\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 35, pp. 1--11.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/35\hfil Second initial boundary value problem] {Existence and smoothness of solutions to second initial boundary value problems for Schr\"odinger systems in cylinders with non-smooth bases} \author[N. M. Hung, N. T. K. Son \hfil EJDE-2008/35\hfilneg] {Nguyen Manh Hung, Nguyen Thi Kim Son} % in alphabetical order \address{Nguyen Manh Hung \newline Department of Mathematics, Hanoi University of Education, Hanoi, Vietnam} \email{hungnmmath@hnue.edu.vn} \address{Nguyen Thi Kim Son \newline Department of Mathematics, Hanoi University of Education, Hanoi, Vietnam} \email{mt02\_02@yahoo.com} \thanks{Submitted December 5, 2007. Published March 12, 2008.} \subjclass[2000]{35D05, 35D10, 35G99} \keywords{Second initial boundary value problem; Schr\"odinger systems; \hfill\break\indent generalized solution; existence; uniqueness; smoothness} \begin{abstract} In this paper, we consider the second initial boundary value problem for strongly general Schr\"odinger systems in both the finite and the infinite cylinders $Q_T, 00$. As we have known, in the first problem, the qualitative properties of solution were indicated by basing on the properties of functions $u\in \overset{\;o}{H}\,_\gamma^{m,0}(Q_T)$, which let us to the Garding inequality (see \cite{h1,h2,h3,l1}). But in the second problem, when the solution space is $H\,_\gamma^{m,0}(Q_T)$ and the second boundary condition is hidden in the integral equality in the definition of generalized solution, the Garding inequality is not valid, so it becomes more complicated to establish the unique solvability of the problem. This difficulty is solved in this paper in section 2, Lemma 2.1. Then based on it, we receive our results on the existence and uniqueness of generalized solution in section 3 and the smoothness with respect to time variable of solutions in the last section. Moreover, the problem becomes more complicated in technics when we consider with non homogeneously initial condition $u(x,0)=\varphi(x)$ in section 3, and the results that we received are more general than those in \cite{h1,h2,h3,l1}, in which the authors just considered the problem with homogeneously initial condition $u(x,0)=0$. \section{Preliminaries}\label{sec 2} Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^n$, and $\overline {\Omega}$, $\partial\Omega$ denote the closure and the boundary of $\Omega$ in $\mathbb{R}^n$. We suppose that $\Gamma=\partial\Omega\backslash\{0\}$ is a smooth manifold and $\Omega$ coincides with the cone $K=\{x: \frac{x}{|x|}\in G\}$ in a neighborhood of the origin point $0$, where $G$ is a smooth domain on the unit sphere $S^{n-1}$ in $\mathbb{R}^n$. We begin by introducing some notations and functional spaces which are used fluently in the rest. Denote $Q_T=\Omega\times (0,T),\ S_T=\Gamma\times (0,T)$, for some $00)$ as the space of all measurable complex functions $u(x,t)$ that have generalized derivatives up to order $m$ with respect to $x$ with the norm $$\|u\|_{H^{m,0}_\gamma(Q_T)}=(\sum_{|\alpha|\le m} \int_{Q_T}|D^\alpha u|^2e^{-2\gamma t}\,dx\,dt)^\frac1{2}.$$ The space $L^\infty (0, T; L_2(\Omega))$ consists of all measurable functions $u :(0, T) \to L_2(\Omega)$, $t\mapsto u (t)$ with the norm $\| u\|_\infty= \mathop{\rm ess\,sup}_{00$ and $\lambda_0$ such that the inequality $(-1)^m B[u,u](t) \ge\mu_0\|u\|^2_{H^m(\Omega)} -\lambda_0\|u\|^2_{L_2(\Omega)}$ is valid for all $u\in H^{m,0}_\gamma(Q_T)$, $\gamma>0$ and almost $t\in (0,T)$. \end{lemma} \begin{proof} It follows from \eqref{e2.2} that $\sum_{|p|=|q|=m}\int_{\Omega}a_{pq}(x,t)\,D^qu \overline{D^pu}dx \ge C_0\sum_{|p|=m}\|D^p u\|^2_{L_2(\Omega)}$ for all $u(x,t)\in H^{m,0}_\gamma(Q_T)$ and almost $t \in (0,T)$, where $C_0$ is a positive number, independent of $u$. Since $a_{pq}$ are bounded, using Cauchy's inequality one has \begin{align*} &C_0\sum_{|p|=m}\|D^p u\|^2_{L_2(\Omega)}\\ &\le \sum_{|p|=|q|=m}\int_{\Omega}a_{pq}\,D^qu \overline{D^pu}dx\\ &=(-1)^mB[u,u](t) -(-1)^m\sum_{|p|+|q|<2m ,\, |p|,|q|\le m } (-1)^{|p|}\int_{\Omega}a_{pq}\,D^qu \overline{D^pu}dx\\ &\le (-1)^mB[u,u](t) +C(\varepsilon)\|u\|^2_{H^{m-1}(\Omega)} + \varepsilon\sum_{|p|=m}\|D^p u\|^2_{L_2(\Omega)}, \end{align*} where $0<\varepsilon0$. This implies $$\sum_{|p|=m}\|D^p u\|^2_{L_2(\Omega)}\le C_1(-1)^mB[u,u](t) + C_2\|u\|^2_{H^{m-1}(\Omega)},\label{e2.6}$$ where $C_1=\frac{1}{C_0-\varepsilon}$, $C_2=\frac{C(\varepsilon)}{C_0-\varepsilon}>0$. Following \cite[Theorem 4.15]{a1}, we have for all $\varepsilon>0$, there exists a constant $C_3(\varepsilon)$ such that the inequality $$\sum_{|p|=k}\|D^p u\|^2_{L_2(\Omega)} \le \varepsilon\sum_{|p|=m}\|D^p u\|^2_{L_2(\Omega)} +C_3(\varepsilon)\|u\|^2_{L_2(\Omega)}\label{e2.7}$$ holds for all $k=1,2,\dots,m-1$, and for all $u\in H^m(\Omega)$. Note that for all $00$, then for almost fixed point $t_1\in (0,T)$ we have $u(x,t_1)\in H^m(\Omega)$ and \eqref{e2.7} is valid for $u(x,t_1)$. Because $\varepsilon, C_3(\varepsilon)$ are independent of $t_1\in (0,T)$, so one gets $$\sum_{|p|=k}\|D^p u(x,t)\|^2_{L_2(\Omega)} \le \varepsilon\sum_{|p|=m}\|D^p u(x,t)\|^2_{L_2(\Omega)} +C(\varepsilon)\|u(x,t)\|^2_{L_2(\Omega)}\label{e2.8}$$ for all $k=1,2,\dots,m-1$, for all $u\in H^{m,0}_\gamma(Q_T)$ and almost $t\in (0,T)$. This follows