# Existence of global-in-time solutions to a system
of fully nonlinear parabolic equations

Takahiro Kosugi 111Tottori University of Environmental Studies, Tottori,
Japan<EMAIL_ADDRESS>and Ryuichi Sato222Fukuoka University, Fukuoka,
Japan<EMAIL_ADDRESS>

###### Abstract

We consider the Cauchy problem for a system of fully nonlinear parabolic
equations. In this paper, we shall show the existence of global-in-time
solutions to the problem. Our condition to ensure the global existence is
specific to the fully nonlinear parabolic system.

Keywords: viscosity solutions, fully nonlinear parabolic systems, global-in-
time solutions, comparison principle
MSC: 35A01, 35D40, 35K45, 35K55

##  1 Introduction

Let us consider the Cauchy problem for a weakly coupled system of nonlinear
parabolic equations

$\left\\{\begin{aligned}
\partial_{t}u_{1}+F_{1}(x,D^{2}u_{1})=|u_{2}|^{p-1}u_{2},\quad
x\in\bm{R}^{N},\ t>0,\\\
\partial_{t}u_{2}+F_{2}(x,D^{2}u_{2})=|u_{1}|^{q-1}u_{1},\quad
x\in\bm{R}^{N},\ t>0,\end{aligned}\right.$ (1.1)

with initial condition

$u_{i}(x,0)=u_{i0}(x),\quad x\in\bm{R}^{N}\mbox{ for }i=1,2,$ (1.2)

where $N\geq 1$, $p,q>0$, $F_{1},F_{2}\in C(\bm{R}^{N}\times S^{N})$ are
uniformly elliptic and homogeneous of order one, and $u_{10}$, $u_{20}\in
BUC(\bm{R}^{N})$ are nonnegative. Here $\partial_{t}u_{i}$ denotes the
derivative $\partial u_{i}/\partial t$ and $D^{2}u_{i}$ denotes the Hessian
matrix of $u_{i}$ in the variable $x$. Throughout this paper, we let $S^{N}$
denote the $N\times N$ real symmetric matrices and let $BUC(\bm{R}^{N})$
denote the set of bounded uniformly continuous functions on $\bm{R}^{N}$.

In [4], Escobedo and Hererro considered the Cauchy problem for a system of
semilinear parabolic equations

$\partial_{t}u_{1}-\bigtriangleup
u_{1}=u_{2}^{p},\quad\partial_{t}u_{2}-\bigtriangleup u_{2}=u_{1}^{q},\quad
x\in\bm{R}^{N},t>0$ (1.3)

with (1.2), where $N\geq 1$, $p,q>0$, and $\bigtriangleup$ denotes the Laplace
operator, that is,

$\bigtriangleup:=\sum_{j=1}^{N}\frac{\partial^{2}}{\partial x_{j}^{2}}.$

The system (1.3) agrees with the case $F_{1}(x,X)=F_{2}(x,X)=-\mathrm{tr}(X)$
for $x\in\bm{R}^{N}$, $X\in S^{N}$. Escobedo and Hererro proved that if $pq>1$
and

$\frac{\max\\{p,q\\}+1}{pq-1}\geq\frac{N}{2},$

then every nontrivial nonnegative solution to (1.3) blows up in a finite time.
On the other hand, if $pq>1$ and

$\frac{\max\\{p,q\\}+1}{pq-1}<\frac{N}{2},$ (1.4)

then there exists a global-in-time solution to (1.3) for some $u_{10}$,
$u_{20}$. These results show that the existence of nonnegative global-in-time
solutions to (1.3) is clarified by the curve

$\frac{\max\\{p,q\\}+1}{pq-1}=\frac{N}{2}.$ (1.5)

This Fujita type result for (1.3) is extended by [6], [18] to the case where
the system with linear but unequal principal parts. In [18], $-\bigtriangleup
u_{1},-\bigtriangleup u_{2}$ are replaced by the linear operators of the form

