# Reconstruction for the time-dependent coefficients of a quasilinear
dynamical Schrödinger equation

Gen Nakamura∗, Tanmay sarkar† and Manmohan Vashisth⋄ ∗Department of
Mathematics, Hokkaido University, Sapporo 060-0810, Japan.
E-mail<EMAIL_ADDRESS>†Department of Mathematics, Indian Institute of
Technology Jammu, Jagti, Jammu - 181221, India.
E-mail<EMAIL_ADDRESS>⋄ Department of Mathematics, Indian
Institute of Technology Jammu, Jagti, Jammu - 181221, India.
E-mail<EMAIL_ADDRESS>
###### Abstract.

We study an inverse problem related to the dynamical Schrödinger equation in a
bounded domain of $\mathbb{R}^{n},n\geq 2$. Since the concerned non-linear
Schrödinger equation possesses a trivial solution, we linearize the equation
around the trivial solution. Demonstrating the well-posedness of the direct
problem under appropriate conditions on initial and boundary data, it is
observed that the solution admits $\varepsilon$-expansion. By taking into
account the fact that the terms $\mathcal{O}(|\nabla u(t,x)|^{3})$ are
negligible in this context, we shall _reconstruct_ the time-dependent
coefficients such as electric potential and vector-valued function associated
with quadratic nonlinearity from the knowledge of input-output map using the
geometric optics solution and Fourier inversion.

###### Key words and phrases:

Schrödinger equation, input-output map, $\varepsilon$-expansion,
reconstruction, geometric optics solution.

###### AMS subject classification. Mathematics Subject Classification:

35R30

##  1\. Introduction

In this paper, we address an inverse problem associated with non-linear
dynamical Schrödinger equation. More precisely, we consider the following
initial boundary value problem (IBVP):

$\displaystyle\begin{cases}i\partial_{t}u(t,x)+\Delta
u(t,x)+q(t,x)u(t,x)=\nabla\cdot\overrightarrow{J}(t,x,\nabla
u),~{}(t,x)\in\Omega_{T},\\\ u(0,x)=\varepsilon\phi(x),\quad x\in\Omega,\\\
u(t,x)=\varepsilon
f(t,x),\quad(t,x)\in\Sigma:=(0,T)\times\partial\Omega,\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{n}~{}(n\geq 2)$ is a bounded and simply
connected domain with smooth boundary $\partial\Omega$. Given time $T>0$, we
denote $\Omega_{T}:=(0,T)\times\Omega$. Furthermore, $\varepsilon>0$ is a
small parameter and
$\nabla:=(\partial_{x_{1}},\partial_{x_{2}},\dots,\partial_{x_{n}})$
represents the gradient operator with respect to the spatial variable
$x:=(x_{1},x_{2},\dots,x_{n})$. In (1.1), real-valued time-dependent bounded
function $q(t,x)$ represents the electric potential and we assume that $q\in
C^{\infty}(\Omega_{T})$.

The vector-valued function $\overrightarrow{J}(t,x,\nabla u)$ in (1.1) is
given by

$\displaystyle\overrightarrow{J}(t,x,\nabla u):=\overrightarrow{P}(t,x,\nabla
u)+\overrightarrow{R}(t,x,\nabla u)$ (1.2)

in which the function $\overrightarrow{P}(t,x,\xi)$ is described by

$\displaystyle\overrightarrow{P}(t,x,\xi):=|\xi|^{2}\overrightarrow{b}(t,x),\quad\xi=(\xi_{1},\xi_{2},\dots,\xi_{n})\in\mathbb{C}^{n},$
(1.3)

with $|\xi|^{2}$ defined using the complex conjugate as

$\quad|\xi|^{2}=\sum_{j=1}^{n}\xi_{j}\overline{\xi_{j}}.$

and $\overrightarrow{b}\in
C^{\infty}\big{(}[0,T]\times\overline{\Omega}\big{)}$ where
$\overline{\Omega}:=\Omega\cup\partial\Omega$. We denote a set of a Fréchet
space $E$-valued $C^{\infty}$ functions over $[0,T]$ flat at $t=0$ by
$C^{\infty}_{(0)}\big{(}(0,T];E\big{)}$ and we assume that
$\displaystyle\overrightarrow{R}(t,x,\xi)\in
C^{\infty}_{(0)}\big{(}(0,T];C^{\infty}(\overline{\Omega}\times H)\big{)},$
where for a constant $h>0$, the set $H$ is defined by
$H:=\\{\xi\in\mathbb{C}^{n}:|\xi|\leq h\\}$. Moreover, there exists a positive
constant $C>0$ such that

$\displaystyle|\partial_{\xi}^{\alpha}\nabla_{x}^{\beta}\overrightarrow{R}(t,x,\xi)|\leq
C|\xi|^{3-|\alpha|},$ (1.4)

holds for all multi-indices $\alpha$ and $\beta$ with $|\alpha|\leq 3$. In
order to state well-posedness results, we define the following Sobolev space
for two non-negative real numbers $r$ and $s$

$\displaystyle H^{r,s}(\Sigma)=L^{2}(0,T;H^{r}(\partial\Omega))\cap
H^{s}(0,T;L^{2}(\partial\Omega))$

equipped with the norm

$\displaystyle\left\|f\right\|_{H^{r,s}(\Sigma)}=\left\|f\right\|_{L^{2}(0,T;H^{r}(\partial\Omega))}+\left\|f\right\|_{H^{s}(0,T;L^{2}(\partial\Omega))}.$

We also set

$\displaystyle H^{r,s}_{0}(\Sigma)=\\{f\in
H^{r,s}(\Sigma):f(0,\cdot)=\partial_{t}f(0,\cdot)=\dots=\partial_{t}^{s-1}f(0,\cdot)=0~{}\text{on}~{}\partial\Omega\\}.$

Let $r\geq\big{[}\frac{n}{2}\big{]}+3$. Then there exists
$\varepsilon_{0}=\varepsilon_{0}(h,r,T)$ such that (1.1) has a unique solution
(see section 2 for more details)

$u\in
X_{r}:=X_{r}([0,T]):=\bigcap_{k=0}^{r}C^{k}\big{(}[0,T];H^{r-k}(\Omega)\big{)}$

in lieu with $H^{0}(\Omega)=L^{2}(\Omega)$, satisfying the compatibility
condition of order $r-1$ and $0<\varepsilon<\varepsilon_{0}$ for any $\phi\in
H^{r}_{0}(\Omega)\cap H^{r+1}(\Omega)$ and $f\in H_{0}^{r,r+1}(\Sigma)$.
Moreover, we define the norm on $X_{r}$ as

$\displaystyle\left\|u\right\|_{X_{r}([0,T])}:=\sup_{t\in[0,T]}\bigg{(}\sum_{k=0}^{r}\left\|u^{(k)}(t,\cdot)\right\|^{2}_{H^{r-k}(\Omega)}\bigg{)}^{1/2},$

where $u^{(k)}:=\frac{\partial^{k}u}{\partial t^{k}}$.

In view of the _unique solvability_ of the IBVP (1.1), we define the input-
output map $\Lambda_{q,\overrightarrow{J}}(\varepsilon\phi,\varepsilon f)$
associated with the dynamical Schrödinger equation as follows:

$\displaystyle\Lambda_{q,\overrightarrow{J}}(\varepsilon\phi,\varepsilon
f):=\bigg{(}u^{\phi,f}|_{t=T},\Big{[}\partial_{\nu}u^{\phi,f}-\nu(x)\cdot\overrightarrow{J}(t,x,\nabla
u^{\phi,f})\Big{]}\Big{|}_{\Sigma}\bigg{)}$ (1.5)

for any $(\phi,f)\in\big{(}H^{r}_{0}(\Omega)\cap H^{r+1}(\Omega)\big{)}\times
H_{0}^{r,r+1}(\Sigma)$, where $0<\varepsilon<\varepsilon_{0}$ and
$u^{\phi,f}(t,x)$ is the solution of IBVP (1.1) and $\nu$ is the unit normal
vector to $\partial\Omega$ at $x\in\partial\Omega$ directed into the exterior
of $\Omega$.

In this paper, we are interested in the following inverse problem associated
with the quasilinear dynamical Schrödinger equation.
Inverse problem: Indentify the electric potential $q=q(t,x)$ and the quadratic
nonlinearity in $\overrightarrow{P}=\overrightarrow{P}(t,x,\nabla u)$ from the
knowledge of input-output map $\Lambda_{q,\overrightarrow{J}}$ defined in
(1.5).

In recent years, there is a growing interest in recovering coefficients
associated with quasilinear equations from the knowledge of boundary
observations. For instance, the coefficients of a quasilinear elliptic
equation in divergence form are reconstructed from the the Dirichlet to
Neumann (DN) map defined on the boundary (see [10] for details). In [11],
similar analysis has been carried out for the perturbed weighted $p$-Laplacian
equation to recover the coefficients from prescribed DN-map.

Significant progress has also been made to recover the time-dependent
coefficients appearing in the quasilinear equations, for instance, refer to
[15, 28, 32] for quasilinear parabolic equation, and [20, 38] and references
therein for hyperbolic equations. In case of magnetic Schrödinger equation,
the problem of recovering coefficients has attracted the attention of several
researchers in recent years. For instance, in [7], inverse problem of
determining the magnetic field and time-dependent electric potential from the
knowledge of DN-map was considered in three or higher dimension. In [17], the
author dealt with a problem of determining time-dependent electromagnetic
potentials appearing in the Schrödinger equation using geometric optics
construction. Further in this direction, inverse problems related to the
linear dynamical Schrödinger equation is considered and corresponding
stability estimates are obtained (one can refer to [1, 3, 5, 30] and
references cited therein for details).

Choulli et al. [14] considered the problem of recovering the time-dependent
scalar potential associated with the Schrödinger equation from boundary
measurements in an unbounded domain. Moreover, determination of eletric or
magnetic potential appearing in the linear dynamical Schrödinger equation in
Riemannian geometry have also been studied (see [2, 6, 17, 31, 34] and
references therein). However, reconstructions of the time-dependent
coefficients for the nonlinear equations have not reached the same stage of
maturity as linear problems. Recently in [35], the determination of time-
dependent coefficients in a semi-linear dyanmical Schrödinger equation is
considered where they showed that the electric potential and coefficient of
non-lnearity can be determined from the knowledg of source-to-solution map.
Inverse initial boundary value problem for a quasilinear hyperbolic equation
in second or higher dimension was studied (refer to [39]), in which unique
determination of time-dependent coefficients is proved from the measurement
given by input-output map.

The following result is our main contribution in this paper:

######  Theorem 1.1.

The electric potential $q(t,x)$ and the coefficient $\overrightarrow{b}(t,x)$
of the quadratic part of the vector-valued function
$\overrightarrow{J}(t,x,\nabla u)$ can be constructively determined from the
knowledge of input-output map
$\Lambda_{q,\overrightarrow{J}}\left(\epsilon\phi,\epsilon f\right)$ measured
for all $(\phi,f)\in\big{(}H^{r}_{0}(\Omega)\cap H^{r+1}(\Omega)\big{)}\times
H_{0}^{r,r+1}(\Sigma)$.

To the best of our knowledge, this is the first result which deals with the
recovery of time-dependent cofficients appearing in a non-linear dynamical
Schrödinger equation from the knowledge of boundary measurements of the
solution. Recently in [35], the problem of recovery of time-dependent
coefficients is considered but they recovered these coefficients from the
knowledge of source-to-solution map while we recovered from the knowledge of
solutions measured on the boundary.

The remaining part of the article is organized as follows. In Section 2, we
will analyze the $\varepsilon$-expansion of the initial-boundary value problem
(1.1) and use the resulting expansion to linearize the input-output map
$\Lambda_{q,\overrightarrow{J}}$. Furthermore, we study the well-posedness of
problem (1.1) and demonstrate that the solution admits
$\varepsilon$-expansion. In Section 3, we reconstruct the electric potential
$q(t,x)$ and the coefficient of nonlinearity $\overrightarrow{b}(t,x)$ from
the knowledge of input-output map $\Lambda_{q,\overrightarrow{J}}$.

