# Determining complex spin mixing conductance and spin diffusion length from
spin pumping experiments in magnetic insulator/heavy metal bilayers

Kuntal Roy<EMAIL_ADDRESS>Department of Electrical Engineering and
Computer Science,
Indian Institute of Science Education and Research Bhopal, Bhopal, Madhya
Pradesh 462066, India

###### Abstract

Magnetic insulators are promising materials for the development of energy-
efficient spintronics. Unlike metallic counterparts, the magnetic insulators
are characterized by imaginary part of the interfacial spin mixing conductance
as well in a bilayer with heavy metals and it is responsible for the field-
like toque in spin-orbit torque devices. Here, we study the underlying
theoretical constructs and develop a general strategy to determine the complex
spin mixing conductance from the experimental results of ferromagnetic
resonance and spin pumping. The results show that the imaginary part of the
spin mixing conductance can be one order more than the real part and it
matches the critical trend of spin mixing conductance with thickness of the
heavy metal. The interpretation of experimental results also indicates that at
small thicknesses the interface contribution becomes significant and bulk
diffusion model cannot explain the results. A thickness-dependent spin
diffusion length is necessary too that is tantamount to Elliott-Yafet spin
relaxation mechanism in the heavy metals. Also, we effectively explain the
experimental results while inserting a copper layer with varying thicknesses
in between the magnetic insulator and the heavy metal using spin-circuit
formalism.

††preprint: AIP/123-QED

Magnetic insulators or pure spin conductors such as yttrium iron garnet
($Y_{3}Fe_{5}O_{12}$, $YIG$) have attracted a lot of attention for the
development of energy-efficient spintronics. Kajiwara  _et al._ (2010); Serga,
Chumak, and Hillebrands (2010); Zhou  _et al._ (2013); Kapelrud and Brataas
(2013); Chumak  _et al._ (2015); Cornelissen  _et al._ (2015, 2016); Avci  _et
al._ (2017); Kosub  _et al._ (2018); Mendil  _et al._ (2019) Different
phenomena such as spin-Hall magnetoresistance (SMR) Chen  _et al._ (2013);
Nakayama  _et al._ (2013) and spin-torque ferromagnetic resonance (ST-FMR)
Jungfleisch  _et al._ (2017) have been observed in magnetic insulators. By
means of direct spin Hall effect (SHE) and a SHE layer, magnetization can be
electrically exited in an adjacent magnetic insulator and the spin current can
be detected by inverse spin Hall effect (ISHE) due to spin pumping. Heinrich
_et al._ (2011); Castel  _et al._ (2012); d’Allivy Kelly  _et al._ (2013); Sun
_et al._ (2013); Hyde  _et al._ (2014); Wang  _et al._ (2014); Chang  _et al._
(2017) In spin pumping mechanism, a precessing magnetization emits spins into
surrounding conductors and it is the reciprocal phenomenon Brataas  _et al._
(2012) of spin momentum transfer, Slonczewski (1996); *RefWorks:155 according
to Onsager’s reciprocity. Onsager (1931a); *RefWorks:1293 Spin pumping
mechanism gives us a methodology to understand and estimate the relevant
parameters in the system. Roy (2017a, 2018) Such understandings can benefit
the device design using SHE, Roy (2014); Ramaswamy  _et al._ (2018) which has
potential for building future spintronic devices, alongwith other promising
energy-efficient emerging devices. Roy (2016)

The Landau-Lifshitz-Gilbert (LLG) equation Landau and Lifshitz (1935);
*RefWorks:161 of magnetization dynamics with phenomenological damping
parameter is modified when the spin pumping contribution is considered and the
experimentally observable quantities are the enhancement of damping and the
ferromagnetic resonance field shift as Tserkovnyak  _et al._ (2005)

$\frac{d\mathbf{m}}{d\tau}=-\gamma_{eff}\mathbf{m}\times\mathbf{H_{eff}}+\alpha_{eff}\left(\mathbf{m}\times\frac{d\mathbf{m}}{d\tau}\right)$
(1)

where

$\frac{\gamma}{\gamma_{eff}}=1-\frac{\hbar\gamma}{4\pi
M_{s}t_{m}}g_{eff,i}^{\uparrow\downarrow},$ (2)
$\alpha_{eff}\frac{\gamma}{\gamma_{eff}}=\alpha+\alpha_{sp},\;\alpha_{sp}=\frac{\hbar\gamma}{4\pi
M_{s}t_{m}}g_{eff,r}^{\uparrow\downarrow},$ (3)

$\mathbf{m}$ is the magnetization and $\mathbf{H_{eff}}$ is the effective
field acting on the magnetization at time $\tau$, $4\pi M_{s}$ is the
saturation magnetization, $t_{m}$ is the thickness of the magnet,
$\gamma_{eff}$ and $\alpha_{eff}$ are the modified (from $\gamma$ and
$\alpha$, respectively) gyromagnetic ratio and damping parameter in the
presence of an adjacent conductor, $\alpha_{sp}$ is the increased damping due
to spin pumping, and
$g_{eff}^{\uparrow\downarrow}=g_{eff,r}^{\uparrow\downarrow}+i\,g_{eff,i}^{\uparrow\downarrow}$
is the complex (reflection) _effective_ spin mixing conductance comprising the
_bare_ interfacial spin mixing conductance
$g^{\uparrow\downarrow}=g_{r}^{\uparrow\downarrow}+i\,g_{i}^{\uparrow\downarrow}$
and both the interface and bulk conductances of the adjacent conductor. While
first principles calculations and experimental results on ferromagnetic
resonance field shift show that the imaginary component of spin mixing
conductance is low for metallic interfaces, magnetic insulators exhibit a
significant imaginary part of the spin mixing conductance. Jia  _et al._
(2011) The imaginary part can be conceived as an effective exchange field
acting on the spin accumulation and it has immense consequence as the field-
like torque in spin-orbit torque devices. Roy, Bandyopadhyay, and Atulasimha
(2012) Magnetic proximity effect Huang  _et al._ (2012); Caminale  _et al._
(2016) has been found to be irrelevant. Liu  _et al._ (2014); Zhu, Ralph, and
Buhrman (2018)

Figure 1:  (a) A precessing magnetization in a magnetic insulator is pumping
_pure_ spin current into the adjacent normal metal (NM) in a ferromagnetic
resonance (FMR) experiment. Due to spin pumping, spin potentials are developed
at the surfaces marked by 1 and 2. (b) In an equivalent spin-circuit diagram,
the voltage source $[V_{SP}]$ acts as a spin battery,
$[G^{\uparrow\downarrow}]$ is the interfacial spin mixing conductance between
the magnetic layer and the NM layer, $[G_{I}]$ is the spin conductance
representing the interfacial spin memory loss, and a $\pi$-network comprising
the conductances $[G_{1}]$ and $[G_{2}]$ represents the bulk NM layer. (c)
Minimized spin-circuit with $[G_{B}]$ representing the spin conductance due to
the bulk NM layer. (d) Minimized spin-circuit with $[G_{eff}]$ representing
the _effective_ spin mixing conductance. Figure 2:  Fitting the thickness $t$
dependence of effective spin-mixing conductance,
$g_{eff}^{\uparrow\downarrow}$ (a) Real component,
$g_{eff,r}^{\uparrow\downarrow}$. (b) Imaginary component,
$g_{eff,i}^{\uparrow\downarrow}$. Experimental data points are taken from the
Ref. Sun  _et al._ , 2013.

