of the solution. \end{proof} The proof of Theorem \ref{thm3.3} is similar to the proof of Theorem \ref{thm3.2} and is ommitted. \begin{remark} \label{rmk3.4} \rm Theorems 2.1, 2.2, 3.2 and 3.3 are invariable by translation. In fact, the particular problem \begin{gather*} u_t+(u+u^2)_x+u_{xxxxt}=0\quad \mbox{in }\widehat{Q}\subset \Omega \times ]T_0,T_1[ \\ u(x,t)=0\quad \mbox{in }\widehat{\Sigma} \\ u_x(x,t)=0\quad \mbox{in }\widehat{\Sigma} \\ u(x,T_0)=u^0(x)\quad \mbox{in }\Omega _{T_0}, \end{gather*} with the change of variable $u(x,t)=\overline{u}(x,t-T_0)$, can be transformed into a problem of type \eqref{I}. \end{remark} \section{Solutions on mixed domains} Here we analyze the case when $\widehat{Q}$ is a mixed domain; i.e., $\widehat{Q}=B_1\cup B_2$ where $B_1=\{(x,t)\in \widehat{Q}: 0