##  2\. $\varepsilon$-expansion of the solution of IBVP

The aim of this section is to demonstrate the unique solvability of magnetic
Schrödinger equation (1.1) via $\varepsilon$-expansion technique. This
expansion technique is quite instrumental in the hyperbolic and elliptic
problems (see [38, 39]) for nonlinear wave equation and (see [10, 29]) for
quasilinear elliptic equation. However, we hereby impose the technique in a
nonlinear dynamical Schrödinger equation where the treatment differs
significantly and hence, the complete details will be provided for existence,
uniqueness and regularity of the solution to equation (1.1).

We shall start with the following convention:

$\displaystyle
v(t,x)=\mathcal{O}(\varepsilon^{3})\Leftrightarrow\left\|v\right\|_{X_{m}([0,T])}=\mathcal{O}(\varepsilon^{3}).$

######  Theorem 2.1.

Let $m\geq\big{[}\frac{n}{2}\big{]}+3$ and
$(\phi,f)\in\big{(}H^{r}_{0}(\Omega)\cap H^{r+1}(\Omega)\big{)}\times
H_{0}^{r,r+1}(\Sigma)$. Then for given $T>0$, there exists
$\varepsilon_{0}=\varepsilon_{0}(h,r,T)$ such that (1.1) has a unique solution
$u(t,x)\in X_{r}([0,T])$ for any $0<\varepsilon<\varepsilon_{0}$, where $h$
and $X_{r}$ are defined in Section 1.

Furthermore, $u(t,x)$ admits $\varepsilon$-expansion

$\displaystyle u(t,x)=\varepsilon
u_{1}(t,x)+\varepsilon^{2}u_{2}(t,x)+\mathcal{O}(\varepsilon^{3})$ (2.1)

where $\varepsilon\rightarrow 0$ and $u_{1}(t,x)$ is a solution to the
homogeneous equation:

$\displaystyle\begin{cases}i\partial_{t}u_{1}(t,x)+\Delta
u_{1}(t,x)+q(t,x)u_{1}(t,x)=0,\quad&(t,x)\in\Omega_{T},\\\
u_{1}(0,x)=\phi(x),\quad&x\in\Omega,\\\
u_{1}(t,x)=f(t,x),\quad&(t,x)\in\Sigma;\end{cases}$ (2.2)

and $u_{2}(t,x)$ is a solution to the non-homogeneous equation:

$\displaystyle\begin{cases}i\partial_{t}u_{2}(t,x)+\Delta
u_{2}(t,x)+q(t,x)u_{2}(t,x)=\nabla_{x}\cdot\Big{(}|\nabla_{x}u_{1}(t,x)|^{2}\overrightarrow{b}(t,x)\Big{)},\quad&(t,x)\in\Omega_{T},\\\
u_{2}(0,x)=0,\quad&x\in\Omega,\\\
u_{2}(t,x)=0,\quad&(t,x)\in\Sigma.\end{cases}$ (2.3)

###  2.1. Well-posedness:

Hereby, we describe the well-posedness of initial-boundary value problems
(2.2) and (2.3). We have the following existence and uniqueness result of
(2.2):

######  Theorem 2.2.

Let $T>0$. Assume that $q\in W^{1,\infty}(\Omega_{T})$, $\phi\in
H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$, and $f\in H^{1,2}_{0}(\Sigma)$. Then,
there exists a unique solution $u_{1}\in C(0,T;H^{1}(\Omega))\cap
C^{1}(0,T;L^{2}(\Omega))$ of the dynamical Schrödinger equation (2.2).
Moreover, $\partial_{\nu}u_{1}\in L^{2}(\Sigma)$ and the following estimate
holds:

$\displaystyle\left\|u_{1}(t,\cdot)\right\|_{H^{1}(\Omega)}+\left\|\partial_{\nu}u_{1}\right\|_{L^{2}(\Sigma)}\leq
C(\left\|\phi\right\|_{H^{2}(\Omega)}+\left\|f\right\|_{H^{1,2}(\Sigma)})$
(2.4)

with the positive constant
$C=C\big{(}T,\left\|q\right\|_{W^{1,\infty}(\Omega_{T})}\big{)}$.

The proof of Theorem 2.2 follows by decomposing the solution $u_{1}$ of (2.2)
as $u_{1}=v+w$, where $v$ and $w$ satisfy the following equations
respectively:

$\begin{aligned} \begin{cases}i\partial_{t}v+\Delta
v=0,\quad&(t,x)\in\Omega_{T}\\\ v(0,x)=0,\quad&x\in\Omega,\\\
v(t,x)=f(t,x),\quad&(t,x)\in\Sigma;\end{cases}\end{aligned}\qquad\qquad\begin{aligned}
\begin{cases}i\partial_{t}w+\Delta
w+q(t,x)w=-q(t,x)v,\quad&(t,x)\in\Omega_{T}\\\
w(0,x)=\phi(x),\quad&x\in\Omega,\\\
w(t,x)=0,\quad&(t,x)\in\Sigma.\end{cases}\end{aligned}$

Afterwards, it can be seen from [4, Appendix A] that

$\displaystyle v\in C^{1}(0,T;H^{1}(\Omega))\quad\text{and
}\quad\left\|v\right\|_{C^{1}(0,T;H^{1}(\Omega))}\leq
C\left\|f\right\|_{H^{1,2}(\Sigma)}.$

In addition, we also have

$\displaystyle\partial_{\nu}v\in L^{2}(\Sigma)\quad\text{and
}\quad\left\|\partial_{\nu}v\right\|_{L^{2}(\Sigma)}\leq
C\left\|f\right\|_{H^{1,2}(\Sigma)}.$

Furthermore, following the argument from [7, Theorem 1.1] we can show that

$\displaystyle w\in C(0,T;H^{1}_{0}(\Omega)\cap H^{2}(\Omega))\cap
C^{1}(0,T;L^{2}(\Omega))\quad\text{and
}\quad\left\|w(t,\cdot)\right\|_{H^{1}_{0}(\Omega)}\leq
C(\left\|\phi\right\|_{H^{2}(\Omega)}+\left\|f\right\|_{H^{1,2}(\Sigma)})$

along with

$\displaystyle\partial_{\nu}w\in L^{2}(\Sigma)\quad\text{and
}\quad\left\|\partial_{\nu}w\right\|_{L^{2}(\Sigma)}\leq
C(\left\|\phi\right\|_{H^{2}(\Omega)}+\left\|f\right\|_{H^{1,2}(\Sigma)}).$

Hence the estimate (2.4) is obtained by combining the above estimates.
Furthermore, we have the following regularity theorem.

######  Theorem 2.3.

Let $T>0$ and $q\in C^{\infty}(\Omega_{T})$. Assume that $\phi\in
H^{r}_{0}(\Omega)\cap H^{r+1}(\Omega)$ and $f\in H_{0}^{r,r+1}(\Sigma)$ with
$r\in\mathbb{N}$. Then the equation (2.2) admits a unique solution

$u_{1}\in\displaystyle\bigcap_{k=0}^{r}C^{k}(0,T;H^{r-k}(\Omega)).$

The proof of regularity results presented in Theorem 2.3 follows from the
decomposition of $u$ into $v$ and $w$ as above. Afterwards, one can refer to
[36, Theorem 12.1] in order to obtain the required regularity for $w$, and
similar approach as in [4, Lemma A.4] for the regularity of $v$.

To demonstrate the well-posedness of IBVP (2.3), we define

$\displaystyle H^{r,s}(\Omega_{T})=L^{2}(0,T;H^{r}(\Omega))\cap
H^{s}(0,T;L^{2}(\Omega))$

equipped with the norm

$\displaystyle\left\|v\right\|_{H^{r,s}(\Omega_{T})}=\left\|v\right\|_{L^{2}(0,T;H^{r}(\Omega))}+\left\|v\right\|_{H^{s}(0,T;L^{2}(\Omega))}.$

We state the following result for general non-homogeneous (source) term
$F=F(t,x)$ (cf. [30, Lemma 2.3])

######  Theorem 2.4.

Consider the following IBVP:

$\displaystyle\begin{cases}i\partial_{t}v(t,x)+\Delta
v(t,x)+q(t,x)v(t,x)=F(t,x),&\quad(t,x)\in\Omega_{T},\\\ v(0,x)=0,&\quad
x\in\Omega,\\\ v(t,x)=0,&\quad(t,x)\in\Sigma\end{cases}$ (2.5)

with $q\in W^{1,\infty}(\Omega_{T})$ and $F\in H^{1}(0,T;L^{2}(\Omega))$, with
$F(0,\cdot)=0$ a.e. in $\Omega$. Then there exists a unique solution

$v\in C(0,T;H^{1}_{0}(\Omega)\cap H^{2}(\Omega))\cap C^{1}(0,T;L^{2}(\Omega))$

of (2.5). In addition, there holds

$\displaystyle\left\|v\right\|_{C(0,T;H^{2}(\Omega))}+\left\|v\right\|_{C^{1}(0,T;L^{2}(\Omega))}\leq
C\left\|F\right\|_{H^{1}(0,T;L^{2}(\Omega))}$ (2.6)

with the positive constant
$C=C(T,|\Omega|,\left\|q\right\|_{W^{1,\infty}(\Omega_{T})})$.

In a similar manner as in Theorem 2.4, we can deduce the following regularity
result:

######  Theorem 2.5.

Suppose that $q\in C^{\infty}(\Omega_{T})$ and $F\in H^{r-1,r}(\Omega_{T})$,
$r\in\mathbb{N}$, and $F$ satisfies the compatibility condition of order $r-1$
at $t=0$ a.e. in $\Omega$. Then there exists a unique solution

$v\in\displaystyle\bigcap_{k=0}^{r}C^{k}(0,T;H_{0}^{r-k}(\Omega))$

of (2.5).

The proof of Theorem 2.5 follows from [36, Theorem 12.2] using parabolic
regularization of (2.5). One can also refer to [30, Lemma 2.3] in which Faedo-
Galerkin technique is applied. As a consequence, well-posedness of the IBVP
(2.3) follows from Theorem 2.5 under the appropriate assumptions mentioned
therein.

######  Corollary 2.6.

Under the hypothesis of Theorem 2.2 and Theorem 2.4, equation (2.3) possess an
unique solution $u_{2}$ and moreover, $\partial_{\nu}u_{2}\in L^{2}(\Sigma)$
with the estimate

$\displaystyle\left\|\partial_{\nu}u_{2}\right\|_{L^{2}(\Sigma)}\leq
C(\left\|\phi\right\|_{H^{1}_{0}(\Omega)\cap
H^{2}(\Omega)}+\left\|f\right\|_{H^{1,2}(\Sigma)}).$

To demonstrate the above estimate, one can follow the similar approach
presented in [7, Appendix A].

######  Remark 2.7.

The DN-map $\Lambda_{q,\overrightarrow{J}}$ defined in (1.5) is well-defined
under the hypothesis of Theorem 2.2 and 2.4. Furthermore, DN-map acts as a
bounded linear map from $H^{2}(\Omega)\times H^{1,2}(\Sigma)$ to
$H^{1}(\Omega)\times L^{2}(\Sigma)$.

Hereby, our aim is to prove Theorem 2.1 through the following Proposition 2.8
and several lemmas described below.