Here, we determine the complex spin mixing conductance and spin diffusion
length from the experimental results of spin pumping in magnetic
insulator/heavy metal bilayers presented in Ref. Sun  _et al._ , 2013.
Usually, the imaginary part of the spin mixing conductance is ignored for
magnetic insulators, Sun  _et al._ (2013); Castel  _et al._ (2012); Wang  _et
al._ (2014) however, the interpretation of the experimental results show that
the imaginary part of the spin mixing conductance can be one order more than
the real part. Also, it turns out to be crucial to match the trend of the real
part of spin mixing conductance. As the determination of the spin diffusion
length ($\lambda$) in the heavy metals is concerned, recent calculations Roy
(2017a, 2018); Tao  _et al._ (2018); Yu  _et al._ (2018); Berger  _et al._
(2018) signify the Elliott-Yafet (EY) spin relaxation mechanism in which
$\lambda$ is dependent on thickness ($t$) since conductivity ($\sigma$) varies
with thickness of the sample. Castel  _et al._ (2012) In particular, when
considering the voltage due to ISHE, Feng  _et al._ (2012); Wang  _et al._
(2014) it clearly depicts the EY spin relaxation Roy (2017a) and not the
Dyakonov-Perel (DP) Freeman  _et al._ (2018) spin relaxation mechanism. Liu,
Buhrman, and Ralph (2011); Zhang  _et al._ (2013); Boone  _et al._ (2015) The
controversy acclaimed in the Ref. Feng  _et al._ , 2012 that two different
constant values of spin diffusion lengths are required to explain the
experimental results of $g_{eff}^{\uparrow\downarrow}$ and a quantity
dependent on $V_{ISHE}$ can be solved by considering a thickness-dependent
$\lambda$. Roy (2017a) There is significant interface contribution at the
magnetic insulator/heavy metal interface as well, which introduces more
scattering for thin samples and lowers the conductivity. The interpretation of
experimental results clearly indicate that at small thicknesses bulk diffusion
model cannot explain such lowering of the conductivity. We employ spin-circuit
representation of spin pumping Roy (2017b) to interpret the experimental
results effectively, in particular for multilayers including a sandwiched
copper layer. Ref. Roy, 2017b reproduces the standard mathematical results of
spin-pumping in the literature, however, it did not consider the interface
contribution. Note that magnon diffusion length is not considered since we are
not dealing with propagating spin waves in magnetic insulators here.

Figure 1(a) shows a schematic diagram of spin pumping by a precessing
magnetization in a magnetic insulator into a normal metal layer having a
length $l$, width $w$, and thickness $t$. The corresponding spin-circuit
representation Roy (2017a) containing voltage source and conductances is shown
in the Fig. 1(b). The voltage source $[V_{SP}]$ that acts as a spin battery
due to a precessing magnetization $\mathbf{m}$ and the spin current due to
spin pumping can be written, respectively as

$\mathbf{{V}_{SP}}=\frac{\hbar}{2e}\left(\mathbf{m}\times\frac{d\mathbf{m}}{d\tau}\right),$
(4)
$\mathbf{{I}_{SP}}=\cfrac{\hbar}{2e}\,\left(2G_{r}\,\mathbf{m}\times\frac{d\mathbf{m}}{d\tau}+2G_{i}\,\frac{d\mathbf{m}}{d\tau}\right).$
(5)

The $[G^{\uparrow\downarrow}]$ is the interfacial _bare_ spin mixing
conductance between the magnetic layer and the NM layer, which can be
represented in $(\mathbf{m},d\mathbf{m}/d\tau,\mathbf{m}\times
d\mathbf{m}/d\tau)$ basis as

$\left[G^{\uparrow\downarrow}\right]=\left[\begin{array}[]{crr}0&0&0\\\
0&2G_{r}&2G_{i}\\\ 0&-2G_{i}&2G_{r}\end{array}\right]$ (6)

where $G_{r(i)}=lw\,(e^{2}/h)\,g_{r(i)}^{\uparrow\downarrow}$. $[G_{I}]$ is
the spin conductance due to interfacial spin memory loss Fert and Lee (1996);
Kovalev, Brataas, and Bauer (2002); Wang  _et al._ (2006); Jiao and Bauer
(2013); Eid  _et al._ (2002); Nguyen, Pratt, and Bass (2014); Rojas-Sánchez
_et al._ (2014); Liu  _et al._ (2014); Chen and Zhang (2015); Freimuth,
Blügel, and Mokrousov (2015); Zhang  _et al._ (2015); Bass (2016); Amin and
Stiles (2016a); *RefWorks:2600; Belashchenko, Kovalev, and van Schilfgaarde
(2016); Ghosh  _et al._ (2017); Lang, Zhou, and Qiu (2018); Lu  _et al._
(2019); Hayashi  _et al._ (2020); Gupta  _et al._ (2020) with parameter
$\delta$ representing the spin flip probability $1-e^{-\delta}$ at the
interface, where $G_{I}=G_{I,r}+i\,G_{I,i}$,
$G_{I,r(i)}=lw\,(\delta/R_{r(i)}^{*})\,sinh(\delta)=(2e^{2}/h)\,g_{I,r(i)}$,
and $R^{*}=R_{r}^{*}+i\,R_{i}^{*}$ is an effective interface resistance
depending on the interface spin polarization. Liu  _et al._ (2014) Note that
$R^{*}$ depends on the conductivity $\sigma$ and therefore thickness $t$ of
the NM layer. Liu  _et al._ (2014)

The bulk diffusion in the NM layer can be represented by a $\pi$-circuit
(which comes after solving the spin diffusion equation in normal metals) as
shown in the Fig. 1(b). Roy (2017b) In the absence of spin memory loss
$\left[G_{1(2)}\right]=G_{1(2)}\left[I_{3\times 3}\right]$, where
$G_{1}=G_{\lambda}tanh(t/2\lambda)$, $G_{2}=G_{\lambda}csch(t/\lambda)$,
$G_{\lambda}=\sigma lw/\lambda$, and $\left[I_{3\times 3}\right]$ is the
${3\times 3}$ identity matrix. All the bulk conductances will be altered by
the spin accumulation in the NM layer with a multiplication factor
$cosh(\delta)$ in the presence of spin memory loss Roy (2017a) giving
$\left[G_{B}\right]=G_{B}\left[I_{3\times 3}\right]$, where
$G_{B}=G_{1}+G_{1}G_{2}/(G_{1}+G_{2})=G_{\lambda}cosh(\delta)\,tanh(t/\lambda)=(2e^{2}/h)\,g_{B}$,
as shown in Fig. 1(c). Fig. 1(d) shows the _effective_ spin mixing conductance
$\left[G_{eff}\right]$ of the system where
$G_{eff,r(i)}=lw\,(2e^{2}/h)\,g_{eff,r(i)}^{\uparrow\downarrow}$.

Figure 3:  $R_{r}^{*}/\delta$ and $R_{i}^{*}/\delta$ with the conductivity of
the $Pt$ layer for $\delta=3.83$, used to fit the results in the Fig. 2.
Figure 4:  (a) A precessing magnetization in a magnetic insulator is pumping
_pure_ spin current into a multilayer of Cu and Pt in a ferromagnetic
resonance (FMR) experiment. There are three surfaces marked by 1,2, and 3
where spin potentials are developed. (b) In an equivalent spin circuit
diagram, the voltage source $[V_{SP}]$ acts as a spin battery,
$[G^{\uparrow\downarrow}_{Cu}]$ is the interfacial spin mixing conductance
between the magnetic layer and the Cu layer, a $\pi$-network comprising the
conductances $[G_{1}^{Cu}]$ and $[G_{2}^{Cu}]$ to represent the Cu layer, and
another $\pi$-network comprising the conductances $[G_{1}^{Pt}]$ and
$[G_{2}^{Pt}]$ to represent the Pt layer. (c) Minimized spin-circuit with
$[G_{B}^{Pt}]$ representing the spin conductance due to the Pt layer. This
spin circuit can be further minimized with the conductances calculated by
series-parallel combination to get the _effective_ spin mixing conductance of
the system. Figure 5:  Thickness $t_{Cu}$ dependence of effective spin-mixing
conductance $g_{eff,r}^{\uparrow\downarrow}$ corresponding to Fig. 4.
Experimental data points are taken from the Ref. Sun  _et al._ , 2013.

