\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 $Q_T, 0<T\leq +\infty$, with non-smooth base
 $\Omega$. Some results on the existence, uniqueness and smoothness
 with respect to time variable of generalized solution of this
 problem are given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

 Boundary value problems for Schr\"odinger equations have been
considered in the books by Lions and Magenes \cite{l2} in
finite cylinders $Q_T=\Omega\times (0,T), (T<+\infty)$, with base
$\Omega$, where $\partial \Omega$ is smooth. Their results are
 restricted to Schr\"odinger type equations, where coefficients
$a_{pq}$ of equations are functions independent of $t$ (except $a_{00}$).
 The first initial boundary value problem for general Schr\"odinger systems,
where the coefficients $a_{pq}(x,t)$ are matrices of functions of
two variables $x$ and $t$ for all $p,q$, was considered in
\cite{h1,h2}, in cases the finite cylinder $Q_T,\,T<+\infty$ or in cases
the infinite cylinder $Q_\infty=\Omega\times (0,+\infty)$ as in
 \cite{h2,h3}.
In this paper, we consider the second initial boundary value problem
for these systems in both the finite and the infinite cylinder
 $Q_T=\Omega\times (0,T)$, where $0<T\le +\infty$ and $\Omega$
is a domain with non-smooth boundary. Our main purpose is to
study the existence, uniqueness and smoothness with respect to
time variable of generalized solution of the mentioned problem.
Such results are investigated in a scale of weighted spaces
$H^{m,0}_\gamma(Q_T)$ for some $\gamma>0$.

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 $0<T\le +\infty$; $\ x=(x_1,\dots ,x_n)\in\Omega,\
u(x,t)=(u_1(x,t),\dots ,u_s(x,t))$ is a vector complex function;
$\alpha=(\alpha_1,\dots ,\alpha_n)$ ($\alpha_i\in\mathbb N$,
$i=1,\dots,n$) is a multi-index; $|\alpha |=\alpha_1+\dots
+\alpha_n$, $ D^\alpha =\partial^{|\alpha|}/\partial
x_1^{\alpha_1}\dots \partial x_n^{\alpha_n}$,
$|D^\alpha u|^2={\sum_{i=1}^s}|D^\alpha u_i|^2$,
 $u_{t^j}=(\partial^j u_1/\partial t^j,\dots ,\partial^j
u_s/\partial t^j)$,  $C_{k}^s = \frac{k!}{ s!(k-s)!}$
($0\le s\le k$).

In this paper we  use the usual functional spaces:
 $C^\infty(\overline{\Omega})$,
$L_2(\Omega), H^m(\Omega)$, $ L_2(Q_T)$, $ H^{l,k}(Q_T)$
 when $T<+\infty$ and $m,l,k\in \mathbb{N} $
(see \cite{h2,h3} for the precise definitions).

Moreover, when $0<T\le +\infty$ we define
$H^{m,0}_\gamma(Q_T)$ $(\gamma>0)$ 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 $\| u\|_\infty= \mathop{\rm ess\,sup}_{0<t<T} \| u (t)\|_{L_2(\Omega)}
 < +\infty$.

For convenience, in the rest of this paper we say that $u(x,t)$
 belongs to some spaces if all of its components belong to that space.
We now introduce a $2m$th-order differential operator
\begin{equation}
L(x,t,D)=\sum_{|p|,|q|=0}^m  D^p(a_{pq}(x,t)\,D^q),\label{e2.1}
\end{equation}
where $a_{pq}$ are $s\times s$ matrices of bounded measurable complex
functions defined on $Q_T$,
$a_{pq}=(-1)^{|p|+|q|}a_{qp}^*$ ($a_{qp}^*$ denotes the transposed
conjugate matric of $a_{qp}$). Moreover, we assume that the
operator $L$ satisfies a hypothesis that given as follows
(see for example \cite{k1,m1,m2}).

For all $(x,t)\in Q_T$ and $(\eta_p)_{|p|=m}\in C^{s.m_1^*}\setminus \{0\}$,
 we have
\begin{equation}
\sum_{|p|=|q|=m}
a_{pq}(x,t)\eta_q\overline{\eta_p}\ge C_0\sum_{|p|=m}|\eta_p|^2,\label{e2.2}
\end{equation}
where $C_0$ is a positive real number, independent of
$(\eta_p)_{|p|=m}$; $m_1^*=\sum_{|p|=m}1$.