######  Proposition 2.8.

Let us consider that the solution $u(x,t)$ of (1.1) assumes the form

$\displaystyle
u(t,x)=\varepsilon\big{\\{}u_{1}(t,x)+\varepsilon\big{(}u_{2}(t,x)+w(t,x)\big{)}\big{\\}},$
(2.7)

where $u_{1}$ and $u_{2}$ are the solutions to the IBVP (2.2) and (2.3)
respectively. Consequently, we look to derive the equation for $w(t,x)$. Then
$w(t,x)$ satisfies

$\displaystyle\begin{cases}i\partial_{t}w-A(w)w=\varepsilon\overrightarrow{G}(t,x,\varepsilon\nabla
w)\cdot\nabla w+\varepsilon F(t,x,\nabla u_{1},\nabla
u_{2}),\quad(t,x)\in\Omega_{T},\\\ w(0,x)=0,\quad x\in\Omega,\\\
w(t,x)=0,\quad(t,x)\in\partial\Omega_{T};\end{cases}$ (2.8)

where the following notations are being incorporated:

$\displaystyle\begin{cases}A(v)w:=-\Delta w-q(t,x)w+\varepsilon E(t,x,\nabla
v)\cdot\nabla^{2}w;\\\ \varepsilon E(t,x,\nabla
w):=\Big{[}2\varepsilon(\overrightarrow{b}\otimes\nabla
u_{1})+2\varepsilon^{2}(\overrightarrow{b}\otimes\nabla
w)+2\varepsilon^{2}(\overrightarrow{b}\otimes\nabla u_{2})+\varepsilon
K(t,x,\varepsilon\nabla w)+\varepsilon^{2}\mathcal{K}(t,x,\varepsilon\nabla
w)\nabla w\Big{]};\\\ \varepsilon F(t,x,\nabla u_{1},\nabla
u_{2}):=2\varepsilon\nabla\cdot((\nabla u_{1}\cdot\nabla
u_{2})\overrightarrow{b})+\varepsilon^{2}\nabla\cdot(|\nabla
u_{2}|^{2}\overrightarrow{b})+\frac{1}{\varepsilon^{2}}\nabla\cdot\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2});\\\
\varepsilon\overrightarrow{G}(t,x,\nabla v)\cdot\nabla
w:=2\varepsilon(\nabla\cdot\overrightarrow{b})(\nabla u_{1}\cdot\nabla
w)+2\varepsilon\nabla^{2}u_{1}\cdot(\overrightarrow{b}\otimes\nabla
w)+\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla v\cdot\nabla w)\\\
\qquad\qquad\qquad\qquad\qquad+2\varepsilon^{2}\nabla^{2}u_{2}\cdot(\overrightarrow{b}\otimes\nabla
w)+2\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla u_{2}\cdot\nabla
w)+\varepsilon(\nabla\cdot K(t,x,\varepsilon\nabla v))\cdot\nabla
w.\end{cases}$

###### Proof.

Since we seek for a solution $u(t,x)$ of (1.1) of the form (2.7), we
substitute $u(t,x)$ from (2.7) in (1.1) to obtain

$\displaystyle\begin{cases}i\partial_{t}w(t,x)+\Delta
w(t,x)+q(t,x)w(t,x)=\frac{1}{\varepsilon^{2}}\nabla\cdot\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\varepsilon^{2}w)\\\
\qquad+2\varepsilon\nabla\cdot\big{(}(\nabla u_{1}\cdot\nabla
u_{2})\overrightarrow{b}\big{)}+2\varepsilon\nabla\cdot\big{(}(\nabla
u_{1}\cdot\nabla w)\overrightarrow{b}\big{)}\\\
\qquad+\varepsilon^{2}\nabla\cdot\Big{(}\big{(}|\nabla u_{2}|^{2}+|\nabla
w|^{2}+2\nabla u_{2}\cdot\nabla
w\big{)}\overrightarrow{b}\Big{)},\qquad(t,x)\in\Omega_{T},\\\ w(0,x)=0,\quad
x\in\Omega,\\\ w(t,x)=0,\quad(t,x)\in\partial\Omega_{T};\end{cases}$ (2.9)

where we have taken into account (2.2) and (2.3). We re-write the equation for
$w(t,x)$ in (2.9) as

$\displaystyle i\partial_{t}w(t,x)+\Delta
w(t,x)+q(t,x)w(t,x)=T_{1}+T_{2}+T_{3}+T_{4},$

where $T_{i},~{}i=1,2,3,4$ represent the right side terms in the same order.
As a result,

$\displaystyle T_{1}:$
$\displaystyle=\frac{1}{\varepsilon^{2}}\nabla\cdot\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\varepsilon^{2}w)$
$\displaystyle=\frac{1}{\varepsilon^{2}}\nabla\cdot\Big{(}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla
u_{2})+\int_{0}^{1}\frac{d}{d\theta}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla
w)~{}d\theta\Big{)}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\nabla\cdot\Big{(}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2})+\varepsilon^{3}K(t,x,\varepsilon\nabla
w)\nabla w\Big{)},$

where

$\displaystyle\varepsilon K(t,x,\varepsilon\nabla
w;\varepsilon):=\int_{0}^{1}\nabla_{\xi}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla w)~{}d\theta$

with $\nabla_{\xi}\overrightarrow{R}(t,x,\xi)=\partial_{\xi_{j}}R_{i},~{}1\leq
i,j\leq n$ and $K=(K_{ij})=(\partial_{\xi_{j}}R_{i})$. Proceeding further

$\displaystyle T_{1}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\nabla\cdot\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2})+\varepsilon K(t,x,\varepsilon\nabla
w)\cdot\nabla^{2}w+\varepsilon^{2}\mathcal{K}(t,x,\varepsilon\nabla w)\nabla
w\cdot\nabla^{2}w+\varepsilon(\nabla\cdot K)\cdot\nabla w,$

where $\nabla^{2}w$ represents the _Hessian_ of $w$ and

$\displaystyle(\nabla\cdot
K)_{j}=\sum_{i=1}^{n}\partial_{i}K_{ij},\quad(\mathcal{K}\nabla
w)_{ij}=\sum_{l=1}^{n}\partial_{\xi_{j}}K_{il}\partial_{l}w.$

$T_{2}$ remains as it is. By denoting the tensor product as $\otimes$, we
obtain

$\displaystyle T_{3}$
$\displaystyle=2\varepsilon\Big{[}(\overrightarrow{b}\otimes\nabla
u_{1})\cdot\nabla^{2}w+\nabla^{2}u_{1}\cdot(\overrightarrow{b}\times\nabla
w)+(\nabla\cdot\overrightarrow{b})(\nabla u_{1}\cdot\nabla w)\Big{]}.$
$\displaystyle T_{4}$ $\displaystyle=\varepsilon^{2}\Big{[}\nabla\cdot(|\nabla
u_{2}|^{2}\overrightarrow{b})+(\nabla\cdot\overrightarrow{b})(\nabla\cdot\nabla
w)+(\overrightarrow{b}\otimes\nabla
u_{2})\cdot\nabla^{2}w+\nabla^{2}u_{2}\cdot(\overrightarrow{b}\otimes\nabla
w)$ $\displaystyle\qquad+(\nabla\cdot\overrightarrow{b})(\nabla w\cdot\nabla
w)+2(\overrightarrow{b}\otimes\nabla w)\cdot\nabla^{2}w\Big{]}.$

With this, the equation (2.9) can be rewritten as

$\displaystyle\begin{cases}i\partial_{t}w-A(w)w=\varepsilon\overrightarrow{G}(t,x,\varepsilon\nabla
w)\cdot\nabla w+\varepsilon F(t,x,\nabla u_{1},\nabla
u_{2}),\quad&(t,x)\in\Omega_{T},\\\ w(0,x)=0,\quad&x\in\Omega,\\\
w(t,x)=0,\quad&(t,x)\in\partial\Omega_{T};\end{cases}$ (2.10)

in which the notations $A(w)w,\varepsilon E(t,x,\nabla w),\varepsilon
F(t,x,\nabla u_{1},\nabla u_{2})$ and
$\varepsilon\overrightarrow{G}(t,x,\nabla w)\cdot\nabla w$, described in the
proposition 2.8, are incorporated. Hence the result follows. ∎

###  2.2. Well-posedness for the semilinear Schrödinger equation

Our aim is to demonstrate the well-posedness of the IBVP (2.10). In this
regard, we consider the following semi-linear equation corresponding to
(2.10):

$\displaystyle\begin{cases}i\partial_{t}w^{sem}-A(U)w^{sem}=\varepsilon\mathcal{G}(t,x,\nabla
w^{sem}),\qquad(t,x)\in\Omega_{T},\\\ w^{sem}(0,x)=0,\quad x\in\Omega,\\\
w^{sem}(t,x)=0,\quad(t,x)\in\partial\Omega_{T};\end{cases}$ (2.11)

where $\mathcal{G}(t,x,\nabla w^{sem})$ is defined by

$\displaystyle\mathcal{G}(t,x,\nabla w^{sem}):=F(t,x,\nabla u_{1},\nabla
u_{2})+\overrightarrow{G}(t,x,\nabla w^{sem})\cdot\nabla w^{sem}$ (2.12)

and $U\in Z(M)$, where the set $Z(M)$ is defined as follows:

$\displaystyle Z(M):=\big{\\{}U\in
X_{r}([0,T]):U(0,x)=0,~{}x\in\Omega;U(t,x)=0,~{}(t,x)\in\partial\Omega_{T};\left\|U\right\|_{X_{r}}\leq
M\big{\\}}$ (2.13)

with $M>0$. For a given function $U(t)$, we prove the unique solvability of
the dynamical semilinear Schrödinger equation (2.11) associated with zero
initial and boundary data. Finally, by proving the map $T(U)=w^{sem}$ is a
contraction, we establish the unique solvability of (2.10).

We start by introducing the operator $L$ for a given function $U(t)$

$\displaystyle Lw:=i\partial_{t}w-A(U)w.$ (2.14)

In the following lemma, we derive the coercivity and continuous dependence
estimates involving the operator $A(U)(\cdot)$, which play an important role
in establishing the well-posedness of (2.10).

######  Lemma 2.9.

(Properties of the operator $A(U)(\cdot)$)
Let us consider that $U\in Z(M)$ and $\varepsilon\in[0,\varepsilon_{0}]$ for a
fixed $\varepsilon_{0}>0$. Then $A(U)w$ has the following properties:

  1. (1)

For any $w\in H_{0}^{1}(\Omega)\cap H^{k+1}(\Omega)$ and $t\in[0,T]$,

$\displaystyle\left\|w\right\|_{H^{k+1}(\Omega)}\leq
C_{1}(\left\|w\right\|_{H^{k-1}(\Omega)}+\left\|A(U)w\right\|_{H^{k-1}(\Omega)}),\quad
k=1,2,\dots,m-1,$ (2.15)

where $C_{1}$ is a positive constant.

  2. (2)

Coercivity estimate: There exist positive constants $C_{b}$ and $C_{e}$ such
that

$\displaystyle-\langle
A(U)w,w\rangle+C_{b}\left\|w\right\|^{2}_{L^{2}(\Omega)}\geq
C_{e}\left\|w\right\|^{2}_{H^{1}(\Omega)},\quad t\in[0,T],$ (2.16)

where $w\in H^{1}_{0}(\Omega)$.

  3. (3)

Continuous dependence: Let $U_{1},U_{2}\in H^{1}(\Omega)$ with
$\left\|U_{1}\right\|_{H^{1}(\Omega)},\left\|U_{2}\right\|_{H^{1}(\Omega)}\leq
M$. There holds:

$\displaystyle\left\|\big{(}A(U_{1})-A(U_{2})\big{)}w\right\|_{L^{2}(\Omega)}\leq\varepsilon\sigma(M,\varepsilon)\cdot\left\|\nabla
U_{1}-\nabla U_{2}\right\|_{L^{2}(\Omega)},\quad t\in[0,T],~{}w\in Z(M);$
(2.17)

where $\sigma:\mathbb{R}^{+}\times\mathbb{R}^{+}\mapsto\mathbb{R}^{+}$ is a
continuous function and $\mathbb{R}^{+}$ is the set of non-negative real
numbers.