From the spin-circuit in the Fig. 1, we can write

$\displaystyle\left[\begin{array}[]{rr}G_{eff,r}&G_{eff,i}\\\
-G_{eff,i}&G_{eff,r}\end{array}\right]^{-1}$
$\displaystyle=\left[\begin{array}[]{rr}G_{r}&G_{i}\\\
-G_{i}&G_{r}\end{array}\right]^{-1}$ (11)
$\displaystyle+\left[\begin{array}[]{cc}G_{I,r}+G_{B}&G_{I,i}\\\
-G_{I,i}&G_{I,r}+G_{B}\end{array}\right]^{-1}$ (14)

and therfore

$\displaystyle
G_{eff,r(i)}=\frac{G_{r}^{n}+G_{IB,r(i)}^{n}}{(G_{r}^{n}+G_{IB,r}^{n})^{2}+(G_{i}^{n}+G_{IB,i}^{n})^{2}}$
(15)

where

$\displaystyle G_{r(i)}^{n}=\frac{G_{r(i)}}{G_{r}^{2}+G_{i}^{2}},$ (16)
$\displaystyle G_{IB,r}^{n}$
$\displaystyle=\frac{G_{I,r}+G_{B}}{(G_{I,r}+G_{B})^{2}+G_{I,i}^{2}},$ (17)
$\displaystyle G_{IB,i}^{n}$
$\displaystyle=\frac{G_{I,i}}{(G_{I,r}+G_{B})^{2}+G_{I,i}^{2}}.$ (18)

From Equation (14), we can write

$\displaystyle G_{r(i)}^{n}=G_{eff,r(i)}^{n}-G_{IB,r(i)}^{n}$ (19)

where

$\displaystyle
G_{eff,r(i)}^{n}=\frac{G_{eff,r(i)}}{G_{eff,r}^{2}+G_{eff,i}^{2}}.$ (20)

From Equation (16), we can write
${G_{r}^{n}}^{2}+{G_{i}^{n}}^{2}=1/(G_{r}^{2}+G_{i}^{2})$ and thus

$\displaystyle G_{r(i)}=\frac{G_{r(i)}^{n}}{{G_{r}^{n}}^{2}+{G_{i}^{n}}^{2}}$
(21)

which allows us to calculate the _bare_ spin mixing conductance
$[G^{\uparrow\downarrow}]$ as given in the Equation (6).

Note that $g_{eff,r}^{\uparrow\downarrow}$ and
$g_{eff,i}^{\uparrow\downarrow}$ are thickness-dependent due to the thickness
dependence of $g_{I,r}$, $g_{I,i}$, and $g_{B}$. The trend of the dependence
depends on how $t/\lambda$ scales with lowering thickness $t$, since $\sigma$
also decreases with decreasing thickness, with $\lambda(t)\propto\sigma(t)$,
according to EY spin relaxation mechanism. It is possible to measure
experimentally both the $g_{eff,i}^{\uparrow\downarrow}$ and
$g_{eff,r}^{\uparrow\downarrow}$ (from Equations (2) and (3), respectively),
and conductivity $\sigma$ with thickness $t$ of the NM layer. Then choosing a
value of $\lambda_{max}$ (at the highest fabricated thickness) and
$(R_{r(i)}^{*},\delta)$ representing interface conductance, the
$g^{\uparrow\downarrow}$ can be calculated from the Equation (21). Using
$\lambda_{max}$, $g^{\uparrow\downarrow}$, $(R_{r(i)}^{*},\delta)$, and the
relation $\lambda(t)\propto\sigma(t)$, we can calculate
$g_{eff}^{\uparrow\downarrow}(t)$ from the Equation (15). We can choose the
$\lambda_{max}$ and $(R_{r(i)}^{*},\delta)$ that give us the best fit with the
experimental data.

Figure 2 shows the fitting of the experimental results presented in the Ref.
Sun  _et al._ , 2013. This fitting is for $\delta=3.83$, $R_{r(i)}^{*}$ in
Fig. 3, $\lambda_{max}=1.24\,nm$, and
$(g_{r}^{\uparrow\downarrow},g_{i}^{\uparrow\downarrow})=(2.95e18,3.11e19)\,m^{-2}$.
Note that $g_{i}^{\uparrow\downarrow}$ is one order more than
$g_{r}^{\uparrow\downarrow}$ and this dictates the trend that there is a peak
of $g_{eff,r}^{\uparrow\downarrow}$ at $t=3\,nm$. Such trend cannot be
achieved for metallic magnets where there is no significant
$g_{i}^{\uparrow\downarrow}$. Roy (2017a) Fig. 3 shows the trend of
$R_{r(i)}^{*}/\delta$ with conductivity $\sigma$ when interface plays the key
role ($t\leq 2.5\,nm$). The plot of $\lambda(t)\propto\sigma(t)$ is given in
the Supplementary Fig. S1. The experimental measurements of $\sigma$ with
thickness $t$ is taken from Ref. Castel  _et al._ , 2012. Using Kittel formula
$f=\gamma\sqrt{H_{r}(H_{r}+4\pi M_{s})}$, with $f=9.5\,GHz$, $H_{r}=2545\,Oe$,
and $\gamma=2.8\,MHz/Oe$, the saturation magnetization is calculated as $4\pi
M_{s}=1978\,G$. Castel  _et al._ (2012) Then, $\gamma_{eff}$ for different
thicknesses are calculated from the FMR field shifts Sun  _et al._ (2013) and
accordingly, $g_{eff,i}^{\uparrow\downarrow}$ for different thicknesses are
determined from the Equation (2).

Figure 4 shows the spin-circuit while incorporating a copper (Cu) layer in
between the magnetic insulator (YIG) and platinum (Pt) layers. The circuit
elements are similar to as in the Fig. 1. The thickness of Pt layer (23 nm) is
quite high compared to Pt’s spin diffusion length, so the $Cu|Pt$ interface
contribution is ignored and due to negligible spin-orbit coupling, the
interface resistance of $YIG|Cu$ is not considered. Since no significant FMR
field shift is observed for this multilayer, Sun  _et al._ (2013) we neglect
the imaginary spin mixing conductance and match the trend of experimental
results of only $g_{eff,r}^{\uparrow\downarrow}$ in the Fig. 5. The
interfacial spin mixing conductance turns out to be
$g_{r}^{\uparrow\downarrow}=3.45e19\,m^{-2}$ and the plot of
$\lambda_{Cu}(t_{Cu})\propto\sigma_{Cu}(t_{Cu})$ is given in the Supplementary
Fig. S2. The conductance $G_{1}^{Cu}$ is smaller compared to $G_{2}^{Cu}$ and
$G_{B}^{Pt}$ and thus can be ignored in Fig. 4(c). Since $G_{2}^{Cu}\propto
csch(t_{Cu}/\lambda_{Cu})\simeq\lambda_{Cu}/t_{Cu}$, for
$\lambda_{Cu}>2t_{Cu}$, we get an inverse trend of
$g_{eff,r}^{\uparrow\downarrow}$ with $t_{Cu}$ in the Fig. 5.

To summarize, we have explained the critical trend of effective complex spin
mixing conductance in magnetic insulator/heavy metal bilayers and explaining
the experimental results with a sandwiched copper layer shows the prowess of
the spin-circuit formalism. The interfacial spin mixing conductances for
different samples of different thicknesses causes the variability in the
experimental data points. Roy (2017a) Similar analysis can be applied to other
magnetic insulators and heavy metals in general.

See the supplementary material for additional plots on thickness-dependent
conductivity and spin diffusion length.

This work was supported by Science and Engineering Research Board (SERB) of
India via sanction order SRG/2019/002166.

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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#  Stability in determination of states for the mean field game equations

1 Hongyu Liu and 2 Masahiro Yamamoto

###### Abstract.

We consider solutions satisfying the Neumann zero boundary condition and a
linearized mean field game system in $\Omega\times(0,T)$, where $\Omega$ is a
bounded domain in $\mathbb{R}^{d}$ and $(0,T)$ is the time interval. We prove
two kinds of stability results in determining the solutions. The first is
Hölder stability in time interval $(\varepsilon,T)$ with arbitrarily fixed
$\varepsilon>0$ by data of solutions in $\Omega\times\\{T\\}$. The second is
the Lipschitz stability in $\Omega\times(\varepsilon,T-\varepsilon)$ by data
of solutions in arbitrarily given subdomain of $\Omega$ over $(0,T)$.