Setting $\eta_p=\xi^p\eta$ with $\xi\in \mathbb{R}^n\setminus
\{0\}$, $\xi^p=\xi_1^{p_1}\dots \xi_n^{p_n}$ and
$\eta\in \mathbb{C}^s\setminus\{0\}$, it follows from
condition \eqref{e2.2} that
$\sum_{|p|=|q|=m}
a_{pq}(x,t)\xi^p\xi^q\eta\overline{\eta}\ge C_0|\xi|^{2m}|\eta|^2$,
for all $(x,t)\in Q_T$, which is equivalent to the strong ellipticity
of the operator $L$.
However, one can see easily that the condition of strong ellipticity
of the operator $L$ does not imply the condition \eqref{e2.2}.

In the cylinder $Q_T$ we consider
{\it the second initial boundary problem for the Schr\"odinger system}
\begin{equation}
i(-1)^{m-1}L(x,t,D)u-u_t=f(x,t),\quad (x,t)\in Q_T,\label{e2.3}
\end{equation}
with initial condition
\begin{equation}
u(x,0)=\varphi(x),\quad  x\in \Omega\label{e2.4},
\end{equation}
where $L(x,t,D)$ is the operator in \eqref{e2.1},
satisfies the condition \eqref{e2.2} and $u, f, \varphi$ are
vector functions.

The function $u(x,t)$ is called {\it generalized solution}
in the space $H^{m,0}_\gamma(Q_T)$ of the second initial boundary
problem for the Schr\"odinger system \eqref{e2.3}
 and initial condition \eqref{e2.4} if and only if
$u(x,t)\in H^{m,0}_\gamma(Q_T)$, satisfying
\begin{equation}
\begin{aligned}
&(-1)^{m-1} i \sum_{| p|, | q| = 0}^m (-1)^{| p|}
\int_{Q_\tau} a_{pq}(x,t) D^q u(x,t) \overline{D^p \eta}(x,t) \,dx\,dt\\
&\quad + \int_{Q_\tau} u(x,t) \overline {\eta_t}(x,t) \,dx\,dt \\
&=- \int_{\Omega}\varphi(x)\eta(x,0)dx
 + \int_{Q_\tau} f(x,t) \overline \eta(x,t) \,dx\,dt
\end{aligned} \label{e2.5}
\end{equation}
for each $0<\tau<T$ and all test functions
$\eta (x, t) \in  H^{m, 1} (Q_\tau)$, $\eta (x, \tau) = 0$.
Set
\[
B[u,v](t)=\sum_{| p|, | q| = 0}^m (-1)^{| p|}
 \int_{\Omega} a_{pq}(x,t) D^q u(x,t) \overline{D^p v(x,t)}dx.
\]
To consider the problem we need to prove the important following lemma.

\begin{lemma} \label{lem2.1}
There exist two constants $\mu_0>0$ 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 $0<\varepsilon<C_0,\ \ C(\varepsilon)>0$.
 This implies
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
holds for all $k=1,2,\dots,m-1$, and for all $ u\in H^m(\Omega)$.
Note that for all $0<T\le +\infty$, if
$ u\in H_\gamma^{m,0}(Q_T)$, $\gamma>0$, 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
 \begin{equation}
\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}
\end{equation}
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
\begin{equation}
\|u\|^2_{H^{m-1}(\Omega)}\le\varepsilon \|u\|^2_{H^m(\Omega)}
+C_4\|u\|_{L_2(\Omega)}\label{e2.9}
\end{equation}
 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
\begin{equation}
(-1)^mB[u,u](t)\geq \mu_0\|u\|^2_{H^m(\Omega)}\label{e2.10}
\end{equation}
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 $0<T\le +\infty$.