###### Proof.

For the properties $(1)$ and $(2)$, one can use the elliptic regularity
argument [39]. For the property $(3)$, we have

$\displaystyle\left\|A(U_{1})w-A(U_{2})w\right\|_{L^{2}(\Omega)}$
$\displaystyle=\left\|\varepsilon(E(t,x,\nabla U_{1})-E(t,x,\nabla
U_{2}))\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$ $\displaystyle\leq
2\varepsilon^{2}\left\|(\overrightarrow{b}\otimes\nabla
U_{1})\cdot\nabla^{2}w-(\overrightarrow{b}\otimes\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$
$\displaystyle\qquad+\varepsilon\left\|K(t,x,\nabla
U_{1})\cdot\nabla^{2}w-K(t,x,\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$
$\displaystyle\qquad+\varepsilon^{2}\left\|(\mathcal{K}(t,x,\nabla
U_{1})\nabla U_{1})\cdot\nabla^{2}w-(\mathcal{K}(t,x,\nabla U_{2})\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

We obtain the following estimates on $I_{1},I_{2}$ and $I_{3}$:

$\displaystyle I_{1}$
$\displaystyle=2\varepsilon^{2}\left\|(\overrightarrow{b}\otimes\nabla
U_{1})\cdot\nabla^{2}w-(\overrightarrow{b}\otimes\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$ $\displaystyle\leq
2\varepsilon^{2}\|{\overrightarrow{b}}\|_{L^{\infty}(\Omega)}\left\|w\right\|_{L^{2}(\Omega)}\left\|\nabla(U_{1}-U_{2})\right\|_{L^{2}(\Omega)};$
$\displaystyle I_{2}$ $\displaystyle=\varepsilon\left\|K(t,x,\varepsilon\nabla
U_{1})\cdot\nabla^{2}w-K(t,x,\varepsilon\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$
$\displaystyle=\varepsilon\left\|\bigg{(}\int_{0}^{1}\big{\\{}\nabla_{\xi}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla
U_{1})-\nabla_{\xi}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla
U_{2})\big{\\}}~{}d\theta\cdot\nabla^{2}w\bigg{)}\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C_{1}(M,\varepsilon)\left\|\nabla(U_{1}-U_{2})\right\|_{L^{2}(\Omega)}^{2};$
$\displaystyle I_{3}$
$\displaystyle=\varepsilon^{2}\left\|(\mathcal{K}(t,x,\nabla U_{1})\nabla
U_{1})\cdot\nabla^{2}w-(\mathcal{K}(t,x,\nabla U_{2})\nabla
U_{2})\cdot\nabla^{2}w\right\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C_{2}(M,\varepsilon)\left\|\nabla(U_{1}-U_{2})\right\|_{L^{2}(\Omega)},$

where we have take into account the estimate (1.4) and the constants $C_{1}$
and $C_{2}$ depend on $M$ and $\varepsilon$ and finally, in the estimates
$I_{2}$ and $I_{3}$, we have incorporated
$\left\|U_{i}\right\|_{H^{1}(\Omega)}\leq M,~{}i=1,2$. ∎

We seek an appropriate estimate of the function $\mathcal{G}(t,x,\nabla w)$.
The resulting estimate will be required to analyze the well-posedness of
(2.10). In the following lemma, we analyze the estimate of
$\mathcal{G}(t,x,\nabla w)$:

######  Lemma 2.10.

[Estimate of $\mathcal{G}(t,x,\nabla w)$]
Let us assume that $r\geq\big{[}\frac{n}{2}\big{]}+3$. Then for any $w\in
Z(M)$, there holds

$\displaystyle\left\|\mathcal{G}(t,x,\nabla w)\right\|_{H^{r-1}(\Omega)}\leq
C\Big{(}1+\left\|w\right\|^{r}_{H^{r-1}(\Omega)}\Big{)},\quad t\in[0,T],$
(2.18)

where the constant $C>0$ is depending only on $M$.

To demonstrate the estimate (2.18) in Lemma 2.10, the following result will be
quite instrumental.

######  Lemma 2.11.

([39, Lemma 2.6])
Let $r\in\mathbb{Z}$ such that $r\geq\big{[}\frac{n}{2}\big{]}+3$ and
$\kappa>0$ be the Sobolev embedding

$H^{[n/2]+1}\hookrightarrow C^{0}(\overline{\Omega})$

constant. For a given $C^{r-1}$ function $f(t,x;z)$ on
$\mathcal{D}=\\{(t,x,z)\in[0,T]\times\Omega\times\mathbb{C}:|z|\leq\kappa
M\\}$, we have

$\displaystyle\left\|f(t,x;z)\right\|_{H^{r-1}(\Omega)}\leq
C_{r-1}M_{r-1}\big{\\{}1+(1+\left\|z(t)\right\|^{r-2}_{H^{r-2}(\Omega)}\left\|z(t)\right\|_{H^{r-1}(\Omega)}\big{\\}},\quad
t\in[0,T],$ (2.19)

where the constant $C_{r-1}$ is depending on $r-1$ and

$\displaystyle M_{r-1}:=\max_{|\beta|\leq
r-1}\sup_{\mathcal{D}}\Big{|}\Big{(}\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}},\dots,\frac{\partial}{\partial
x_{n}},\frac{\partial}{\partial z}\Big{)}^{\beta}f(t,x;z)\Big{|}.$

###### Proof of the Lemma 2.10.

By the Lemma 2.11 and the fact that $\partial^{\alpha}_{x}w\in X_{p-|\alpha|}$
for $w\in X_{p}$ for $p\in\mathbb{N}\cup\\{0\\}$ and multi-index $|\alpha|\leq
p$, we obtain

$\displaystyle\left\|\mathcal{G}(t,\cdot,\nabla
w)\right\|_{H^{r-1}(\Omega)}\leq
C_{r-1}M_{r-1}\Big{\\{}1+\big{(}1+\left\|w(t)\right\|^{r-2}_{H^{r-2}(\Omega)}\big{)}\left\|w(t)\right\|_{H^{r-1}(\Omega)}\Big{\\}}.$

Further, we observe that

$\displaystyle
1+\Big{(}1+\left\|w(t)\right\|^{r-2}_{H^{r-2}(\Omega)}\Big{)}\left\|w(t)\right\|_{H^{r-1}(\Omega)}$
$\displaystyle\leq
1+\Big{(}1+\left\|w(t)\right\|^{r-2}_{H^{r-2}(\Omega)}\Big{)}\Big{(}1+\left\|w(t)\right\|_{H^{r-1}(\Omega)}\Big{)}$
$\displaystyle=1+\Big{(}1+\left\|w(t)\right\|_{H^{r-1}(\Omega)}\Big{)}^{r-1}$
$\displaystyle\leq
C\Big{(}1+\left\|w(t)\right\|_{H^{r-1}(\Omega)}^{r-1}\Big{)},$

where $C$ is a positive constant. Hence the result follows. ∎

######  Theorem 2.12.

Let $r\in\mathbb{N}$ such that $r\geq\big{[}\frac{n}{2}\big{]}+3$. Then there
exists $\varepsilon_{1}>0$ such that semilinear IBVP (2.11) has a unique
solution $w^{sem}\in Z(M)$ for each $\varepsilon\in(0,\varepsilon_{1})$.
Furthermore,

$\displaystyle\left\|w^{sem}\right\|_{X_{r}}\leq\varepsilon
C(M,\varepsilon_{1})\exp(KT),\quad\varepsilon\in(0,\varepsilon_{1}),$ (2.20)

where $K=K(M,\varepsilon)$ is a positive constant.

To demonstrate the result in Theorem 2.12, we require a suitable energy
estimate which will be provided by the following lemma (cf. [39, Section 2]):

######  Lemma 2.13.

Consider the IBVP:

$\displaystyle\begin{cases}L[v]=S,\quad(t,x)\in\Omega_{T},\\\ v(0,x)=0,\quad
x\in\Omega,\\\ v(t,x)=0,\quad(t,x)\in\partial\Omega_{T},\end{cases}$ (2.21)

where the operator $L$ is defined by (2.14) and $S\in X_{r-1}$. Moreover, we
assume that $S$ satisfies the compatibility condition of order $r-1$ at $t=0$
a.e. in $\Omega$. Then

  1. (i)

there exists a unique solution $v\in X_{r}$ to (2.21) ;

  2. (ii)

there holds the following energy estimate:

$\displaystyle\left\|v(t)\right\|^{2}_{H^{r}(\Omega)}\leq
C_{r}\int_{0}^{T}\left\|S(t)\right\|^{2}_{H^{r-1}(\Omega)}~{}dt,\quad
t\in[0,T],$ (2.22)

where the general constant $C_{r}$ depending on $r$.

###### Proof of Theorem 2.12.

Existence of solution (2.11): We represent the equation (2.11) as follows:

$\displaystyle\begin{cases}L[w^{sem}]=\varepsilon\mathcal{G}(t,x,\nabla
w^{sem}),\quad&(t,x)\in Q_{T},\\\ w^{sem}(0,x)=0,\quad&x\in\Omega,\\\
w^{sem}(t,x)=0,\quad&(t,x)\in\partial\Omega_{T}.\end{cases}$ (2.23)

We introduce the sequence of functions $\\{w^{sem}_{j}\\}$ by

$\displaystyle L[w^{sem}_{1}]$ $\displaystyle=\varepsilon\mathcal{G}(t,x,0);$
$\displaystyle L[w^{sem}_{2}]$
$\displaystyle=\varepsilon\mathcal{G}(t,x,\nabla w^{sem}_{1});$
$\displaystyle\vdots$ $\displaystyle L[w^{sem}_{j}]$
$\displaystyle=\varepsilon\mathcal{G}(t,x,\nabla
w^{sem}_{j-1}),~{}j=2,3,\cdots.$

Our aim is to show that for any small $\varepsilon>0$,

$\displaystyle w^{sem}_{j}\in X_{r}~{}\text{ and
}~{}\sup_{t\in[0,T]}\|w^{sem}_{j}\|_{H^{r}(\Omega)}\leq M,\quad
j=1,2,3,\cdots.$

Since it follows from (2.12) that

$\displaystyle\mathcal{G}(t,x,0)=F(t,x,\nabla u_{1},\nabla u_{2})$

and as a consequence, for $\varepsilon$ small enough

$\displaystyle\sup_{t\in[0,T]}\left\|w^{sem}_{1}(t)\right\|_{H^{r}(\Omega)}\leq
M.$

Furthermore, due to the following fact: if

$\displaystyle\sup_{t\in[0,T]}\left\|w^{sem}_{j-1}(t)\right\|_{H^{r}(\Omega)}\leq
M,$

then we have for $t\in[0,T]$

$\displaystyle\left\|w^{sem}_{j}(t)\right\|_{H^{r}(\Omega)}^{2}$
$\displaystyle\leq\varepsilon^{2}\int_{0}^{T}\left\|\mathcal{G}(t,x,\nabla
w^{sem}_{j-1})\right\|_{H^{r-1}(\Omega)}^{2}~{}dt$
$\displaystyle\leq\varepsilon^{2}C\int_{0}^{T}\Big{(}1+\left\|w^{sem}_{j-1}(t)\right\|_{H^{r}(\Omega)}^{r-1}\Big{)}^{2}~{}dt,$

and subsequently, if we choose $\varepsilon$ small enough such that it
satisfies

$\displaystyle\varepsilon\leq\min\bigg{\\{}\frac{M}{\sqrt{CT}(1+M^{r-1})},\frac{1}{2}\bigg{\\}},$

then it results into (by using induction principle)

$\displaystyle\sup_{t\in[0,T]}\left\|w^{sem}_{j}(t)\right\|_{H^{r}(\Omega)}\leq
M,\quad j\geq 2.$

Hence $\\{w_{j}\\}$ is bounded and observe that

$\displaystyle L[w^{sem}_{j+1}-w^{sem}_{j}]$
$\displaystyle=\varepsilon\Big{\\{}\mathcal{G}(t,x,\nabla
w^{sem}_{j})-\mathcal{G}(t,x,\nabla w^{sem}_{j-1})\Big{\\}}$
$\displaystyle=\varepsilon\bigg{\\{}\int_{0}^{1}\nabla_{q}\mathcal{G}\Big{(}t,x,\nabla
w^{sem}_{j-1}+\theta\nabla(w^{sem}_{j}-w^{sem}_{j-1})\Big{)}~{}d\theta\bigg{\\}}\cdot\nabla(w^{sem}_{j}-w^{sem}_{j-1}).$

Thanks to the Lemma 2.13 and Lemma 2.10, we have

$\displaystyle\left\|w^{sem}_{j+1}(t)-w^{sem}_{j}(t)\right\|_{H^{r}(\Omega)}^{2}\leq
C\varepsilon^{2}\int_{0}^{T}\left\|w^{sem}_{j}(s)-w^{sem}_{j-1}(s)\right\|_{H^{r}(\Omega)}^{2}~{}ds,\quad
t\in[0,T],$

and subsequqently, by the choice of $\varepsilon$, it is evident that
$\\{w^{sem}_{j}(t)\\}$ is a Cauchy sequence with respect to the norm
$\sup_{t\in[0,T]}\left\|\cdot\right\|_{H^{r}(\Omega)}$. Let $w^{sem}(t)$ be
the limit of the Cauchy sequence $\\{w^{sem}_{j}(t)\\}$. As a consequence, by
standard regularity argument, we can demonstrate that $w^{sem}\in X_{r}$ and
$w^{sem}$ satisfies the equation (2.23). Hence the existence of solution of
(2.23) follows.