that for all $0<\varepsilon<1$, there exists $C_4=C_4(\varepsilon)$ such that the following inequality holds $$\|u\|^2_{H^{m-1}(\Omega)}\le\varepsilon \|u\|^2_{H^m(\Omega)} +C_4\|u\|_{L_2(\Omega)}\label{e2.9}$$ for all $u\in H^{m,0}_\gamma(Q_T)$, almost $t\in (0,T)$. Hence, from \eqref{e2.6} and \eqref{e2.9} we have \begin{align*} \|u\|^2_{H^{m}(\Omega)} &\le C_1(-1)^mB[u,u](t) + (C_2+1)\|u\|^2_{H^{m-1}(\Omega)}\\ &\le C_1(-1)^mB[u,u](t)+ (C_2+1)[\varepsilon\|u\|^2_{H^m(\Omega)} +C_4\|u\|^2_{L_2(\Omega)}] \end{align*} for all $0<\varepsilon<\min\{1,C_0,\frac{1}{C_2+1}\}$. So we obtain $(-1)^mB[u,u](t)\ge\mu_0\|u\|^2_{H^m(\Omega)} -\lambda_0\|u\|^2_{L_2(\Omega)},$ where $\mu_0=[1-(C_2+1)\varepsilon](C_0-\varepsilon)>0$, $\lambda_0=C_4(C_2+1)(C_0-\varepsilon)$. This proves the lemma. \end{proof} From lemma 2.1, using the transformation $u(x,t)=e^{i\lambda_0 t}v(x,t)$ if necessary, we can assume that the operator $L(x,t,D)$ satisfies $$(-1)^mB[u,u](t)\geq \mu_0\|u\|^2_{H^m(\Omega)}\label{e2.10}$$ for all $u\in H^{m,0}_\gamma(Q_T)$, almost $t\in (0,T)$. \section{The uniqueness and existence theorems}\label{sec 3} In this section we investigate the unique solvability of the second initial boundary value problem for the system \eqref{e2.3} with non homogeneously initial condition \eqref{e2.4} in the space $H^{m,0}_{\gamma }(Q_T),\ \gamma>0$, where $00$ arbitrary. \end{theorem} \begin{proof} Suppose that the problem has two solutions $u_1,u_2$ in $H^{m,0}_{\gamma }(Q_T)$. Put $u=u_1-u_2$. For all $0<\tau 0).\label{e3.2} Putting$v_p(x,t)={\int_t^0}D^pu(x,s)ds$,$0\gamma_0=\frac{m^*\mu}{2\mu_0}$the second initial boundary value problem for \eqref{e2.3}--\eqref{e2.4} has a generalized solution$u(x,t)$in the space$H^{m,0}_{\gamma}(Q_T)$and the following estimate holds $$\|u\|^2_{H^{m,0}_{\gamma}(Q_T)}\le C\,\big[\|\varphi\|^2_{H^m(\Omega)} +\|f(.,0)\|^2_{L_2(\Omega)}+\|f\|^2_\infty+\|f_t\|^2_\infty\big],$$ where the constant$C$only depends on$\mu ,\mu_0$. \end{theorem} \begin{proof} Let$\{\varphi_k(x)\}_{k=1}^\infty$be a basis of$H^m(\Omega)$, which is orthonormal in$L_2(\Omega)$. We find an approximate solution$u^N(x,t)$in the form$u^N(x,t)=\sum_{k=1}^N\,C_k^N(t)\varphi_k(x)$, where$\{C_k^N(t)\}_{k=1}^N$satisfies \begin{gather} (-1)^{m-1}i \sum_{|p|,|q|=0}^m(-1)^{|p|} \int_\Omega a_{pq}D^qu^N\overline{D^p\varphi_l}dx- \int_\Omega u_t^N\overline{\varphi_l}dx =\int_\Omega f\overline{\varphi_l}dx,\label{e3.3} \\ C_l^N(0)=\int_\Omega\varphi(x)\varphi_l(x)dx,\quad l=1,\dots, N . \label{e3.4} \end{gather} From (i), (ii) and \eqref{e3.3}--\eqref{e3.4} it follows that coefficients$C^N_k(t)$are defined uniquely and$\|u^N(x,0)\|^2_{H^m(\Omega)}\le \|\varphi(x)\|^2_{H^m(\Omega)}$for all$N=1,2,\dots$. After multiplying \eqref{e3.3} by$\frac{d\overline{C_l^N(t)}}{dt}$, taking sum with respect to$l$from 1 to$N$, we get $$(-1)^m\sum_{|p|,|q|=0}^m(-1)^{|p|} \int_\Omega a_{pq}D^qu^N\overline{D^pu_t^N}dx- i\int_\Omega u_t^N\overline{u_t^N}dx =i\int_\Omega f\overline{u_t^N}dx. \label{e3.5}$$ Adding this equality and its complex conjugate, we have $$(-1)^m\ \sum_{|p|,|q|=0}^m(-1)^{|p|}\ \int_{\Omega}\ a_{pq}\frac{\partial}{\partial t} (D^qu^N\overline{D^pu^N})dx= -2\,\mathop{\rm Im}\,\int_{\Omega}f\overline{u_t^N}dx.$$ So for all$0<\tau0$. Therefore, $$\|u^N(x,\tau)\|_{H^m(\Omega)}^2 \le C_1\Big[\|\varphi\|^2_{H^m(\Omega)} +\|f(x,0)\|^2_{L_2(\Omega)}+\|f\|_\infty^2 +\|f_t\|_\infty^2\Big]e^{\frac{m^*\mu +\varepsilon}{\mu_0-\varepsilon}\tau}. \label{e3.6}$$ For each$\gamma>\gamma_0=\frac{\mu m^*}{2\mu_0} =\inf_{(0,\mu_0)} \frac{m^*\mu +\varepsilon}{2(\mu_0-\varepsilon)}$we can choose$\varepsilon\in(0,\mu_0)$such that$\gamma>\frac{m^*\mu+\varepsilon}{2(\mu_0-\varepsilon)}$; i.e.,$-2\gamma+ \frac{m^*\mu+\varepsilon}{(\mu_0-\varepsilon)}<0$. Multiplying \eqref{e3.6} with$e^{-2\gamma \tau}$, then integrating with respect to$\tau$from 0 to$T$, we obtain $$\|u^N\|^2_{H^{m,0}_{\gamma}(Q_T)} \le C\big[\|\varphi\|^2_{H^m(\Omega)}+\|f(.,0)\|^2_{L_2(\Omega)} +\|f\|_\infty^2+ \|f_t\|_\infty^2\big],\label{e3.7}$$ where$C>0$independent of$N$. Since the sequence$\{u^N\}$is uniformly bounded in$H^{m,0}_\gamma(Q_T)$, we can take a subsequence, denoted also by$\{u^N\}$for convenience, which converges weakly to a vector function$u(x,t)$in$H^{m,0}_\gamma(Q_T)$. We will prove that$u(x,t)$is a generalized solution of the problem. Since $M=\bigcup_{N=1}^\infty \{\sum_{l=1}^N d_l(t)\varphi_l(x), d_l(t) \in H^1(0,\tau),\ d_l(\tau)=0,\ \forall l=1,2,\dots ,N\}$ is dense in the space of test functions$\hat{H}^{m,1}(Q_\tau)=\{\eta (x,t) \in H^{m, 1} (Q_\tau), \eta (x, \tau) = 0\}$for all$0<\tau \gamma_0 = \frac{m^* \mu}{ 2 \mu_0}$the generalized solution$u(x, t)$of the second problem for \eqref{e2.3'}--\eqref{e2.4'} has the generalized derivatives with respect to$t$up to order$h$in the space$ H^{m,0}_{(2h+1)\gamma} (Q_T)$and the following estimate holds $$\| u_{t^h}\|^2_{H^{m, 0}_{(2h+1)\gamma} (Q_T)} \le C\sum_{k=0}^{h+1} \| f_{t^k} \|^2_\infty ,$$ where the constant$C$does not depend on$u$and$f$. \end{theorem} \begin{proof} Let$\{\varphi_k(x)\}_{k=1}^\infty$be a basis of$H^m(\Omega)$, which is orthonormal in$L_2(\Omega)$. For each natural number$N$, we set$u^N(x,t)=\sum_{k=1}^N\,C_k^N(t)\varphi_k(x)$, where$\{C_k^N(t)\}_{k=1}^N$is the solution of the ordinary differential system $$(-1)^{m-1}i \sum_{|p|,|q|=0}^m(-1)^{|p|} \int_\Omega