Denote $m^*=\sum_{|p|=1}^m1$, we begin by studying the uniqueness theorem.

\begin{theorem} \label{thm3.1}
Assume that for a positive constant $\mu$,
$$
\sup\{|\frac{\partial a_{pq}}{\partial t}|: (x,t)
\in Q_{T}, 0 \leq |p|,|q| \leq m\} \leq \mu\,.
$$
Then the second initial boundary value problem for \eqref{e2.3}
with non homogeneously initial condition \eqref{e2.4} has at
most one generalized solution in $H^{m,0}_{\gamma }(Q_T)$
for all $\gamma>0$ 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 $0<\tau<T$ and
$b\in (0,\tau)$ we set
 $$
\eta(x,t)  =\begin{cases}
\int_b^t\,u(x,s)ds ,  &0\le t\le b,\\
0 , &b<t\le \tau.\end{cases}
$$
It is easy to check that
$\eta(x,t) \in H^{m,1}(Q_\tau), \eta(x,\tau)=0$ and
$\eta_t(x,t)=u(x,t)$ for all $(x,t)\in Q_b$.
 It follows from \eqref{e2.5} that
\begin{equation}
(-1)^m\ \sum_{|p|,|q|=0}^m(-1)^{|p|}\
\int_{Q_b}a_{pq}D^q\eta_t\overline{D^p\eta} \,dx\,dt
+i\int_{Q_b}|\eta_t|^2\,dx\,dt=0.\label{e3.1}
\end{equation}
Adding this equation with its complex conjugate,
using $ a_{pq}=(-1)^{|p|+|q|} a_{qp}^*$ and
integrating by parts with respect to $t$, we get
 $$
B[\eta,\eta](0)= -\sum_{|p|,|q|=0}^m(-1)^{|p|}\
\int_{Q_b}\frac{\partial a_{pq}}{\partial t}D^q\eta\overline{D^p\eta}
\,dx\,dt.
$$
Since $|\frac{\partial a_{pq}}{\partial t}|$ are bounded,
using the Cauchy inequality and \eqref{e2.10}, we obtain
\begin{equation}
\|\eta(x,0)\|^2_{H^m(\Omega)}\leq C\|\eta(x,t)\|^2_{H^{m,0}(Q_b)},
\quad (C=\mu m^*/\mu_0> 0).\label{e3.2}
\end{equation}
Putting  $v_p(x,t)={\int_t^0}D^pu(x,s)ds$, $0<t<b$, $0\le|p|\le m$,
so we have
$D^p\eta(x,t)=\int_b^t\,D^pu(x,s)ds =v_p(x,b)-v_p(x,t)$,
$D^p\eta(x,0)=v_p(x,b)$.
Substituting those into \eqref{e3.2}, one gets
\[
\sum_{|p|=0}^m\int_\Omega\ |v_p(x,b)|^2dx
\le 2Cb\ \sum_{|p|=0}^m\int_\Omega\ |v_p(x,b)|^2dx+
2C\ \sum_{|p|=0}^m\int_{Q_b}\ |v_p(x,t)|^2\,dx\,dt.
\]
 Setting $J(t)={\sum_{|p|=0}^m\int_\Omega}\ |v_p(x,t)|^2dx$, we have
$(1-2Cb)J(b)\le 2 C\ \int_0^b\,J(t)dt$,
or $J(b)\le 4C\ \int_0^b\,J(t)dt$, for all $b\in [0,\frac 1{4C}]$.
This implies that $J(t)\equiv 0$ on $[0,\frac{1}{4C}]$
by Gronwall-Bellman's inequality. It follows $u_1\equiv u_2$ on
$[0,\frac{1}{4C}]$, where $C$ does not depend on $\tau$.
Using similar  arguments for two functions $u_1, u_2$ on
$[\frac{1}{4C},\tau]$, we can show that after finite steps
we get $u_1\equiv u_2$ on $[0,\tau]$.
Since $0<\tau<T$ is arbitrary, so $u_1\equiv u_2$ on $(0,T)$.
The  theorem is proved.
\end{proof}

Now, we establish the existence of generalized solution of the
 mentioned problem by Galerkin's approximate method.

\begin{theorem} \label{thm3.2}
Assume that:
\begin{itemize}
\item[(i)] For a positive constant $\mu$,
$|a_{pq}(x,0)|;\ \big|\frac{\partial a_{pq}}{\partial t}\big|\le \mu$,
for all $0\le |p|, |q|\le m$; and all $(x,t)\in Q_{T}$;