Uniqueness of the solution (2.11): We prove the estimate (2.20). We fix
$t\in(0,T)$ and we consider the equation (2.23) by taking inner product with
$w^{sem}(t,\cdot)$ to obtain

$\begin{split}i\Big{\langle}\partial_{t}w^{sem}(t,\cdot),w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}-\Big{\langle}A(U)w^{sem}(t,\cdot),&w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}\\\
&=\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}.\end{split}$ (2.24)

Taking the imaginary part of both sides in the above identity we obtain

$\displaystyle\frac{d}{ds}\left\|w^{sem}(s,\cdot)\right\|^{2}_{L^{2}(\Omega)}=2\text{
Im }\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(s,\cdot)\Big{\rangle}_{L^{2}(\Omega)},\qquad s\in(0,T).$

Integrating the above identity over $(0,t)$ resulting into

$\begin{split}\left\|w^{sem}(t,\cdot)\right\|^{2}_{L^{2}(\Omega)}&=2\text{ Im
}\int_{0}^{t}\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(s,\cdot)\Big{\rangle}_{L^{2}(\Omega)}~{}ds\\\ &\leq
2\int_{0}^{t}\Big{|}\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(s,\cdot)\Big{\rangle}_{L^{2}(\Omega)}\Big{|}~{}ds\end{split}$
(2.25)

since $w^{sem}(0,\cdot)=0$. Taking the real part in equation (2.24) we get

$\displaystyle\Big{\langle}A(U)w^{sem}(t,\cdot),w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}=-\text{
Re }\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}.$

Thanks to the coercivity estimate (2.16), we have the following inequality

$\displaystyle\left\|w^{sem}(t,\cdot)\right\|^{2}_{H^{1}(\Omega)}\leq
C\Big{|}\Big{\langle}\varepsilon F(t,x,\nabla
w^{sem}),w^{sem}(t,\cdot)\Big{\rangle}_{L^{2}(\Omega)}\Big{|}.$ (2.26)

The quadratic term $|\nabla w^{sem}(s,\cdot)|^{2}$ present in $F$ is estimated
using Sobolev embedding as follows: for any $s\in[0,T]$,

$\begin{split}\int_{\Omega}|\nabla
w^{sem}(s)|^{2}|w(s)|~{}dx&\leq\sup_{\Omega}|\nabla
w^{sem}(s)|\cdot\left\|w^{sem}(s)\right\|_{L^{2}(\Omega)}\cdot\left\|\nabla
w^{sem}(s)\right\|_{L^{2}(\Omega)}\\\ &\leq
C\left(\left\|w^{sem}(s)\right\|^{2}_{L^{2}(\Omega)}+\left\|w^{sem}(s)\right\|^{2}_{H^{1}(\Omega)}\right).\end{split}$
(2.27)

Combining the estimates (2.25), (2.26), (2.27) and Lemma 2.9 yield the
following estimate for sufficiently small $\varepsilon>0$

$\displaystyle\left\|w^{sem}(t,\cdot)\right\|^{2}_{H^{1}(\Omega)}+\left\|w^{sem}(t,\cdot)\right\|^{2}_{L^{2}(\Omega)}\leq
C_{1}\varepsilon^{2}+C_{2}^{\varepsilon}\int_{0}^{t}\left(\left\|w^{sem}(s,\cdot)\right\|^{2}_{H^{1}(\Omega)}+\left\|w^{sem}(s,\cdot)\right\|^{2}_{L^{2}(\Omega)}\right)~{}ds$
(2.28)

for any $t\in[0,T]$. In (2.28), $C_{1}$ is a general positive constant and the
constant $C_{2}^{\varepsilon}$ is also positive. In addition,
$C_{2}^{\varepsilon}$ is bounded with respect to $\varepsilon$.

Afterwards, the Gronwall’s inequality along with the estimate (2.28) yield the
estimate (2.20). The uniqueness of the solution of (2.11) immediately follows
from the estimate (2.20). Hence the Theorem 2.12 follows. ∎

###### Proof of Theorem 2.1.

To establish the well-posedness described of the initial boundary value
problem (2.10) in the Proposition 2.8, we use the idea of fixed point
argument. We have observed in the Lemma 2.1 that given $U\in Z(M)$, there
exists a unique solution $w$ of (2.11) and $w\in Z(M)$ provided $\varepsilon$
is small enough. This fact ensures that the following mapping

$\displaystyle T:Z(M)\rightarrow Z(M)~{}\text{is defined by }T(U)=w,$

where $w$ is the solution of (2.11), is well-defined. We will prove that the
map $T$ is a contraction map.

Let $U_{1}$ and $U_{2}$ be such that $T(U_{i})=w_{i},~{}i=1,2$, where $w_{i}$
is the solution of semi-linear equation (2.11) for $U_{i}\in Z(M)$. Let us
define

$\displaystyle W:=w_{1}-w_{2},\qquad V:=U_{1}-U_{2}.$

Then the variable $W$ will satisfy the following initial boundary value
problem

$\begin{split}\begin{cases}&i\partial_{t}W-B(U_{1})W-(B(U_{1})-B(U_{2})w_{2}=\varepsilon\overrightarrow{G}(t,x,\nabla
w_{1})\cdot\nabla W\\\
&\qquad\qquad\qquad+\varepsilon\left(\overrightarrow{G}(t,x,\nabla
w_{1})-\overrightarrow{G}(t,x,\nabla w_{2})\right)\cdot\nabla
w_{2},\quad(t,x)\in\Omega_{T}\\\ &W(0,x)=0,\quad x\in\Omega,\\\
&W(t,x)=0,\quad(t,x)\in\partial\Omega_{T}.\end{cases}\end{split}$ (2.29)

Multiplying (2.29) by $2W$ and integrate over $[0,t]\times\Omega$, we obtain

$\displaystyle i\left\|W(t)\right\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=2\int_{0}^{t}\langle
B(U_{1})W,W\rangle~{}d\tau+2\int_{0}^{t}\Big{\langle}\left(B(U_{1})-B(U_{2})\right)w_{2},W\Big{\rangle}~{}d\tau$
$\displaystyle+2\varepsilon\int_{0}^{t}\Big{\langle}$
$\displaystyle\overrightarrow{G}(t,x,\nabla w_{1})\cdot\nabla
W,W\Big{\rangle}~{}d\tau+2\varepsilon\int_{0}^{t}\bigg{\langle}\left(\overrightarrow{G}(t,x,\nabla
w_{1})-\overrightarrow{G}(t,x,\nabla w_{2})\right)\cdot\nabla
w_{2},W\bigg{\rangle}~{}d\tau$ $\displaystyle=:I_{1}+I_{2}$
$\displaystyle+I_{3}+I_{4}.$

Considering the imaginary part on both sides, we have

$\displaystyle\left\|W(t)\right\|_{L^{2}(\Omega)}^{2}$ $\displaystyle=\text{Im
}(I_{1})+\text{Im }(I_{2})+\text{Im }(I_{3})+\text{Im }(I_{4})$
$\displaystyle\leq|I_{1}|+|I_{2}|+|I_{3}|+|I_{4}|.$

Since $w_{2}\in Z(M)$, the continuous dependence estimate in (2.17) implies

$\displaystyle\left\|\left(B(U_{1})-B(U_{2})\right)w_{2}\right\|_{L^{2}(\Omega)}\leq\varepsilon\sigma(M,\varepsilon)\left\|\nabla
V\right\|_{L^{2}(\Omega)}.$ (2.30)

As a consequence, the term $I_{2}$ can be estimated as follows:

$\displaystyle I_{2}$
$\displaystyle=2\int_{0}^{t}\Big{\langle}\left(B(U_{1})-B(U_{2})\right)w_{2},W\Big{\rangle}~{}d\tau$
$\displaystyle\leq
C\varepsilon^{2}\sup_{t\in[0,T]}\left\|V\right\|_{H^{1}(\Omega)}^{2}+2\int_{0}^{t}\left\|W(s)\right\|_{L^{2}(\Omega)}^{2}~{}ds.$

Afterwards, we shall estimate $I_{3}$ and $I_{4}$ as follows:

$\displaystyle I_{3}$
$\displaystyle=2\varepsilon\int_{0}^{t}\Big{\langle}\overrightarrow{G}(t,x,\nabla
w_{1})\cdot\nabla W,W\Big{\rangle}~{}d\tau$
$\displaystyle=2\int_{0}^{t}\bigg{\langle}2\varepsilon\nabla^{2}u_{1}\cdot(\overrightarrow{b}\otimes\nabla
W)+2\varepsilon(\nabla\cdot\overrightarrow{b})(\nabla u_{1}\cdot\nabla W)$
$\displaystyle\qquad\qquad+\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla
w_{1}\cdot\nabla
W)+2\varepsilon^{2}\nabla^{2}u_{2}\cdot(\overrightarrow{b}\otimes\nabla W)$
$\displaystyle\qquad\qquad+2\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla
u_{2}\cdot\nabla W)+\varepsilon(\nabla\cdot K)\cdot\nabla
W,W\bigg{\rangle}~{}d\tau$ $\displaystyle\leq
K_{\varepsilon}\int_{0}^{t}\Big{(}\left\|\nabla
W\right\|_{L^{2}(\Omega)}^{2}+\left\|W\right\|_{L^{2}(\Omega)}^{2}\Big{)}~{}d\tau,$

and

$\displaystyle I_{4}$
$\displaystyle=\Big{\langle}\left(\overrightarrow{G}(t,x,\nabla
w_{1})-\overrightarrow{G}(t,x,\nabla w_{2})\right)\cdot\nabla
w_{2},W\Big{\rangle}~{}d\tau$
$\displaystyle=2\varepsilon\int_{0}^{t}\Big{\langle}\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla
w_{1}\cdot\nabla w_{2})-\varepsilon^{2}(\nabla\cdot\overrightarrow{b})(\nabla
w_{2}\cdot\nabla w_{2})$
$\displaystyle\qquad\qquad\qquad+\varepsilon(\nabla\cdot K)\cdot\nabla
w_{2}-\varepsilon(\nabla\cdot K)\cdot\nabla w_{2},W\Big{\rangle}~{}d\tau$
$\displaystyle\leq
C\varepsilon^{2}+2\int_{0}^{t}\left\|W(s)\right\|_{L^{2}(\Omega)}^{2}~{}d\tau,$

where we have used the following estimate

$\displaystyle\nabla\cdot\Big{(}\varepsilon K(t,x,\varepsilon\nabla
w_{1})\Big{)}-\nabla\cdot\Big{(}\varepsilon K(t,x,\varepsilon\nabla
w_{2})\Big{)}$
$\displaystyle\quad=\nabla\cdot\left(\int_{0}^{1}\Big{\\{}\nabla_{\xi}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla
w_{1})-\nabla_{\xi}\overrightarrow{R}(t,x,\varepsilon\nabla
u_{1}+\varepsilon^{2}\nabla u_{2}+\theta\varepsilon^{2}\nabla
w_{2})\Big{\\}}~{}d\theta\right)$ $\displaystyle\quad\leq C\varepsilon^{2}.$

Combining the above estimates and using coercivity property, we have

$\displaystyle\left\|W(t)\right\|^{2}_{H^{1}(\Omega)}\leq\varepsilon^{2}C\sup_{t\in[0,T]}\left\|V(t)\right\|^{2}_{H^{1}(\Omega)}+K_{\varepsilon}\int_{0}^{t}\left\|W(\tau)\right\|^{2}_{H^{1}(\Omega)}~{}d\tau$
(2.31)

with a constant $K_{\varepsilon}>0$ bounded with respect to $\varepsilon$ and
a general constant $C>0$.