1 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
SAR, P.R. China. E-mail<EMAIL_ADDRESS>
2 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,
Meguro, Tokyo 153-8914, Japan. E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, be a smooth bounded
domain and $\nu=\nu(x)$ be the outward unit normal vector to $\partial\Omega$
at $x\in\partial\Omega$. We set $\partial_{\nu}v=\nabla v\cdot\nu$ and

$Q:=\Omega\times(0,T).$

For an arbitrarily given domain $D\subset\mathbb{R}^{d+1}$ in the
$(x,t)$-space, we set

$\|u\|_{H^{2,1}(D)}:=\left(\|u\|^{2}_{L^{2}(D)}+\|\nabla
u\|^{2}_{L^{2}(D)}+\sum_{i,j=1}^{d}\|\partial_{i}\partial_{j}u\|^{2}_{L^{2}(D)}+\|\partial_{t}u\|^{2}_{L^{2}(D)}\right)^{\frac{1}{2}}.$

We consider a system of linearized mean field game equations:

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}u+\Delta u=Q_{1}u+Sv+F,\\\
&\partial_{t}v-\Delta v=Q_{2}(u,v)+\rho_{0}\Delta u+G\quad\mbox{in
$Q$,}\end{array}\right.$

where $\rho_{0}$ is a constant (not necessarily positive) and

$None$ $\left\\{\begin{array}[]{rl}&|Q_{1}(u)(x,t)|\leq C(|u(x,t)|+|\nabla
u(x,t)|),\\\ &|Q_{2}(u,v)(x,t)|\leq C(|u(x,t)|+|v(x,t)|+|\nabla
u(x,t)|+|\nabla v(x,t)|).\end{array}\right.$

We assume

$None$ $|S(v)(x,t)|\leq C|v(x,t)|,\quad(x,t)\in Q,$

or

$None$ $\|Sv(\cdot,t)\|_{L^{2}(\Omega)}\leq
C\|v(\cdot,t)\|_{L^{2}(\Omega)},\quad 0<t<T.$

Moreover we let

$None$ $\partial_{\nu}u=\partial_{\nu}v=0\quad\mbox{on
$\partial\Omega\times(0,T)$}.$

For small $\varepsilon>0$, we set
$Q_{\varepsilon}:=\Omega\times(\varepsilon,\,T)$.

Let $u,v\in H^{2,1}(Q)$ satisfy (1.1)-(1.2) with (1.3) or (1.4). Then
Theorem 1.1.
We assume

$\|\nabla v(\cdot,0)\|_{L^{(}\Omega)}\leq M,$

where $M>0$ is a given constant. For any $\varepsilon\in(0,T)$, there exist
constants $C=C(\varepsilon,M)>0$ and $\theta=\theta(\varepsilon,M)\in(0,1)$
such that

$\|u\|_{H^{2,1}(\Omega\times(\varepsilon,T))}+\|v\|_{H^{2,1}(\Omega\times(\varepsilon,T))}\leq
C(\|u(\cdot,T)\|_{H^{1}(\Omega)}+\|v(\cdot,T)\|_{L^{2}(\Omega)})^{\theta}$

provided that $\|u(\cdot,T)\|_{H^{1}(\Omega)}+\|v(\cdot,T)\|_{L^{2}(\Omega)}$
is sufficiently small.

We refer to Lasry and Lions [9] for a convenient reference on the background
of the mean field game equations (1.1)-(1.5) considered in this paper. As
earlier works by estimates of Carleman’s type, we refer to Klibanov [5],
Klibanov and Averboukh [6], Klibanov, Li and Liu [7], [8]. The stability
estimate in Theorem 1.1 was proved in Klibanov, Li and Liu [8] for a nonlinear
mean field game equations by a different Carleman estimate which is not
attached with second large parameter $\lambda>0$ such as in Theorem 2.1 stated
below. Thus the argument in [8] requires extra estimation for $\rho_{0}\Delta
u$ in the second equation in (1.1). Thanks to the second large parameter
$\lambda>0$, our proof is direct and is applicable to other cases where
$\rho_{0}\Delta$ is replaced by arbitrary, not necessarily elliptic, second-
order partial differential operators. However, we here consider only a simple
case (1.1) for describing our methodology. On the other hand, the constant
$C>0$ in the conclusion of Theorem 1.1 must be larger than [8], Theorem 1.1
immediately implies the uniqueness for the backward problem for (1.1). The
theorem follows directly from the Carleman-type estimate (Theorem 2.1). Our
inverse problem is related to backward parabolic equations in time, for which
we can refer to many works and other methods may be available, and here we are
limited only to two monographs: Ames and Straughan [1], Payne [11].

In Klibanov and Averboukh [6], the Lipschitz stability is proved in
determining $u,v$ in $\Omega\times(0,T)$ by data
$u(\cdot,T),v(\cdot,T),v(\cdot,0)$ in $\Omega$, while for the Hölder stability
in $\Omega\times(\varepsilon,T)$ with arbitrary $\varepsilon>0$ and the
uniqueness in $\Omega\times(0,T)$, we do not need data $v(\cdot,0)$ in
$\Omega$. Moreover Theorem 2.1 stated below readily implies such a global
Lipschitz stability:

$\|u\|_{H^{2,1}(Q)}+\|v\|_{H^{2,1}(Q)}\leq
C(\|u(\cdot,T)\|_{H^{1}(\Omega)}+\|\nabla
v(\cdot,0)\|_{L^{2}(\Omega)}+\|v(\cdot,T)\|_{L^{2}(\Omega)}),$

which can estimate the stronger norm $\|v\|_{H^{2,1}(Q)}$ than [6].

As for a different kind of state determination, Klibanov, Li and Liu [7]
proved conditional Hölder stability.

We can choose the weight $\varphi(t)=e^{-\lambda t}$ with large constant
$\lambda>0$ in Theorem 2.1, and follow the proof of Theorem 1.1, so that we
can obtain the Hölder stability in estimating
$\|u\|_{H^{2,1}(\Omega\times(\varepsilon,T))}+\|v\|_{H^{2,1}(\Omega\times(\varepsilon,T))}$
by $u(\cdot,0)$ and $v(\cdot,0)$ in $\Omega$. In Klibanov [5], the global
Lipschitz stability by data $u(\cdot,0),u(\cdot,T),v(\cdot,0)$ in $\Omega$,
but our result can prove the Hölder stability in determining $u,v$ in
$\Omega\times(\varepsilon,T)$ only by $u(\cdot,0),v(\cdot,0)$ in $\Omega$.
Moreover our key Carleman estimate readily produces the global Lipschitz
stability if we can use $u(\cdot,0),u(\cdot,T),v(\cdot,0)$ in $\Omega$.

##  2\. Proof of Theorem 1.1

We prove a key estimate of Carleman type with two large parameters. We set

$\varphi(t)=e^{\lambda t},$

where a constant $\lambda>0$ is chosen later.
Theorem 2.1.
There exists a constant $C>0$ such that

$\displaystyle\int_{Q}\biggl{\\{}|\partial_{t}u|^{2}+|\Delta
u|^{2}+s\lambda\varphi|\nabla u|^{2}+s^{2}\lambda^{2}\varphi^{2}|u|^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{s\varphi}\left(|\partial_{t}v|^{2}+|\Delta
v|^{2}\right)+\lambda|\nabla
v|^{2}+s\lambda^{2}\varphi|v|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle\int_{\Omega}(s^{2}\lambda|u(x,0)|^{2}+s|\nabla
u(x,0)|^{2})e^{2s}dx$ $\displaystyle+$
$\displaystyle\int_{\Omega}(s\lambda|v(x,0)|^{2}e^{2s}+|\nabla
v(x,T)|^{2}e^{2s\varphi(T)})dx$ $\displaystyle\leq$ $\displaystyle
C\int_{\Omega}(s\lambda\varphi(T)|v(x,T)|^{2}e^{2s\varphi(T)}+|\nabla
v(x,0)|^{2}e^{2s})dx$ $\displaystyle+$ $\displaystyle
C\int_{\Omega}(s\lambda^{2}\varphi(T)^{2}|u(x,T)|^{2}+s\varphi(T)|\nabla
u(x,T)|^{2})e^{2s\varphi(T)}dx$

for all large $s,\lambda>0$.