\item[(ii)] $f,f_t\in L^\infty(0,T ;\,L_2(\Omega))$,
$f(.,0)\in L_2(\Omega)$;

\item[(iii)] $\varphi\in H^m(\Omega)$.
\end{itemize}
Then for every  $\gamma>\gamma_0=\frac{m^*\mu}{2\mu_0}$ 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
\begin{equation}
(-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}
\end{equation}
 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 $0<\tau<T$, by integrating with respect to $t$ from
0 to $\tau$, and integrating by parts, we obtain
\begin{align*}
 (-1)^m B[u^N,u^N](\tau)
&=(-1)^m\sum_{|p|,|q|=0}^m(-1)^{|p|}
\int_{Q_\tau}\frac{\partial a_{pq}}{\partial t}D^qu^N
 \overline{D^pu^N}\,dx\,dt \\
&\quad +(-1)^mB[u^N,u^N](0)-2\,\mathop{\rm Im}\int_{\Omega}f(x,0)
 \overline{u^N}(x,0)dx\\
&\quad +2\mathop{\rm Im}\Big[\int_{\Omega}f(x,\tau)\overline{u^N}(x,\tau)dx-
\int_{Q_\tau}f_t\overline{u^N}\,dx\,dt\Big].
\end{align*}
This implies by Cauchy's inequality and \eqref{e2.10} that for all
$0<\varepsilon<\mu_0$,
\begin{align*}
\|u^N(x,\tau)\|_{H^m(\Omega)}^2
&\le\frac{m^*\mu +\varepsilon}{\mu_0-\varepsilon}
 \int_0^\tau\|u^N(x,t)\|_{H^m(\Omega)}^2dt
 +\frac{m^*\mu +\varepsilon}{\mu_0-\varepsilon}
 \|u^N(x,0)\|_{H^m(\Omega)}^2\\
& \quad + \frac 1{\varepsilon(\mu_0-\varepsilon)}
\big[\|f(x,0)\|^2_{L_2(\Omega)}+\|f\|_\infty^2+
\tau\|f_t\|_\infty^2\big].
\end{align*}
Applying Gronwall-Bellman's inequality, one gets
\begin{align*}
&\|u^N(x,\tau)\|_{H^m(\Omega)}^2\\
&\le C_1\Big[\|u^N(x,0)\|^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},
\end{align*}
where $C_1=\max\{ \frac{m^*\mu}{\mu_0-\varepsilon},\
\frac{1}{\varepsilon(\mu_0-\varepsilon)}\}>0$.
Therefore,
 \begin{equation}
 \|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}
\end{equation}
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
\begin{equation}
\|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}
\end{equation}
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
$0<\tau<T$ so it suffices to
show that $u(x,t)$ satisfies \eqref{e2.5}  for all $\eta(x,t)\in M$.
Note that the denseness of the set $M$ in the space
$\hat{H}^{m,1}(Q_\tau)$ can be proved easily by using lemma 2.1
and arguments analogous as that
used in the first problem (see in \cite{h1,h2}).