We introduce a metric $\rho$ in the space $Z(M)$ as follows:

$\displaystyle\rho(f,g):=\max_{t\in[0,T]}\big{\\{}\left\|f(t)-g(t)\right\|_{H^{1}(\Omega)}\big{\\}}.$

Finally using the Gronwall’s inequality, we have

$\displaystyle\rho(T(U_{1}),T(U_{2}))\leq\varepsilon
Ce^{K_{\varepsilon}T}\rho(U_{1},U_{2})$

with the constant $C>0$. As a result, we assert that the map $T$ is a
contraction mapping for $\varepsilon>0$ small enough. We conclude that for
each $\varepsilon>0$ small enough, there exists $w\in Z(M)$ such that

$\displaystyle T(w)=w,\quad\left\|w\right\|_{X_{m}}=O(\varepsilon)~{}\text{ as
}\varepsilon\rightarrow 0.$ (2.32)

Hence the Theorem 2.1 follows. ∎

##  3\. Proof of Theorem 1.1

In this section our aim is give the reconstruction formula for determining the
coefficents $q(t,x)$ and $\overrightarrow{b}(t,x)$ from the knowledge of
input-output map $\Lambda_{q,\overrightarrow{J}}(\varepsilon\phi,\varepsilon
f)$given for all $\phi$ and $f$. Based on the analysis presented in the
earlier section, we consider the $\varepsilon$-expansion of the solution to
(1.1) as

$\displaystyle u(t,x)=\varepsilon
u_{1}(t,x)+\varepsilon^{2}u_{2}(t,x)+O(\varepsilon^{3}),$ (3.1)

where $u_{1}(t,x)$ and $u_{2}(t,x)$ satisfy the equations (2.2) and (2.3)
respectively.

To begin with, we linearize the input-output map incorporating the
representation of $u$ in (3.1),

$\displaystyle\Lambda_{q,\overrightarrow{J}}(\varepsilon\phi,\varepsilon f)$
$\displaystyle=\left(u\Big{|}_{t=T},\Big{[}\partial_{\nu}u-\nu\cdot\overrightarrow{J}(t,x,\nabla_{x}u)\Big{]}\Big{|}_{\partial\Omega_{T}}\right)$
$\displaystyle=\bigg{(}\Big{[}\varepsilon
u_{1}+\varepsilon^{2}u_{2}+O(\varepsilon^{3})\Big{]}\Big{|}_{t=T},\partial_{\nu}\Big{[}\varepsilon
u_{1}+\varepsilon^{2}u_{2}+O(\varepsilon^{3})\Big{]}\Big{|}_{\partial\Omega_{T}}$
$\displaystyle\qquad\qquad\qquad\qquad-\Big{[}\nu\cdot\overrightarrow{J}\big{(}t,x,\varepsilon
u_{1}+\varepsilon^{2}u_{2}+O(\varepsilon^{3})\big{)}\Big{]}\Big{|}_{\partial\Omega_{T}}\bigg{)}$
$\displaystyle:=\varepsilon\overrightarrow{g_{1}}+\varepsilon^{2}\overrightarrow{g_{2}}+O(\varepsilon^{3}),$

where $\overrightarrow{g_{1}}$ and $\overrightarrow{g_{2}}$ are given by

$\displaystyle\overrightarrow{g_{1}}$
$\displaystyle:=\left(u_{1}\big{|}_{t=T},\partial_{\nu}u_{1}\big{|}_{\partial\Omega_{T}}\right)$
(3.2) $\displaystyle\overrightarrow{g_{2}}$
$\displaystyle:=\left(u_{2}\big{|}_{t=T},\big{[}\partial_{\nu}u_{2}-\nu\cdot\overrightarrow{b}(t,x)|\nabla_{x}u_{1}|^{2}\big{]}\big{|}_{\partial\Omega_{T}}\right).$
(3.3)

Now let us define

$\displaystyle\Lambda_{q}(\phi,f)$
$\displaystyle:=\left(u_{1}\big{|}_{t=T},\partial_{\nu}u_{1}\big{|}_{\partial\Omega_{T}}\right)=\mbox{known}$
(3.4) $\displaystyle\Lambda_{\overrightarrow{b}}(\phi,f)$
$\displaystyle:=\left(u_{2}\big{|}_{t=T},\big{[}\partial_{\nu}u_{2}-\nu\cdot\overrightarrow{b}(t,x)|\nabla_{x}u_{1}|^{2}\big{]}\big{|}_{\partial\Omega_{T}}\right)=\mbox{known}.$
(3.5)

Now that the map $\Lambda_{q}(\phi,f)$ defined by (3.4) is input-outpur map
for (2.2). Using the knowledge of $\Lambda_{q}(\phi,f)$ given for all
$(\phi,f)\in\big{(}H_{0}^{r}(\Omega)\cap H^{r+1}(\Omega)\big{)}\times
H_{0}^{r,r+1}(\Sigma)$, in Subsetcion 3.1, we give the reconstruction formula
for determining $q(t,x)$ in $\Omega_{T}$.

###  3.1. Reconstruction of electric potential

Our aim is to reconstruct $q(t,x)$ from $\Lambda_{q}(\phi,f)$ for all
$(\phi,f)\in\big{(}H_{0}^{r}(\Omega)\cap H^{r+1}(\Omega)\big{)}\times
H_{0}^{r,r+1}(\Sigma)$ and the equation (2.2). We multiply the equation (2.2)
by $v(t,x)$, where $v(t,x)$ is assumed to be the solution of

$\displaystyle-i\partial_{t}v+\Delta v=0$ (3.6)

and integrate over $\Omega_{T}$ to obtain

$\displaystyle\int_{0}^{T}\int_{\Omega}\big{[}i\partial_{t}u_{1}(t,x)+\Delta
u_{1}(t,x)+q(t,x)u_{1}(t,x)\big{]}v(t,x)~{}dxdt=0.$

Now integration by parts yields

$\displaystyle\int_{\Omega}i[u_{1}(T,x)v(T,x)-u_{1}(0,x)$ $\displaystyle
v(0,x)]~{}dx-\int_{0}^{T}\int_{\Omega}iu_{1}\partial_{t}v~{}dxdt+\int_{0}^{T}\int_{\Omega}u_{1}\nabla
v~{}dxdt$
$\displaystyle+\int_{0}^{T}\int_{\partial\Omega}(\partial_{\nu}u_{1}v-\partial_{\nu}vu_{1})~{}dsdt+\int_{0}^{T}\int_{\Omega}q(t,x)u_{1}v~{}dxdt=0$

which results into

$\begin{split}\int_{0}^{T}\int_{\Omega}q(t,x)u_{1}(t,x)v(t,x)~{}dxdt=&-\int_{\Omega}iu_{1}(T,x)v(T,x)~{}dx-\int_{\Omega}iu_{1}(0,x)v(0,x)]~{}dx\\\
&-\int_{0}^{T}\int_{\partial\Omega}\partial_{\nu}u_{1}(t,x)v(t,x)~{}dsdt-\int_{0}^{T}\int_{\partial\Omega}\partial_{\nu}v(t,x)u_{1}(t,x)~{}dsdt.\end{split}$
(3.7)

In the right hand side of (3.7), first and third terms are known from the
measurement whereas second and fourth terms are known thanks to the input
data. As a consequence,

$\displaystyle\int_{0}^{T}\int_{\Omega}q(t,x)u_{1}(t,x)v(t,x)~{}dxdt=\text{known},$
(3.8)

where $v(t,x)$ satisfies the PDE (3.6).

In order to reconstruct $q(t,x)$ from (3.8), we shall choose geometric optics
solution for $u_{1}$ and $v$.

Observe that

$\displaystyle(i\partial_{t}+\Delta)[e^{-i(\lambda^{2}t+\lambda
x\cdot\omega)}A_{1}(t,x)]=e^{-i(\lambda^{2}t+\lambda
x\cdot\omega)}\left\\{i\partial_{t}A_{1}+\Delta
A_{1}-2i\lambda\omega\cdot\nabla_{x}A_{1}\right\\},$

where $\omega$ is an unit vector. By equating coefficient of $\lambda$ to
zero, we obtain the transport equation

$\displaystyle\omega\cdot\nabla_{x}A_{1}=0.$ (3.9)

As a consequence, we seek the solution $u_{1}$ of the form

$\displaystyle u_{1}(t,x)=e^{-i(\lambda^{2}t+\lambda
x\cdot\omega)}\left[A_{1}(t,x)+R_{1}(t,x)\right].$ (3.10)

Using [5], we choose

$\displaystyle u_{1}(t,x)=e^{-i(\lambda^{2}t+\lambda
x\cdot\omega)}[1+R_{1}(t,x)]$ (3.11)

with

$\displaystyle\lambda\left\|R_{1}\right\|_{L^{2}}+\left\|\nabla_{x}R_{1}\right\|_{L^{2}}\leq
C.$ (3.12)

In the similar manner, we recall the adjoint equation

$\displaystyle-i\partial_{t}v+\Delta v=0~{}\text{ in }\Omega_{T}.$ (3.13)

We seek for the solution $v(t,x)$ of (3.13) in the following form:

$\displaystyle v(t,x)=e^{i(\lambda^{2}t+\lambda
x\cdot\omega)}\left[A_{2}(t,x)+R_{2}(t,x)\right].$ (3.14)

It is straightforward to derive that we shall choose $A_{2}(t,x)$ such that

$\displaystyle\omega\cdot\nabla_{x}A_{2}=0.$ (3.15)

As a consequence, we will consider

$\displaystyle A_{2}(t,x)=e^{i(\tau t+\xi\cdot x)},$ (3.16)

where $\xi\cdot\omega=0$. Moreover, using [5] the following estimates can be
obtained

$\displaystyle\left\|R_{2}\right\|_{L^{2}}\leq\frac{C}{\lambda},\quad\left\|\nabla_{x}R_{2}\right\|_{L^{2}}\leq
C$

for some positive constant $C$ independent of $\lambda$. Hence we choose the
geometric optics solution $v(t,x)$ as

$\displaystyle v(t,x)=e^{i(\lambda^{2}t+\lambda x\cdot\omega)}\left(e^{i(\tau
t+\xi\cdot x)}+R_{2}(t,x)\right)$ (3.17)

with $\omega\perp\xi$ and

$\displaystyle\lambda\left\|R_{2}\right\|_{L^{2}(\Omega_{T})}+\left\|R_{2}\right\|_{L^{2}(\Omega_{T})}\leq
C.$ (3.18)

Substituting the expressions for $u_{1}(t,x)$ from (3.11) and for $v(t,x)$
from (3.17) in (3.8), we obtain

$\displaystyle\int_{0}^{T}\int_{\Omega}q(t,x)\left(e^{i(\tau t+\xi\cdot
x)}+R_{2}(t,x)\right)\left(1+R_{1}(t,x)\right)~{}dxdt=0$

which after simplification gives

$\displaystyle\int_{0}^{T}\int_{\Omega}q(t,x)\bigg{(}e^{i(\tau t+\xi\cdot
x)}+e^{i(\tau t+\xi\cdot
x)}R_{1}(t,x)+R_{2}(t,x)+R_{1}(t,x)R_{2}(t,x)\bigg{)}~{}dxdt=\text{known}$

for all $\xi\perp\omega$. Now using (3.12) and (3.18) and letting
$\lambda\rightarrow\infty$, we get

$\displaystyle\int_{0}^{T}\int_{\Omega}q(t,x)e^{i(\tau t+\xi\cdot
x)}~{}dxdt=\text{known},\quad\text{for all }\xi\perp\omega.$

Consequently, we have

$\displaystyle\int_{R^{1+n}}\chi_{\Omega_{T}}(t,x)q(t,x)e^{i(\tau t+\xi\cdot
x)}~{}dxdt=\text{known}.$

As a result, we have the following

$\displaystyle{\widehat{\displaystyle\chi_{\Omega_{T}}q}}(\tau,\xi)=\text{known
},\quad\text{for all }(\tau,\xi)\in\mathbb{R}^{1+n}$ (3.19)

which results into $\displaystyle\chi_{\Omega_{T}}(t,x)q(t,x)$ is known for
all $(t,x)\in\mathbb{R}^{1+n}$. Thus we conclude that $q(t,x)$ is known for
all $(t,x)\in\Omega_{T}$.