It is essential that the constant $\lambda>0$ in $e^{\lambda t}$ is activated
as additional large parameter with $s>0$. The weight
$e^{s\varphi}=e^{se^{\lambda t}}$ is double exponential, so that the constant
in the conclusion of Theorem 1.1 may be quite large, but the proof is much
easier. This weight was considered in Murray and Protter [10] for a different
problem.

The proof follows from Theorem 9.1 in Yamamoto [12], and for convenience, I
will give a sketch of the proof in Section 4.

Now we proceed to the proof of Theorem 1.1. We fix $\lambda>0$ sufficiently
large and we can delete the $\lambda$-dependence in Theorem 2.1:

$\displaystyle\int_{Q}\biggl{\\{}|\Delta u|^{2}+\frac{1}{s}|\Delta
v|^{2}+|\partial_{t}u|^{2}+\frac{1}{s}|\partial_{t}v|^{2}$ $\displaystyle+$
$\displaystyle(s|\nabla u|^{2}+|\nabla
v|^{2}+s^{2}|u|^{2}+s|v|^{2}\biggr{\\}}e^{2s\varphi}dxdt$ $\displaystyle\leq$
$\displaystyle C\int_{\Omega}e^{Cs}(s|v(x,T)|^{2}+s|u(x,T)|^{2}+|\nabla
u(x,T)|^{2})dx$ $None$ $+C\int_{\Omega}|\nabla v(x,0)|^{2}e^{2s}dx$

for all large $s>0$.

We consider in $\Omega\times(\varepsilon,T)$, and note

$e^{2s\varphi(t)}\geq e^{2se^{\lambda\varepsilon}}=e^{2s(\mu_{0}+1)},$

where we set

$\mu_{0}:=e^{\lambda\varepsilon}-1>0,$

By $\|\nabla v(\cdot,0)\|_{L^{2}(\Omega)}\leq M$, the estimate (2.1) implies

$e^{2s(\mu_{0}+1)}(\|u\|_{H^{2,1}(\Omega\times(\varepsilon,T))}^{2}+\|v\|_{H^{2,1}(\Omega\times(\varepsilon,T))}^{2})\leq
Cse^{Cs}D^{2}+CsM^{2}e^{2s}.$

Hence,

$None$
$\|u\|_{H^{2,1}(\Omega\times(\varepsilon,T))}^{2}+\|v\|_{H^{2,1}(\Omega\times(\varepsilon,T))}^{2}\leq
Cse^{Cs}D^{2}+CM^{2}e^{-2s\mu_{0}}\leq Ce^{C_{1}s}D^{2}+CM^{2}e^{-2s\mu_{0}}$

for all $s\geq s_{0}$: large constant. Here and henceforth we set

$D:=(\|u(\cdot,T)\|^{2}_{H^{1}(\Omega)}+\|v(\cdot,T)\|_{L^{2}(\Omega)})^{\frac{1}{2}}.$

Replacing $s:=s+s_{0}$ and further choosing a large constant $C>0$, we have
(2.2) for all $s>0$. Without loss of generality, we can assume $D<M$.

Making the right-hand side of (2.2) large, as one possibility we choose $s>0$
satisfying

$e^{C_{1}s}D^{2}=e^{-2s\mu_{0}}M^{2}.$

Then, we have

$s=\frac{2}{C_{1}+2\mu_{0}}\log\frac{M}{D}.$

Then (2.2) yields the conclusion of Theorem 1.1 with

$\theta=\frac{2\mu_{0}}{C_{1}+2\mu_{0}}\in(0,1).$

Thus the proof of Theorem 1.1 is complete.

##  3\. Lipschitz stability in determination of state without initial and
final values

We consider

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}u+\Delta u=Q_{1}u+Sv+F,\\\
&\partial_{t}u-\Delta u=Q_{2}(u,v)+\rho_{0}\Delta u+G\quad\mbox{in
$Q$}\end{array}\right.$

and

$None$ $\partial_{\nu}u=\partial_{\nu}v=0\quad\mbox{on
$\partial\Omega\times(0,T)$}.$

Moreover we assume (1.2) for $Q_{1}$ and $Q_{2}$, and

$|Sv(x,t)|\leq C|v(x,t)|,\quad(x,t)\in Q.$

Let $\omega\subset\Omega$ be an arbitrarily chosen subdomain.

Then we can prove
Theorem 3.1.
For arbitrarily given $\varepsilon>0$, we can find a constant
$C_{\varepsilon}>0$ such that

$\|u\|_{H^{2,1}(\Omega\times(\varepsilon,T-\varepsilon))}+\|v\|_{H^{2,1}(\Omega\times(\varepsilon,T-\varepsilon))}\leq
C_{\varepsilon}(\|F\|_{L^{2}(Q)}+\|G\|_{L^{2}(Q)})+C_{\varepsilon}(\|u\|_{L^{2}(\omega\times(0,T))}+\|v\|_{L^{2}(\omega\times(0,T))})$

and

$\|u(\cdot,t)\|_{L^{2}(\Omega)}+\|v(\cdot,t)\|_{L^{2}(\Omega)}\leq
C_{\varepsilon}(\|F\|_{L^{2}(Q)}+\|G\|_{L^{2}(Q)})+C_{\varepsilon}(\|u\|_{L^{2}(\omega\times(0,T))}+\|v\|_{L^{2}(\omega\times(0,T))})$

for any $\varepsilon\leq t\leq T-\varepsilon$.

This is the Lipschitz stability in the time interval
$(\varepsilon,\,T-\varepsilon)$.

Corollary.
In Theorem 3.1, we further assume that

$Sv(x,t)=q(x,t)v(x,t),\quad(x,t)\in Q\quad\mbox{supp $q\cap\omega$ has
interior points.}$

Then we can replace the right-hand side of the conlusions of Theorem 3.1 by

$C(\|F\|_{L^{2}(Q)}+\|G\|_{L^{2}(Q)})+C\|u\|_{H^{2,1}(\omega\times(0,T))}.$

In the corollary, we note that we do not need any extra data of $v$.

Proof. We set $t_{0}:=\frac{T}{2}$.

We know (e.g., Imanuvilov [3]) that there exists $\eta\in
C^{2}(\overline{\Omega})$ such that

$\eta|_{\partial\Omega}=0,\quad\eta>0\quad\mbox{in
$\Omega$},\quad|\nabla\eta|>0\quad\mbox{on
$\overline{\Omega\setminus\omega}$}.$

Set

$\varphi(x,t):=\frac{e^{\lambda\eta(x)}}{t(T-t)},\quad\alpha(x,t):=\frac{e^{\lambda\eta(x)}-e^{2\lambda\|\eta\|_{C(\overline{\Omega})}}}{t(T-t)}.$

Then
Lemma 3.1.

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\left(|\partial_{t}u|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}u|^{2}\right)+s\varphi|\nabla
u|^{2}+s^{3}\varphi^{3}|u|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}|(\partial_{t}\pm\Delta)u|^{2}e^{2s\alpha}dxdt+C\int_{\omega\times(0,T)}s^{3}\varphi^{3}|u|^{2}e^{2s\alpha}dxdt$

for all $s\geq s_{0}$ and $u\in H^{2,1}(Q)$ satisfying $\partial_{\nu}u=0$ on
$\partial\Omega\times(0,T)$.
Proof of Lemma 3.1.