Taking $\eta(x,t)\in M$ arbitrarily, there exists $N_0$ such that
$\eta$ can be written in the form
$\eta(x,t)=\sum_{l=1}^{N_0} d_l(t)\varphi_l(x), d_l(t)\in
H^1(0,\tau),\ d_l(\tau)=0,\ \forall l=1,\dots ,N_0$. Multiplying
\eqref{e3.3} (with $N\ge N_0$) by $d_l(t)$, taking sum  with respect to
l from 1 to $N$, then integrating with respect to t from 0 to
$\tau$, we obtain
$$
(-1)^{m-1}i\ \sum_{|p|,|q|=0}^m(-1)^{|p|}\
\int_{Q_\tau}\ a_{pq}D^qu^N\overline{D^p\eta}\,dx\,dt
-\int_{Q_\tau}\,u_t^N\overline{\eta}\,dx\,dt=
\int_{Q_\tau}\,f\overline{\eta}\,dx\,dt.
$$
It is easy to check that
$\int_{Q_\tau}u_t^N\overline{\eta}\,dx\,dt
=-\int_\Omega \varphi(x)\overline{\eta(x,0)}dx
-\int_{Q_\tau}u^N\overline{\eta_t}\,dx\,dt$,
so one has
\begin{align*}
& (-1)^{m-1}i\hspace{-0.25cm}\sum_{|p|,|q|=0}^m(-1)^{|p|}
\int_{Q_\tau}a_{pq}D^qu^N\overline{D^p\eta}\,dx\,dt
+\int_{Q_\tau}u^N\overline{\eta}_t\,dx\,dt\\
&=-\int_\Omega \varphi(x)\overline{\eta(x,0)}dx
 +\int_{Q_\tau}f\overline{\eta}\,dx\,dt.
\end{align*}
Passing to the limit for the weakly convergent subsequence, we get
\begin{align*}
&(-1)^{m-1}i\sum_{|p|,|q|=0}^m(-1)^{|p|}
\int_{Q_\tau}a_{pq}D^qu\overline{D^p\eta}\,dx\,dt
+\int_{Q_\tau}u\overline{\eta}_t\,dx\,dt
\\
&=-\int_\Omega \varphi(x)\overline{\eta(x,0)}dx
+\int_{Q_\tau}f\overline{\eta}\,dx\,dt.
\end{align*}
Hence $u(x,t)$ is a generalized solution of the second initial
boundary value problem for the system \eqref{e2.3}--\eqref{e2.4}.
Moreover, the weak convergence of the subsequence of $\{u^N(x,t)\}$
and \eqref{e3.7} imply that this solution satisfies the
inequality
\begin{align*}
\|u\|^2_{H^{m,0}_\gamma(Q_T)}
&\le \liminf_{N\to\infty} \|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],
\end{align*}
where $C$ only depends on $\mu,\mu_0$.
This completes the proof.
\end{proof}

\section{Smoothness of generalized solutions with respect to time}\label{sec 4}

In this section, we consider the second initial boundary value
 problem for the system
\begin{gather}
i(-1)^{m-1}L(x,t,D)u-u_t=f(x,t),\quad (x,t)\in Q_T,\label{e2.3'}\\
u(x,0)=0,\quad  x\in \Omega\label{e2.4'},
\end{gather}
We will prove that the smoothness with respect to time variable
of generalized solution of the second initial boundary value
problem for the Schr\"odinger system \eqref{e2.3'}--\eqref{e2.4'}
 depends on only the smoothness with respect to time variable
of the coefficients and the right side of the systems.
Indeed, we have the following theorem.

\begin{theorem} \label{thm4.1}
Suppose that
\begin{itemize}
\item[(i)] for some positive constant $\mu$,
 $\{|a_{pq}(x,0)|,\ |\frac{\partial^k a_{pq}}{ \partial t^k}(x,t)|\}
\le \mu$, for all $0 \le |p|,|q| \le m$,  all $(x, t) \in  Q_T$, all
$1 \le k \le h+1$;

\item[(ii)] $f_{t^k} \in L^\infty (0, T; L_2 (\Omega))$, for all
$0 \le k \le h +1$, $f(x,0)=0$,
if $h\geq 2$ then we assume that $f_{t^k} (x, 0) = 0$, for all
$1 \le k \le h-1$, all $x\in \Omega$.

\end{itemize}
Then for every  $\gamma > \gamma_0 = \frac{m^* \mu}{ 2 \mu_0}$
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
\begin{equation}
\| u_{t^h}\|^2_{H^{m, 0}_{(2h+1)\gamma} (Q_T)}
\le C\sum_{k=0}^{h+1} \| f_{t^k} \|^2_\infty ,
\end{equation}
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
 \begin{equation}
(-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}
\end{equation}
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
\begin{equation}
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}
\end{equation}
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{equation}
\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}
\end{equation}
By using (ii) and induction on $k$, 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<\tau<T,\; \forall N=1,2,\dots ,\label{e4.6}
\\
\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{e4.7}
\end{gather}
are valid with $0 < \varepsilon < \mu_0$,
$\lambda_h=\frac{(2h+1)m^*\mu + \varepsilon}{ \mu_0 - \varepsilon}$;
$C$ does not depend on $N$, $f$.

From the inequalities \eqref{e3.6}--\eqref{e3.7}
 (with $\varphi(x)=0, f(x,0)=0$), we can see easily that
\eqref{e4.6}--\eqref{e4.7} hold for $h=0$, and $\{u^N\}$
convergent weakly to the solution $u$ of the problem
in $H^{m,0}_\gamma(Q_T)$.