###  3.2. Reconstruction of the coefficient of quadratic nonlinearity

We begin this subsection by proving the following lemma which will be helpful
in giving the reconstruction formula in determining $\overrightarrow{b}(t,x)$
for $(t,x)\in\Omega_{T}$.

######  Lemma 3.1.

Let $n\geq 2$ and $N\geq\frac{n}{2}+2$. Then there exist
$\left(v_{j}\right)_{1\leq j\leq n}$ solutions to
$L_{q}v:=-i\partial_{t}v+\Delta v+q(t,x)v=0$, such that $v_{j}\in
L^{2}\left((0,T);H^{N}(\Omega)\right)$ for $1\leq j\leq n$ and
$\displaystyle\det\left(\left(\frac{\partial v_{i}}{\partial
x_{j}}(t,x)\right)\right)_{1\leq i,j\leq n}\neq 0$, a.e. in $\Omega_{T}$.

###### Proof.

We follow the arguments similar to the one used in [29] for elliptic case and
in [39] for the wave equation. Let us begin with choosing
$\omega_{j}\in\mathbb{S}^{n-1}$ for $1\leq j\leq n$ such that the set
$\\{\omega_{1},\omega_{2},\cdots,\omega_{n}\\}$ is a linearly independent
subset of $\mathbb{R}^{n}$. Also let us extend the potential $q(t,x)$ to
function in $C_{0}^{\infty}(\mathbb{R}^{1+n})$. Now for $1\leq j\leq n$, we
construct the asymptotic solutions $v_{j}(t,x)$ to $L_{q}v=0$ in
$\mathbb{R}^{1+n}$, taking the following form

$\displaystyle v_{j}(t,x)=e^{i\left(\lambda^{2}t+\lambda
x\cdot\omega_{j}\right)}\sum_{k=0}^{N}\frac{A_{jk}(t,x)}{\left(2i\lambda\right)^{k}}+R_{j}(t,x),\
\ \mbox{with}\ N>\frac{n}{2}+2\ \mbox{and}\ \lambda>>1.$ (3.20)

It is easy to see that

$\displaystyle\begin{aligned} L_{q}v_{j}&=e^{i\left(\lambda^{2}t+\lambda
x\cdot\omega_{j}\right)}\Bigg{[}\left(L_{q}+2i\lambda\mathcal{L}_{j}\right)\Bigg{(}A_{j0}(t,x)+\frac{A_{j1}(t,x)}{2i\lambda}+\frac{A_{j2}(t,x)}{\left(2i\lambda\right)^{2}}+\cdots\\\
&\ \
\quad+\frac{A_{jN}(t,x)}{\left(2i\lambda\right)^{N}}+e^{-\left(\lambda^{2}t+\lambda
x\cdot\omega_{j}\right)}R_{j}(t,x)\Bigg{)}\Bigg{]}\end{aligned}$

where $\mathcal{L}_{j}:=\omega_{j}\cdot\nabla_{x}$ is the transport operator.
Arranging the terms in descending power of $\lambda$, we get

$\displaystyle\begin{aligned}
&2i\lambda\mathcal{L}_{j}A_{j0}+\left(\mathcal{L}_{j}A_{j1}+L_{q}A_{j0}\right)+\frac{1}{2i\lambda}\left(\mathcal{L}_{j}A_{j2}+L_{q}A_{j1}\right)+\cdots+\frac{1}{\left(2i\lambda\right)^{N-2}}\left(\mathcal{L}_{j}A_{j,N-1}+L_{q}A_{j,N-2}\right)\\\
&\ \ \
+\frac{1}{\left(2i\lambda\right)^{N-1}}\left(\mathcal{L}_{j}A_{jN}+L_{q}A_{j,N-1}\right)+\frac{1}{\left(2i\lambda\right)^{N}}L_{q}A_{jN}+e^{-i\left(\lambda^{2}t+\lambda
x\cdot\omega_{j}\right)}L_{q}R_{j}=0.\end{aligned}$

Comparing the various powers of $\lambda$, we have the transport equations for
$A_{jk}$ for $0\leq k\leq N$ given by

$\displaystyle\begin{aligned} &\mathcal{L}_{j}A_{j0}(t,x)=0\\\
&\mathcal{L}_{j}A_{jk}(t,x)=-L_{q}A_{j,k-1}(t,x),\ \ 1\leq k\leq
N.\end{aligned}$ (3.21)

We take $A_{j0}=1,$ as a solution to first equation in (3.21) and using this
we find $A_{jk}(t,x)$ for $1\leq k\leq N$ by solving the second equation in
(3.21). Using $A_{jN}(t,x)$ we take $R_{j}(t,x)$ as the solution to the
following equation

$\displaystyle\begin{aligned}
\begin{cases}L_{q}R_{j}(t,x)=-\frac{1}{\left(2i\lambda\right)^{N}}e^{i\left(\lambda^{2}t+\lambda
x\cdot\omega\right)}L_{q}A_{jN}(t,x),\
(t,x)\in(0,\infty)\times\mathbb{R}^{n}\\\
R_{j}(0,x)=0.\end{cases}\end{aligned}$ (3.22)

After solving the Cauchy problem given by (3.22) for $R_{j}$, we get that
$R_{j}\in C\left((0,\infty);H^{N}(\mathbb{R}^{n})\right)$. Now after
restricting $R_{j}$ to $\Omega_{T}$ and using the Sobolev embedding theorem,
we have that $R_{j}\in W^{1,\infty}(\Omega_{T})$ and $\lVert
R_{j}\rVert_{W^{1,\infty}(\Omega_{T})}\leq C$ for some constant $C>0$
independent of $\lambda$. Now let us define $\alpha_{j}(t,x)$ and
$\widetilde{R}_{j}(t,x)$ by

$\alpha_{j}(t,x):=e^{i\left(\lambda^{2}t+\lambda x\cdot\omega_{j}\right)}\ \
\mbox{and}\ \ \widetilde{R}_{j}(t,x):=e^{i\left(\lambda^{2}t+\lambda
x\cdot\omega_{j}\right)}\sum_{k=1}^{N}\frac{A_{jk}(t,x)}{\left(2i\lambda\right)^{k}}+R_{j}(t,x).$

Using these in (3.20), we get

$\displaystyle\begin{aligned}
v_{j}(t,x)=\alpha_{j}(t,x)+\widetilde{R}_{j}(t,x)\ \ \mbox{and}\ \
\nabla_{x}v_{j}=i\lambda\alpha_{j}(t,x)\omega_{j}+\nabla_{x}\widetilde{R}_{j}(t,x).\end{aligned}$

Now consider the matrix

$\displaystyle\begin{aligned} A(t,x,\lambda):=\left(\left(\frac{\partial
v_{i}}{\partial x_{j}}\right)\right)_{1\leq i,j\leq
n}=\begin{bmatrix}i\lambda\alpha_{1}(t,x)\omega_{11}+\partial_{1}R_{1}&i\lambda\alpha_{1}(t,x)\omega_{12}+\partial_{2}R_{1}&\cdots&i\lambda\alpha_{1}(t,x)\omega_{1n}+\partial_{n}R_{1}\\\
i\lambda\alpha_{2}(t,x)\omega_{21}+\partial_{1}R_{2}&i\lambda\alpha_{1}(t,x)\omega_{22}+\partial_{2}R_{2}&\cdots&i\lambda\alpha_{1}(t,x)\omega_{2n}+\partial_{n}R_{2}\\\
\vdots&\vdots&\cdots&\vdots\\\
i\lambda\alpha_{n}(t,x)\omega_{n1}+\partial_{1}R_{n}&i\lambda\alpha_{1}(t,x)\omega_{n2}+\partial_{2}R_{n}&\cdots&i\lambda\alpha_{n}(t,x)\omega_{1n}+\partial_{n}R_{n}\\\
\end{bmatrix}\end{aligned}$

where $\omega_{ij}$ denote the j’th component in
$\omega_{i}\in\mathbb{S}^{n-1}$ and $\partial_{i}:=\frac{\partial}{\partial
x_{i}}$ for $1\leq i\leq n$. Our aim is to show that Det$A(t,x,\lambda)\neq 0$
for almost everywhere in $\Omega_{T}$ for $\lambda>>1$. Now using the fact
that $\lVert\nabla_{x}\widetilde{R}_{j}\rVert_{L^{\infty}(\Omega_{T})}\leq C$
for some constant $C>0$ independent of $\lambda$, we get

$\displaystyle\begin{aligned}
\lim_{\lambda\rightarrow\infty}\Bigg{\lVert}\frac{\mbox{Det}A(t,x,\lambda)}{\lambda^{n}}-\mbox{Det}\begin{bmatrix}i\alpha_{1}(t,x)\omega_{11}&\i\alpha_{1}(t,x)\omega_{12}&\cdots&i\alpha_{1}(t,x)\omega_{1n}\\\
i\alpha_{2}(t,x)\omega_{21}&\i\alpha_{2}(t,x)\omega_{22}&\cdots&i\alpha_{2}(t,x)\omega_{2n}\\\
\vdots&\vdots&\cdots&\vdots\\\
i\alpha_{n}(t,x)\omega_{n1}&\i\alpha_{n}(t,x)\omega_{n2}&\cdots&i\alpha_{n}(t,x)\omega_{nn}\end{bmatrix}\Bigg{\rVert}_{L^{2}(\Omega_{T})}=0.\end{aligned}$

This gives us

$\displaystyle\begin{aligned}
\frac{\mbox{Det}A(t,x,\lambda)}{\lambda^{n}}\rightarrow\mbox{Det}\begin{bmatrix}i\alpha_{1}(t,x)\omega_{11}&\i\alpha_{1}(t,x)\omega_{12}&\cdots&i\alpha_{1}(t,x)\omega_{1n}\\\
i\alpha_{2}(t,x)\omega_{21}&\i\alpha_{2}(t,x)\omega_{22}&\cdots&i\alpha_{2}(t,x)\omega_{2n}\\\
\vdots&\vdots&\cdots&\vdots\\\
i\alpha_{n}(t,x)\omega_{n1}&\i\alpha_{n}(t,x)\omega_{n2}&\cdots&i\alpha_{n}(t,x)\omega_{nn}\end{bmatrix}\neq
0,\ \mbox{as}\ \lambda\rightarrow\infty\ \mbox{in}\
L^{2}(\Omega_{T}).\end{aligned}$

Due to the convergence of Det$A(t,x,\lambda)$ in $L^{2}(\Omega_{T})$ as
$\lambda\rightarrow\infty$, we can have a subsequence of Det$A(t,x,\lambda)$
still denote the same such that

$\displaystyle\begin{aligned}
\lim_{\lambda\rightarrow\infty}\frac{\mbox{Det}A(t,x,\lambda)}{\lambda^{n}}=\mbox{Det}\begin{bmatrix}i\alpha_{1}(t,x)\omega_{11}&\i\alpha_{1}(t,x)\omega_{12}&\cdots&i\alpha_{1}(t,x)\omega_{1n}\\\
i\alpha_{2}(t,x)\omega_{21}&\i\alpha_{2}(t,x)\omega_{22}&\cdots&i\alpha_{2}(t,x)\omega_{2n}\\\
\vdots&\vdots&\cdots&\vdots\\\
i\alpha_{n}(t,x)\omega_{n1}&\i\alpha_{n}(t,x)\omega_{n2}&\cdots&i\alpha_{n}(t,x)\omega_{nn}\end{bmatrix}\neq
0,\ \mbox{pointwise for a.e.}\ (t,x)\in\Omega_{T}.\end{aligned}$

Hence, Det$A(t,x,\lambda)\neq 0$ for $\lambda>>1$, a.e. $(t,x)\in\Omega_{T}$.
Thus, we have that
$\nabla_{x}v_{1}(t,x),\nabla_{x}v_{2}(t,x),\cdots,\nabla_{x}v_{n}(t,x)$ are
linearly independent for a.e. $(t,x)\in\Omega_{T}$. The proof of lemma is
complete. ∎