a_{pq}D^qu^N\overline{D^p\varphi_l}dx- \int_\Omega u_t^N\overline{\varphi_l}dx =\int_\Omega f\overline{\varphi_l}dx,\label{e4.3}$$ with$ C_l^N(0)=0$,$l=1,\dots, N$. From (i), (ii), it follows that coefficients$C^N_k(t)$, defined uniquely by \eqref{e4.3}, have derivatives up to order$h+1$and$u^N(x,0)=0$. We will prove that $$D^p u^N_{t^k} (x, 0) = 0,\quad \forall 0 \le k \le h,\; 0 \le |p| \le m, \forall x\in\Omega. \label{e4.4}$$ Indeed, it is clear that \eqref{e4.4} holds for$k=0$. Differentiating \eqref{e4.3}$(k-1)$times with respect to$t$, multiplying by$ {\frac{d^k }{ dt^k}} \big(\overline {C_l^N (t)}\big)$, then taking sum with respect to$l$from$1$to$N, we obtain \begin{aligned} &-i \int_\Omega \big|u_{t^k}^N \big|^2 dx + (-1)^m \sum_{| p|, | q| = 0}^m (-1)^{| p|} \int_\Omega a_{pq} D^q u^N_{t^{k-1}} \overline{D^p u_{t^k}^N} dx\\ &= (-1)^{m-1}\sum_{| p|, | q| = 0}^m (-1)^{| p|} \sum_{s = 0}^{k-2} C_{k-1}^s \int_\Omega \frac{\partial^{k-s-1}a_{pq}}{ \partial t^{k-s-1}} D^q u_{t^s}^N \overline{D^p u_{t^k}^N} dx\\ &\quad + i \int_\Omega f_{t^{k-1}} \overline{u_{t^k}^N} dx. \end{aligned} \label{e4.5} By using (ii) and induction onk$, we obtain \eqref{e4.4} holds for all$0 \le k \le h$. In the following part, we shall prove the inequalities \begin{gather} \big\| u_{t^h}^N (x, \tau)\big\|^2_{H^m(\Omega)} \le Ce^{{\lambda_h}\tau} \sum_{k=0}^{h+1} \| f_{t^k}\|^2_\infty , \quad \forall 0<\tau0$, using Cauchy's inequality and \eqref{e2.10}, we have \begin{align*} &[\mu_0-(\mu m^*(2^h-1)+1)\varepsilon_1] \|u_{t^h}^N(x,\tau)\|^2_{H^m(\Omega)} \\ &\le[(2h+1)m^*\mu+((2^{h+1}-2-h)\mu m^*+1)\varepsilon_1] \int_0^\tau \big\| u^N_{t^h} (x, t) \big\|^2_{H^m (\Omega)} dt\text{}\vspace{-0.7cm} \\ & +C\Big[\sum_{k=0}^{h-1} \big\| u^N_{t^k} \big\|^2_{H^{m, 0} (Q_\tau)} + \sum_{k=0}^{h-1} \big\| u^N_{t^k}(x,\tau)\big\|^2_{H^m (\Omega)} + \|f_{t^h}\|^2_\infty+\tau \big\| f_{t^{h+1}}\big\|_\infty^2\Big], \end{align*} where $C=\max\{\frac{2\mu m^*M}{\varepsilon_1},\frac1{\varepsilon_1}\}$, $M=\max_{s=\overline{0,h-1}} C^s_h$. Set $\varepsilon=((2^{h+1}-2-h)\mu m^*+1)\varepsilon_1 \ge((2^{h}-1)\mu m^*+1)\varepsilon_1>0$ for $h>0$. This implies that for all $0<\varepsilon<\mu_0$, \begin{align*} &\|u_{t^h}^N(x,\tau)\|^2_{H^m(\Omega)}\\ & \le \frac{(2h+1)m^*\mu+\varepsilon}{\mu_0-\varepsilon} \int_0^\tau \big\| u^N_{t^h} (x, t) \big\|^2_{H^m (\Omega)} dt\\ &+C_1\Big[\sum_{k=0}^{h-1} \big\| u^N_{t^k} \big\|^2_{H^{m, 0} (Q_\tau)} + \sum_{k=0}^{h-1} \big\| u^N_{t^k}(x,\tau)\big\|^2_{H^m (\Omega)} + \|f_{t^h}\|^2_\infty+\tau \big\| f_{t^{h+1}}\big\|_\infty^2\Big], \end{align*} where $C_1$ is a positive constant. Using the induction assumption, one has $$\|u_{t^h}^N(x,\tau)\|^2_{H^m(\Omega)} \le \lambda_h\int_0^\tau \big\| u^N_{t^h} (x, t) \big\|^2_{H^m (\Omega)} dt +C_2e^{\lambda_{h-1}\tau} (1+\tau)\sum_{k=0}^{h+1} \big\| f_{t^k}\big\|^2_\infty,$$ where $C_2=\text{const}>0$. Applying Gronwall-Bellman's inequality, we obtain $$\|u_{t^h}^N(x,\tau)\|^2_{H^m(\Omega)} \le C_3\,e^{\lambda_h\tau}\sum_{k=0}^{h+1} \big\| f_{t^k}\big\|^2_\infty, \label{e4.8}$$ where $C_3$ is a positive constant. We can choose $0<\varepsilon<\mu_0$ such that $(2h+1)\gamma>\frac{\lambda_h}{2}$ for all $\gamma>\gamma_0=\frac{\mu m^*}{2\mu_0}$, because $\inf_{0<\varepsilon<\mu_0} \frac{(2h+1)m^*\mu+\varepsilon}{2(\mu_0-\varepsilon)} =\frac{(2h+1)m^*\mu}{2\mu_0}<(2h+1)\gamma\,.$ After multiplying \eqref{e4.8} with $e^{-2(2h+1)\,\gamma\, \tau}$, then integrating with respect to $\tau$ from 0 to $T$, we have the inequality $$\big\| u^N_{t^h}\big\|^2_{H^{m, 0}_{(2h+1)\gamma}(Q_T)} \le C \sum_{k=0}^{h+1} \big\| f_{t^k}\big\|^2_\infty\label{e4.9},$$ where $C$ is a positive number, independent of $N$, $f$. Hence \eqref{e4.6}--\eqref{e4.7} hold for $h$. Since $\big\{u^N_{t^h}\big\}$ is bounded in $H^{m, 0}_{(2h+1)\gamma}(Q_T)$ for all $\gamma>\gamma_0$, we can choose a subsequence which converges weakly to a vector function $u^{(h)}$ in $H^{m, 0}_{(2h+1)\gamma}(Q_T)$. On the other hand, one has $\int_{Q_T}u^N_{t^h}v\,dx\,dt=-(1)^h\int_{Q_T}u^Nv_{t^h}\,dx\,dt,\quad \forall v\in C^\infty_0(Q_T).$ Passing $N\to \infty$, it follows that $\int_{Q_T}u^{(h)}v\,dx\,dt=-(1)^h\int_{Q_T}uv_{t^h}\,dx\,dt$, for all $v\in C^\infty_0(Q_T)$; i.e., $u$ has generalized derivatives up to order $h$ with respect to $t$ and $u_{t^h}=u^{(h)}$. Furthermore, by passing \eqref{e4.9} to the limit for the weakly convergent subsequence, we obtain $$\big\| u_{t^h}\big\|^2_{H^{m, 0}_{(2h+1)\gamma}(Q_T)} \le C \sum_{k=0}^{h+1} \big\| f_{t^k}\big\|^2_\infty.\label{e4.10}$$ The theorem is proved. \end{proof} \begin{remark} \label{rm4.1} \rm We also have the same results of the smoothness with respect to time variable of the solution of the system \eqref{e2.3}--\eqref{e2.4} if the initial function $\varphi (x)$ is required to be in $H^m(\Omega)$ space and the coefficients $a_{pq}$ and the right side $f$ are required to satisfy some suitable conditions. \end{remark} \noindent {\bf Acknowledgement.} The authors would like to thank the referee for his/her helpful comments and suggestions. \begin{thebibliography}{00} \bibitem{a1} Adams, R. A.; {\it Sobolev Spaces}, Academic Press, 1975. \bibitem{h1} Hung, N. M.; {\it The first initial boundary value problem for Schr\"odinger systems in non-smooth domains}, Diff. 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