The case for $\partial_{t}-\Delta$ is known; see e.g., Fursikov and Imanuvilov
[2], Imanuvilov [3].

We consider the case $\partial_{t}u(x,t)+\Delta u(x,t)=F(x,t)$ for
$x\in\Omega$ and $0<t<T$. Setting $w(x,t):=u(x,T-t)$ for $0<t<T$. Then
$\partial_{t}w(x,t)=-(\partial_{t}u)(x,T-t)$ and so

$\partial_{t}w(x,t)-\Delta w(x,t)=-(\partial_{t}u)(x,T-t)-\Delta
u(x,T-t)=-F(x,T-t)$

and $\partial_{\nu}w=0$ on $\partial\Omega\times(0,T)$.

Hence the Carleman estimate for $\partial_{t}-\Delta$ yields

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\left(|\partial_{t}w(x,t)|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}w|^{2}\right)+s\varphi|\nabla
w|^{2}+s^{3}\varphi^{3}|w|^{2}\biggr{\\}}e^{2s\alpha(x,t)}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}|F(x,T-t)|^{2}e^{2s\alpha(x,t)}dxdt+C\int_{\omega\times(0,T)}s^{3}\varphi^{3}|w|^{2}e^{2s\alpha(x,t)}dxdt,$

that is, using $\varphi(x,t)=\varphi(x,T-t)$ and $\alpha(x,t)=\alpha(x,T-t)$,
we obtain

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi(x,T-t)}\left(|\partial_{t}u(x,T-t)|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}u(x,T-t)|^{2}\right)$
$\displaystyle+$ $\displaystyle s2\varphi(x,T-t)|\nabla
u(x,T-t)|^{2}+s^{3}\varphi^{3}|u(x,T-t)|^{2}\biggr{\\}}e^{2s\alpha(x,T-t)}dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}|F(x,T-t)|^{2}e^{2s\alpha(x,T-t)}dxdt+C\int_{\omega\times(0,T)}s^{3}\varphi^{3}|u(x,T-t)|^{2}e^{2s\alpha(x,T-t)}dxdt.$

Changing the variables $t\mapsto\xi$ by $\xi:=T-t$, we can complete the proof
of Lemma 3.1. $\blacksquare$

Moreover
Lemma 3.2 (Imanuvilov and Yamamoto [4]).

$\displaystyle\int_{Q}\biggl{\\{}|\partial_{t}u|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}u|^{2}+s^{2}\varphi^{2}|\nabla
u|^{2}+s^{4}\varphi^{4}|u|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}s\varphi|(\partial_{t}\pm\Delta)u|^{2}e^{2s\alpha}dxdt+C\int_{\omega\times(0,T)}s^{4}\varphi^{4}|u|^{2}e^{2s\alpha}dxdt$

for all large $s\geq s_{0}$ and $u\in H^{2,1}(Q)$ satisfying
$\partial_{\nu}u=0$ on $\partial\Omega\times(0,T)$.

We apply Lemmata 3.2 and 3.1 to $u$ and $v$ respectively by replacing
$F:=F+Q_{1}u+Sv$ and $G:=G+Q_{2}(u,v)+\rho_{0}\Delta u$:

$\displaystyle\int_{Q}\biggl{\\{}|\partial_{t}u|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}u|^{2}+s^{2}\varphi^{2}|\nabla
u|^{2}+s^{4}\varphi^{4}|u|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}s\varphi|F|^{2}e^{2s\alpha}dxdt+C\int_{Q}s\varphi(|\nabla
u|^{2}+|u|^{2}+|v|^{2})e^{2s\alpha}dxdt$ $None$
$+C\int_{\omega\times(0,T)}s^{4}\varphi^{4}|u|^{2}e^{2s\alpha}dxdt$

and

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\left(|\partial_{t}v|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}v|^{2}\right)+s\varphi|\nabla
v|^{2}+s^{3}\varphi^{3}|v|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle C\int_{Q}|G|^{2}e^{2s\alpha}dxdt+C\int_{Q}(|u|^{2}+|\nabla
u|^{2}+|v|^{2}+|\nabla v|^{2})e^{2s\alpha}dxdt$ $None$ $+C\int_{Q}|\Delta
u|^{2}e^{2s\alpha}dxdt+C\int_{\omega\times(0,T)}s^{3}\varphi^{3}|v|^{2}e^{2s\alpha}dxdt$

for all $s\geq s_{0}$.

Applying (3.3) to the third term on the right-hand side of (3.4), we obtain

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\left(|\partial_{t}v|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}v|^{2}\right)+s\varphi|\nabla
v|^{2}+s^{3}\varphi^{3}|v|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}(s\varphi|F|^{2}+|G|^{2})e^{2s\alpha}dxdt+C\int_{Q}s\varphi(|u|^{2}+|\nabla
u|^{2}+|v|^{2})e^{2s\alpha}dxdt$ $\displaystyle+$ $\displaystyle
C\int_{\omega\times(0,T)}s^{4}\varphi^{4}|u|^{2}e^{2s\alpha}dxdt$
$\displaystyle+$ $\displaystyle C\int_{Q}(|u|^{2}+|\nabla
u|^{2}+|v|^{2}+|\nabla v|^{2})e^{2s\alpha}dxdt$ $\displaystyle+$
$\displaystyle
C\int_{\omega\times(0,T)}s^{3}\varphi^{3}|v|^{2}e^{2s\alpha}dxdt.$

For large $s>0$, we can absorb the terms of lower-powers in $s$, so that we
can obtain

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\left(|\partial_{t}v|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}v|^{2}\right)+s\varphi|\nabla
v|^{2}+s^{3}\varphi^{3}|v|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}(s\varphi|F|^{2}+|G|^{2})e^{2s\alpha}dxdt+C\int_{Q}s\varphi(|u|^{2}+|\nabla
u|^{2})e^{2s\alpha}dxdt$ $None$
$+C\int_{\omega\times(0,T)}(s^{4}\varphi^{4}|u|^{2}+s^{3}\varphi^{3}|v|^{2})e^{2s\alpha}dxdt$

for all large $s>0$. Adding (3.3) and (3.5), and absorbing the terms of lower
powers in $s\varphi$, we reach
Theorem 3.2 (global Carleman estimate for the mean field game system).

$\displaystyle\int_{Q}\biggl{\\{}|\partial_{t}u|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}u|^{2}+s^{2}\varphi^{2}|\nabla
u|^{2}+s^{4}\varphi^{4}|u|^{2}$ $\displaystyle+$
$\displaystyle\frac{1}{s\varphi}\left(|\partial_{t}v|^{2}+\sum_{i,j=1}^{d}|\partial_{i}\partial_{j}v|^{2}\right)+s\varphi|\nabla
v|^{2}+s^{3}\varphi^{3}|v|^{2}\biggr{\\}}e^{2s\alpha}dxdt$ $\displaystyle\leq$
$\displaystyle
C\int_{Q}(s\varphi|F|^{2}+|G|^{2})e^{2s\alpha}dxdt+C\int_{\omega\times(0,T)}(s^{4}\varphi^{4}|u|^{2}+s^{3}\varphi^{3}|v|^{2})e^{2s\alpha}dxdt$

for all large $s>0$.

Thanks to Theorem 3.2, we can readily complete the proof of Theorem 3.1.
Indeed, since

$\alpha(x,t)\geq\frac{e^{\lambda\eta(x)}-e^{2\lambda\|\eta\|_{C(\overline{\Omega})}}}{\varepsilon(T-\varepsilon)}\geq\frac{-C_{1}}{\varepsilon(T-\varepsilon)}=:-C_{2}<0$

for all $x\in\overline{\Omega}$ and $\varepsilon\leq t\leq T-\varepsilon$, we
see that

$e^{2s\alpha(x,t)}\geq e^{-2sC_{2}},\quad
x\in\overline{\Omega},\,\varepsilon\leq t\leq T-\varepsilon.$

Thus Theorem 3.2 completes the proof of Theorem 3.1. $\blacksquare$

##  4\. Proof of Theorem 2.1.

First Step.