Recontruction for $\overrightarrow{b}(t,x)$: Since $q(t,x)$ is known in
$\Omega_{T}$, hence we consider $u_{1}^{\phi,f}$ is a solution to (2.2).
Consequently, $u_{2}(t,x)$ satisfies

$\displaystyle\begin{cases}i\partial_{t}u_{2}(t,x)+\Delta
u_{2}(t,x)+q(t,x)u_{2}(t,x)=\nabla\cdot\left(\overrightarrow{b}(t,x)|\nabla_{x}u_{1}^{\phi,f}(t,x)|^{2}\right),\quad(t,x)\in\Omega_{T},\\\
u_{2}(0,x)=0,\quad x\in\Omega,\\\
u_{2}(t,x)=0,\quad(t,x)\in\partial\Omega_{T}.\end{cases}$ (3.23)

By multiplying (3.23) by $v(t,x)$ where $v(t,x)$ satisfies

$\displaystyle-i\partial_{t}v(t,x)+\Delta
v(t,x)+q(t,x)v(t,x)=0,\quad(t,x)\in\Omega_{T}$ (3.24)

and integrate over $\Omega_{T}$, we obtain

$\displaystyle\begin{aligned}
-\int_{0}^{T}\int_{\Omega}i\partial_{t}v(t,x)u_{2}(t,x)~{}dxdt+\int_{0}^{T}\int_{\Omega}\Delta
u_{2}(t,x)v(t,x)~{}dxdt+\int_{0}^{T}\int_{\Omega}q(t,x)u_{2}(t,x)v(t,x)~{}dxdt\\\
=\int_{0}^{T}\int_{\Omega}\nabla\cdot(\overrightarrow{b}(t,x)\lvert\nabla_{x}u_{1}^{\phi,f}(t,x)\rvert^{2})v(t,x)~{}dxdt.\end{aligned}$

Further, integration by parts yields

$\displaystyle\int_{0}^{T}\int_{\Omega}\Big{(}-i\partial_{t}v(t,x)u_{2}(t,x)+\Delta
v(t,x)u_{2}(t,x)+q(t,x)u_{2}(t,x)v(t,x)\Big{)}~{}dxdt$ $\displaystyle\ \ \ \ \
\
+\int_{0}^{T}\int_{\partial\Omega}\left(\partial_{\nu}u_{2}v-\partial_{\nu}vu_{2}\right)(t,x)~{}dS_{x}dt=-\int_{0}^{T}\int_{\Omega}\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x)\lvert\nabla_{x}u_{1}^{\phi,f}(t,x)\rvert^{2}~{}dxdt$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\qquad\qquad\qquad\qquad\qquad+\int_{0}^{T}\int_{\partial\Omega}\nu(x)\cdot\overrightarrow{b}(t,x)|\nabla_{x}u_{1}^{\phi,f}(t,x)|^{2}v(t,x)~{}dS_{x}dt$

which results into

$\displaystyle\int_{0}^{T}\int_{\Omega}\left(\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x)\right)|\nabla_{x}u_{1}^{\phi,f}(t,x)|^{2}~{}dxdt$
$\displaystyle=\int_{0}^{T}\int_{\partial\Omega}\nu(x)\cdot\overrightarrow{b}(t,x)\lvert\nabla_{x}u_{1}^{\phi,f}(t,x)\rvert^{2}v(t,x)~{}dS_{x}dt$
$\displaystyle\ \ \ \
-\int_{0}^{T}\int_{\partial\Omega}\partial_{\nu}u_{2}(t,x)v(t,x)~{}dS_{x}dt.$

As a consequence,

$\displaystyle\int_{0}^{T}\int_{\Omega}(\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x))|\nabla_{x}u_{1}^{\phi,f}(t,x)|^{2}~{}dxdt=\text{known}$

for all $v(t,x)$ satisfying equation (3.24). Now choosing
$u_{1}^{\phi_{1}\pm\phi_{2},f_{1}\pm f_{2}}$ the solutions to (2.2) when
$\phi=\phi_{1}\pm\phi_{2}$ and $f=f_{1}\pm f_{2}$ and subtracting the two sets
of equations, we get

$\displaystyle\begin{aligned}
\int_{0}^{T}\int_{\Omega}\left(\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x)\right)\nabla_{x}u_{1}^{\phi_{1},f_{1}}\cdot\overline{\nabla_{x}u_{1}^{\phi_{2},f_{2}}}(t,x)\
dxdt=\mbox{known}\end{aligned}$

for all $v(t,x)$ solution to (3.24) and $u_{1}^{\phi_{i},f_{i}}$ solutions to
(2.2) when $\phi=\phi_{i}$ and $f=f_{i}$ for $i=1,2$. Now denote
$B_{v}(t,x):=\chi_{\Omega_{T}}(t,x)\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x)$
where $\chi_{\Omega_{T}}$ is the characteristic function of $\Omega_{T}$.
Using this in previous equation, we get

$\displaystyle\begin{aligned}
\int_{\mathbb{R}^{1+n}}B_{v}(t,x)\nabla_{x}u_{1}^{\phi_{1},f_{1}}\cdot\overline{\nabla_{x}u_{1}^{\phi_{2},f_{2}}}(t,x)\
dxdt=\mbox{known}\end{aligned}$ (3.25)

for all $u_{1}^{\phi_{i},f_{i}}$, for $i=1,2$, solutions to (2.2) when
$\phi=\phi_{i}$ and $f=f_{i}$. Now choosing

$\displaystyle\begin{aligned}
&u_{1}^{\phi_{1},f_{1}}(t,x)=e^{-i\left(\lambda^{2}t+\lambda
x\cdot\omega\right)}\left(e^{i\left(\tau t+\xi\cdot
x\right)}+R_{1}(t,x)\right)\ \mbox{and}\ \\\
&u_{1}^{\phi_{2},f_{2}}(t,x)=e^{-i\left(\lambda^{2}t+\lambda
x\cdot\omega\right)}\left(1+R_{2}(t,x)\right)\end{aligned}$

where $\xi\perp\omega$ and $R_{i}^{\prime}s$ satisfies $\lambda\lVert
R_{i}\rVert_{L^{2}(\Omega_{T})}+\lVert\nabla_{x}R_{i}\rVert_{L^{2}(\Omega_{T})}\leq
C$ for some constant $C$ independent of $\lambda$. Using these expressions of
$u_{1}^{\phi_{1},f_{1}}(t,x)$ and $u_{1}^{\phi_{2},f_{2}}(t,x)$ in (3.25), we
have

$\displaystyle\begin{aligned}
\int_{\mathbb{R}^{1+n}}&B_{v}(t,x)\Big{(}e^{i\left(\tau t+\xi\cdot
x\right)}i\xi\cdot\overline{\nabla_{x}R_{2}}+\nabla_{x}R_{1}\cdot\overline{\nabla_{x}R_{2}}+i\lambda\omega\cdot\nabla_{x}R_{1}+i\lambda\overline{R}_{2}\omega\cdot\nabla_{x}R_{1}-i\lambda
e^{i\left(\tau
t+\xi\cdot\omega\right)}\omega\cdot\overline{\nabla_{x}R_{2}}\\\ &\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ +\lambda^{2}e^{i\left(\tau t+\xi\cdot
x\right)}+\lambda^{2}\overline{R}_{2}+\lambda^{2}R_{1}\overline{R}_{2}\Big{)}(t,x)\
dxdt=\mbox{known}\end{aligned}$

for all $\lambda>0,\tau\in\mathbb{R}$ and $\xi\in\mathbb{R}^{n}$ such that
$\xi\cdot\omega=0$. Now dividing by $\lambda^{2}$ and taking
$\lambda\rightarrow\infty$, we get

$\displaystyle\int_{\mathbb{R}^{1+n}}B_{v}(t,x)e^{-i\left(\tau t+\xi\cdot
x\right)}\ dxdt=\mbox{known for all $\tau\in\mathbb{R}$ and
$\xi\in\omega^{\perp}$.}$

This gives that $\widehat{B}_{v}(\tau,\xi)=\mbox{known}$ for all
$\tau\in\mathbb{R}$ and $\xi\in\omega^{\perp}$ for any
$\omega\in\mathbb{S}^{n-1}$. But
$\cup_{\omega\in\mathbb{S}^{n-1}}\\{\xi\in\mathbb{R}^{n}:\
\xi\in\omega^{\perp}\\}=\mathbb{R}^{n}$ therefore we have that
$\widehat{B}_{v}(\tau,\xi)$ is known for $(\tau,\xi)\in\mathbb{R}^{1+n}$.
Hence by using the Fourier inversion, we have that
$\overrightarrow{b}(t,x)\cdot\nabla_{x}v(t,x)$ is known in $\Omega_{T}$ for
all $v(t,x)$ solution to (3.24). Using Lemma 3.1, we choose
$v_{1},v_{2},v_{3},\cdots,v_{n}$ solutions to (3.24) such that
$\nabla_{x}v_{1},\nabla_{x}v_{2},\cdots,\nabla_{x}v_{n}$ are linearly
independent a.e. in $\Omega_{T}$. Using these choices of $v_{j}$ for $1\leq
j\leq n$, we get that $\overrightarrow{b}(t,x)\cdot\nabla_{x}v_{j}(t,x)=$
known for a.e. $(t,x)\in\Omega_{T}$ and for each $1\leq j\leq n$. After
denoting this known value by $F_{v_{j}}(t,x)$, we get the following system of
equations

$\displaystyle\begin{aligned} \begin{bmatrix}\vspace{1mm}\frac{\partial
v_{1}}{\partial x_{1}}&\frac{\partial v_{1}}{\partial x_{2}}&\frac{\partial
v_{1}}{\partial x_{3}}&\cdots&\frac{\partial v_{1}}{\partial x_{n}}\\\
\vspace{1mm}\frac{\partial v_{2}}{\partial x_{1}}&\frac{\partial
v_{2}}{\partial x_{2}}&\frac{\partial v_{2}}{\partial
x_{3}}&\cdots&\frac{\partial v_{2}}{\partial x_{n}}\\\
\vspace{1mm}\vdots&\vdots&\vdots&\cdots&\vdots\\\ \frac{\partial
v_{n}}{\partial x_{1}}&\frac{\partial v_{n}}{\partial x_{2}}&\frac{\partial
v_{n}}{\partial x_{3}}&\cdots&\frac{\partial v_{n}}{\partial x_{n}}\\\
\end{bmatrix}\begin{bmatrix}\vspace{2mm}b_{1}(t,x)\\\
\vspace{2mm}b_{2}(t,x)\\\ \vspace{2mm}\vdots\\\
b_{n}(t,x)\end{bmatrix}=\begin{bmatrix}\vspace{2mm}F_{v_{1}}(t,x)\\\
\vspace{2mm}F_{v_{2}}(t,x)\\\ \vspace{2mm}\vdots\\\
F_{v_{n}}(t,x)\end{bmatrix},\ \ \mbox{for a.e.}\
(t,x)\in\Omega_{T}.\end{aligned}$

Now since the matrix $A(t,x):=\left(\left(\frac{\partial v_{i}}{\partial
x_{j}}\right)\right)_{1\leq i,j\leq n}$ is invertible for a.e.
$(t,x)\in\Omega_{T}$, therefore we obtain that

$\overrightarrow{b}(t,x):=A^{-1}(t,x)\overrightarrow{F}(t,x),\ \ \mbox{for
a.e.}\ (t,x)\in\Omega_{T},$

where

$\displaystyle\overrightarrow{F}(t,x):=\begin{bmatrix}\vspace{2mm}F_{v_{1}}(t,x)\\\
\vspace{2mm}F_{v_{2}}(t,x)\\\ \vspace{2mm}\vdots\\\
F_{v_{n}}(t,x)\end{bmatrix}.$

This gives the recontruction for $\overrightarrow{b}(t,x)$ for a.e.
$(t,x)\in\Omega_{T}$ but $\overrightarrow{b}\in C^{\infty}(\Omega_{T})$
therefore $\overrightarrow{b}(t,x)$ is known for all $(t,x)\in\Omega_{T}$.
Thus the proof of Theorem 1.1 is complete.

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