Now let \eqref{e4.6}--\eqref{e4.7} be true for $h-1$.
We will prove that these also hold for $h$.
Integrating \eqref{e4.5}, for $k=h+1$, with respect to $t$
from $0$ to $\tau$, we get
\begin{align*}
& -i \int_{Q_\tau}\big| u_{t^{h+1}}^N\big|^2 \,dx\,dt
+ (-1)^m \sum_{| p|, | q | = 0}^m (-1)^{| p|}
 \int_{Q_\tau} a_{pq} D^qu^N_{t^h} \overline{D^p u^N_{t^{h+1}}} \,dx\,dt\\
&= (-1)^{m-1} \sum_{| p|, | q| = 0}^m (-1)^{| p|}
 \sum_{s = 0}^{h-1} C_h^s \int_{Q_\tau} \frac{\partial^{h-s}
a_{pq}}{ \partial t^{h-s}} D^q u^N_{t^s}
\overline{D^p u^N_{t^{h+1}}} \,dx\,dt\\
&\quad + i \int_{Q_\tau} f_{t^h} \overline{u_{t^{h+1}}^N} \,dx\,dt.
\end{align*}
Adding this equation with its complex conjugate then integrating
by parts with respect to $t$, using (ii), \eqref{e4.4}, we obtain
\begin{align*}
&(-1)^m B[u^N_{t^h}(x,\tau),u^N_{t^h}(x,\tau)]\\
&=(-1)^m \sum_{| p|, | q | = 0}^m (-1)^{| p|}
 \int_{Q_\tau}\frac{\partial a_{pq}}{\partial t}D^qu^N_{t^h}
 \overline{D^p u^N_{t^{h}}} \,dx\,dt\\
&\quad+(-1)^m 2\mathop{\rm  Re}\sum_{| p|, | q| = 0}^m (-1)^{| p|}.h.
 \int_{Q_\tau}\frac{\partial a_{pq}}{\partial t}D^qu^N_{t^h}
 \overline{D^p u^N_{t^{h}}} \,dx\,dt\\
&\quad + (-1)^{m}2\mathop{\rm  Re} \sum_{| p|, | q| = 0}^m (-1)^{| p|}
 \sum_{s = 0}^{h-1} C_h^s \int_{Q_\tau} \frac{\partial^{h-s+1}
 a_{pq}}{ \partial t^{h-s+1}} D^q u^N_{t^s} \overline{D^p u^N_{t^{h}}}
 \,dx\,dt\\
&\quad +(-1)^m2\mathop{\rm  Re} \sum_{| p|, | q| = 0}^m (-1)^{| p|}
 \sum_{s = 0}^{h-2} C_h^s \int_{Q_\tau}
\frac{\partial^{h-s} a_{pq}}{ \partial t^{h-s}} D^q u^N_{t^{s+1}}
 \overline{D^p u^N_{t^{h}}} \,dx\,dt\\
&\quad -(-1)^m\;2\mathop{\rm  Re} \sum_{| p|, | q| = 0}^m (-1)^{| p|}
 \sum_{s = 0}^{h-1} C_h^s \int_{\Omega}
\frac{\partial^{h-s} a_{pq}}{ \partial t^{h-s}}(x,\tau)
 D^q u^N_{t^s}(x,\tau) \overline{D^p u^N_{t^{h}}(x,\tau)} dx \\
&\quad - 2\mathop{\rm Im}\int_{\Omega} f_{t^h}(x,\tau)
\overline{u_{t^{h}}^N(x,\tau)} dx\ -2\mathop{\rm Im}
\int_{Q_\tau} f_{t^{h+1}} \overline{u_{t^{h}}^N} \,dx\,dt.
\end{align*}
For all $\varepsilon_1>0$, 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
\begin{equation}
\|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,
\end{equation}
where $C_2=\text{const}>0$. Applying Gronwall-Bellman's inequality,
we obtain
\begin{equation}
\|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}
\end{equation}
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
 \begin{equation}
\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},
\end{equation}
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
\begin{equation}
\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}
\end{equation}
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.

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\end{document}