We consider

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}v-\Delta
v=\widetilde{G}\quad\mbox{in $Q$},\\\ &\partial_{\nu}v=0\quad\mbox{on
$\partial\Omega\times(0,T)$}.\end{array}\right.$

Then we can prove

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}(|\partial_{t}v|^{2}+|\Delta
v|^{2})+\lambda|\nabla
v|^{2}+s\lambda^{2}\varphi|v|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$
$\displaystyle\int_{\Omega}(s\lambda|v(x,0)|^{2}e^{2s}+|\nabla
v(x,T)|^{2}e^{2s\varphi(T)})dx$ $None$ $\leq
C\int_{Q}|\widetilde{G}|^{2}e^{2s\varphi}dxdt+C\int_{\Omega}(s\lambda\varphi(T)|v(x,T)|^{2}e^{2s\varphi(T)}+|\nabla
v(x,0)|^{2}e^{2s})dx$

for all large $s,\lambda>1$ (Yamamoto [12]).
Second Step.
Next we consider

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}u+\Delta
u=\widetilde{F}\quad\mbox{in $Q$},\\\ &\partial_{\nu}u=0\quad\mbox{on
$\partial\Omega\times(0,T)$}.\end{array}\right.$

Then we can prove

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}(|\partial_{t}u|^{2}+|\Delta
u|^{2})+\lambda|\nabla
u|^{2}+s\lambda^{2}\varphi|u|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle\int_{\Omega}(s\lambda|u(x,0)|^{2}+|\nabla
u(x,0)|^{2})e^{2s}dx$ $None$ $\leq
C\int_{Q}|\widetilde{F}|^{2}e^{2s\varphi}dxdt+C\int_{\Omega}(s\lambda\varphi(T)|u(x,T)|^{2}+|\nabla
u(x,T)|^{2})e^{2s\varphi(T)}dx$

for all large $s,\lambda>1$.
Proof of (4.4). The proof is similar to (4.2). We set $w_{1}:=e^{s\varphi}u$
and $P_{1}w_{1}=e^{s\varphi}(\partial_{t}+\Delta)(e^{-s\varphi}w_{1})$. Then

$P_{1}w_{1}=\partial_{t}w_{1}-s\lambda\varphi w_{1}+\Delta
w_{1}=\widetilde{F}e^{s\varphi}$

and so

$\displaystyle\|P_{1}w_{1}\|^{2}_{L^{2}(Q)}\geq\int_{Q}|\partial_{t}w_{1}|^{2}dxdt+2\int_{Q}(\partial_{t}w_{1})\Delta
w_{1}dxdt-\int_{Q}2s\lambda\varphi w_{1}(\partial_{t}w_{1})dxdt$
$\displaystyle=:$
$\displaystyle\int_{Q}|\partial_{t}w_{1}|^{2}dxdt+\widetilde{I_{1}}+\widetilde{I_{2}}.$

Now we estimate

$\displaystyle\widetilde{I_{1}}=-2\int_{Q}\nabla(\partial_{t}w_{1})\cdot\nabla
w_{1}dxdt=-\int_{Q}\partial_{t}(|\nabla w_{1}|^{2})dxdt$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left[|\nabla
w_{1}|^{2}\right]^{0}_{T}dx=\int_{\Omega}(|\nabla w_{1}(x,0)|^{2}-|\nabla
w_{1}(x,T)|^{2})dx$

and

$\displaystyle\widetilde{I_{2}}=-\int_{Q}2w_{1}(\partial_{t}w_{1})s\lambda\varphi
dxdt=-\int_{Q}\partial_{t}(|w_{1}|^{2})s\lambda\varphi dxdt$ $\displaystyle=$
$\displaystyle\int_{\Omega}\left[s\lambda\varphi|w_{1}|^{2}\right]^{0}_{T}dx+\int_{Q}s\lambda^{2}\varphi|w_{1}|^{2}dxdt.$

Consequently,

$None$
$\int_{Q}|\partial_{t}w_{1}|^{2}dxdt+\int_{Q}s\lambda^{2}\varphi|w_{1}|^{2}dxdt+\int_{\Omega}(s\lambda|w_{1}(x,0)|^{2}+|\nabla
w_{1}(x,0)|^{2})dx$ $\leq
C\int_{Q}|\widetilde{F}|^{2}e^{2s\varphi}dxdt+C\int_{\Omega}(s\lambda\varphi(T)|w_{1}(x,T)|^{2}+|\nabla
w_{1}(x,T)|^{2})dx.$

Next we consider

$w_{1}P_{1}w_{1}=w_{1}\partial_{t}w_{1}-s\lambda\varphi|w_{1}|^{2}+w_{1}\Delta
w_{1}\quad\mbox{in $Q$}.$

Integrating over $Q$, we obtain

$\int_{Q}\frac{1}{2}\partial_{t}(|w_{1}|^{2})dxdt-\int_{Q}s\lambda\varphi|w_{1}|^{2}dxdt-\int_{Q}|\nabla
w_{1}|^{2}dxdt=\int_{Q}w_{1}(P_{1}w_{1})dxdt.$

Hence

$\displaystyle-\frac{1}{2}\int_{\Omega}\left[|w_{1}|^{2}\right]^{T}_{0}dx+\int_{Q}s\lambda\varphi|w_{1}|^{2}dxdt+\int_{Q}|\nabla
w_{1}|^{2}dxdt$ $\displaystyle=$
$\displaystyle-\int_{Q}w_{1}(P_{1}w_{1})dxdt\leq\int_{Q}|w_{1}||P_{1}w_{1}|dxdtdxdt,$

that is, the multiplication by $\lambda$ yields

$\displaystyle\lambda\int_{Q}|\nabla
w_{1}|^{2}dxdt+\int_{Q}s\lambda^{2}\varphi|w_{1}|^{2}dxdt+\int_{\Omega}\lambda\left[|w_{1}|^{2}\right]^{0}_{T}dx\leq$
$\displaystyle
C\int_{Q}\lambda^{2}|w_{1}|^{2}dxdt+C\int_{Q}|\widetilde{F}|^{2}e^{2s\varphi}dxdt.$

Here we used

$\int_{Q}\lambda|w_{1}||P_{1}w_{1}|dxdt\leq\frac{1}{2}\int_{Q}\lambda^{2}|w_{1}|^{2}dxdt+\frac{1}{2}\int_{Q}|P_{1}w_{1}|^{2}dxdt.$

Adding (4.5), we see

$\displaystyle\int_{Q}|\partial_{t}w_{1}|^{2}dxdt+\int_{Q}\left(2s\lambda^{2}\varphi-C\lambda^{2}\right)|w_{1}|^{2}dxdt+\int_{Q}\lambda|\nabla
w_{1}|^{2}dxdt$ $\displaystyle+$
$\displaystyle\int_{\Omega}\\{(s\lambda+\lambda)|w_{1}(x,0)|^{2}+|\nabla
w_{1}(x,0)|^{2}\\}dx$ $\displaystyle\leq$ $\displaystyle
C\int_{Q}|\widetilde{F}|^{2}e^{2s\varphi}dxdt+C\int_{\Omega}\\{(s\lambda\varphi(T)+\lambda)|w_{1}(x,T)|^{2}+|\nabla
w_{1}(x,T)|^{2}\\}dx.$

Using

$\frac{1}{s\varphi}|\partial_{t}v|^{2}e^{2s\varphi}\leq\frac{1}{s\varphi}|\partial_{t}w|^{2}+Cs\lambda^{2}\varphi|w|^{2},$

we can complete the proof of (4.4). $\blacksquare$
Third Step.
We set $u_{1}:=\varphi^{\frac{1}{2}}u$. Then
$\partial_{t}u_{1}=\frac{1}{2}\lambda\varphi^{\frac{1}{2}}u+\varphi^{\frac{1}{2}}\partial_{t}u$,
and so

$None$ $\partial_{t}u_{1}+\Delta
u_{1}=\varphi^{\frac{1}{2}}\widetilde{F}+\frac{1}{2}\lambda\varphi^{\frac{1}{2}}u.$

Therefore the application of (4.4) to (4.6) yields

$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}\varphi\left(\left|\partial_{t}u+\frac{1}{2}\lambda
u\right|^{2}+|\Delta u|^{2}\right)+\lambda\varphi|\nabla
u|^{2}+s\lambda^{2}\varphi^{2}|u|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle\int_{\Omega}(s\lambda|u_{1}(x,0)|^{2}+|\nabla
u_{1}(x,0)|^{2})e^{2s}dx$ $\displaystyle=$
$\displaystyle\int_{Q}\biggl{\\{}\frac{1}{s\varphi}(|\partial_{t}u_{1}|^{2}+|\Delta
u_{1}|^{2})+\lambda|\nabla
u_{1}|^{2}+s\lambda^{2}\varphi|u_{1}|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle\int_{\Omega}(s\lambda|u_{1}(x,0)|^{2}+|\nabla
u_{1}(x,0)|^{2})e^{2s}dx$ $\displaystyle\leq$ $\displaystyle
C\int_{Q}|\widetilde{F}|^{2}\varphi
e^{2s\varphi}dxdt+C\int_{Q}\lambda^{2}\varphi|u|^{2}e^{2s\varphi}dxdt$ $None$
$+C\int_{\Omega}(s\lambda\varphi(T)|u_{1}(x,T)|^{2}+|\nabla
u_{1}(x,T)|^{2})e^{2s\varphi(T)}dx.$

Hence, choosing $s>0$ large, we can absorb the second term on the right-hand
side of (4.7) into the left-hand side, we obtain

$None$ $\int_{Q}(|\partial_{t}u|^{2}-C\left|\frac{1}{2}\lambda
u\right|^{2}+|\Delta u|^{2}+s\lambda\varphi|\nabla
u|^{2}+s^{2}\lambda^{2}\varphi^{2}|u|^{2})e^{2s\varphi}dxdt$ $\displaystyle+$
$\displaystyle\int_{\Omega}(s^{2}\lambda|u(x,0)|^{2}+s|\nabla
u(x,0)|^{2})e^{2s}dx$ $\displaystyle\leq$ $\displaystyle
C\int_{Q}s\varphi|\widetilde{F}|^{2}e^{2s\varphi}dxdt+C\int_{\Omega}(s^{2}\lambda\varphi(T)^{2}|u(x,T)|^{2}+s\varphi(T)|\nabla
u(x,T)|^{2})e^{2s\varphi(T)}dx$ $\displaystyle=$ $\displaystyle
C\int_{Q}s\varphi|\widetilde{F}|^{2}e^{2s\varphi}dxdt+CD(u(\cdot,T)).$

Here and henceforth, for short descriptions, we set

$D(u(\cdot,T)):=\int_{\Omega}(s^{2}\lambda\varphi(T)^{2}|u(x,T)|^{2}+s\varphi(T)|\nabla
u(x,T)|^{2})e^{2s\varphi(T)}dx$

with fixed large $\lambda,s>0$.
Fourth Step.
We set $\widetilde{F}:=Q_{1}u+Sv$ and
$\widetilde{G}:=Q_{2}(u,v)+\rho_{0}\Delta u$. We note that

$\displaystyle|\widetilde{F}(x,t)|\leq C(|Sv(x,t)|+|u(x,t)|+|\nabla u(x,t)|),$
$\displaystyle|\widetilde{G}(x,t)|\leq C(|u(x,t)|+|\nabla
u(x,t)|+|v(x,t)|+|\nabla v(x,t)|+|\Delta u(x,t)|)$

for $(x,t)\in Q$. Moreover, in the case (1.4), we estimate

$\int_{Q}s\varphi|Sv(x,t)|^{2}e^{2s\varphi}dxdt=\int^{T}_{0}s\varphi
e^{2s\varphi}\left(\int_{\Omega}|Sv(x,t)|^{2}dx\right)dt=\int^{T}_{0}s\varphi
e^{2s\varphi}\|Sv(\cdot,t)\|_{L^{2}(\Omega)}^{2}dt$ $\leq
C\int^{T}_{0}s\varphi
e^{2s\varphi}\|v(\cdot,t)\|^{2}_{L^{2}(\Omega)}dt=C\int_{Q}s\varphi|v|^{2}e^{2s\varphi}dxdt.$

Therefore,

$None$ $\int_{Q}s\varphi|\widetilde{F}|^{2}e^{2s\varphi}dxdt\leq
C\int_{Q}s\varphi(|v|^{2}+|u|^{2}+|\nabla u|^{2})e^{2s\varphi}dxdt.$

Moreover (4.8) implies

$\int_{Q}|\widetilde{G}|^{2}e^{2s\varphi}dxdt\leq C\int_{Q}(|u|^{2}+|\nabla
u|^{2}+|v|^{2}+|\nabla v|^{2}+|\Delta u|^{2})e^{2s\varphi}dxdt$ $None$ $\leq
C\int_{Q}s\varphi|\widetilde{F}|^{2}e^{2s\varphi}dxdt+C\int_{Q}(|v|^{2}+|\nabla
v|^{2})e^{2s\varphi}dxdt+CD(u(\cdot,T)).$

Adding (4.2) and (4.8), and applying (4.9) and (4.10), we obtain

$\displaystyle\int_{Q}\biggl{\\{}|\partial_{t}u|^{2}+|\Delta
u|^{2}+s\lambda\varphi|\nabla
u|^{2}+s^{2}\lambda^{2}\varphi^{2}|u|^{2}+\frac{1}{s\varphi}(|\partial_{t}v|^{2}+|\Delta
v|^{2})+\lambda|\nabla
v|^{2}+s\lambda^{2}\varphi|v|^{2}\biggr{\\}}e^{2s\varphi}dxdt$
$\displaystyle+$
$\displaystyle\int_{\Omega}(s\lambda|v(x,0)|^{2}e^{2s}+|\nabla
v(x,T)|^{2}e^{2s\varphi(T)})dx$ $\displaystyle+$
$\displaystyle\int_{\Omega}(s^{2}\lambda|u(x,0)|^{2}+s|\nabla
u(x,0)|^{2})e^{2s}dx$ $\displaystyle\leq$ $\displaystyle
C\int_{Q}(s\varphi|\widetilde{F}|^{2}+|\widetilde{G}|^{2})e^{2s\varphi}dxdt+C\int_{\Omega}(s\lambda\varphi(T)|v(x,T)|^{2}e^{2s\varphi(T)}+|\nabla
v(x,0)|^{2}e^{2s})dx+CD(u(\cdot,T))$ $\displaystyle\leq$ $\displaystyle
C\int_{Q}s\varphi(|v|^{2}+|u|^{2}+|\nabla
u|^{2})e^{2s\varphi}dxdt+C\int_{Q}(|v|^{2}+|\nabla v|^{2})e^{2s\varphi}dxdt$
$\displaystyle+$ $\displaystyle
CD(u(\cdot,T))+C\int_{\Omega}(s\lambda\varphi(T)|v(x,T)|^{2}e^{2s\varphi(T)}+|\nabla
v(x,0)|^{2}e^{2s})dx.$

Choosing $s,\lambda>0$ large and noting that the powers of the terms $|v|^{2}$
are $s\lambda^{2}\varphi$ on the left-hand side and $s\varphi$ and $1$ on the
right-hand side, etc., we can absorb the first and the second terms on the
right-hand side into the left-hand side, we complete the proof of Theorem 2.1.
$\blacksquare$

Acknowledgements. The authors thanks Professor M.V. Klibanov (University of
North Carolina at Charlotte) for valuable comments. The work was supported by
Grant-in-Aid for Scientific Research (A) 20H00117 of Japan Society for the
Promotion of Science.

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