$\displaystyle L_{1}u_{1}=-\sum_{j,k=1}^{N}\frac{\partial}{\partial
x_{j}}\left(a^{jk}\frac{\partial u_{1}}{\partial x_{k}}\right),\quad
L_{2}u_{2}=-\sum_{j,k=1}^{N}\frac{\partial}{\partial
x_{j}}\left(b^{jk}\frac{\partial u_{2}}{\partial x_{k}}\right),$

where the coefficients $a^{jk}$, $b^{jk}$ are sufficiently smooth, uniformly
elliptic and symmetric. In particular, the system with constant diffusion
coefficients $L_{1}=-d_{1}\bigtriangleup u_{1}$, $L_{2}=-d_{2}\bigtriangleup
u_{1}$, $d_{1}$,$d_{2}>0$ is considered in [6] (see also [5] for another
context). The Fujita exponent for the system

$\partial_{t}u_{1}+L_{1}u_{1}=u_{2}^{p},\quad\partial_{t}u_{2}+L_{2}u_{2}=u_{1}^{q},\quad
x\in\bm{R}^{N},\ t>0$

is also given by (1.5). Namely, the Fujita exponent is given by (1.5) if the
principal parts are linear.

Let us introduce results for a single equation. Setting $u_{1}=u_{2}=u$,
$F_{1}=F_{2}=F$ and $p=q>1$, then (1.1) becomes a single nonlinear parabolic
equation

$\partial_{t}u+F(x,D^{2}u)=u^{p},\quad x\in\bm{R}^{N},\ t>0.$ (1.6)

Typical examples of $F$ are given below. When $F(D^{2}u)=-\bigtriangleup u$,
(1.6) is the Fujita equation. In [7], Fujita considered the Cauchy problem for
(1.6) with $F(x,D^{2}u)=-\bigtriangleup u$. He proved that the critical
exponent for the existence of nonnegative global-in-time solutions is given by

$\frac{1}{p-1}=\frac{N}{2}.$

More precisely, if $1<p<p_{F}:=1+2/N$, then all positive solutions blow-up in
a finite time, while if $p>p_{F}$, then there exists a positive global-in-time
solution of (1.6). (Readers are referred to [3] for a survey of blow-up
problems.) When $F$ is fully nonlinear, the critical exponent for the
existence of global-in-time solutions to (1.6) was obtained in [16, 17]. We
employ the viscosity solutions to treat fully nonlinear equations. To give
prrecise examples and state the existence of viscosity solutions, we suppose
precise assumptions on $F_{i}$’s. The definition of viscosity solutions is
given in the next section.

For $i=1,2$, we assume that $F_{i}:\bm{R}^{N}\times S^{N}\to\bm{R}$ satisfies
the following properties.

  1. (i)

$F_{i}$ is continuous in $\bm{R}^{N}\times S^{N}$, that is,

$\displaystyle F_{i}\in C(\bm{R}^{N}\times S^{N}).$ (1.7)

  2. (ii)

There exist constants $0<\lambda_{i}\leq\Lambda_{i}$ such that

$\displaystyle\mathcal{P}_{i}^{-}(X-Y)\leq
F_{i}(x,X)-F_{i}(x,Y)\leq\mathcal{P}_{i}^{+}(X-Y)$ (1.8)

for $(x,X,Y)\in\bm{R}^{N}\times S^{N}\times S^{N}$, where
$\mathcal{P}^{\pm}_{i}$ are the Pucci extremal operators defined by

$\displaystyle\mathcal{P}^{+}_{i}(X)$
$\displaystyle=\mathcal{P}^{+}_{\lambda_{i},\Lambda_{i}}(X):=\max\\{\mathrm{tr}[-AX]\
|\lambda_{i}I\leq A\leq\Lambda_{i}I,\ A\in S^{N}\\},$
$\displaystyle\mathcal{P}^{-}_{i}(X)$
$\displaystyle=\mathcal{P}^{-}_{\lambda_{i},\Lambda_{i}}(X):=\min\\{\mathrm{tr}[-AX]\
|\lambda_{i}I\leq A\leq\Lambda_{i}I,\ A\in S^{N}\\},$

for $X\in S^{N}$.

  3. (iii)

$F_{i}$ is Lipshitz continuous in $x$. Namely, there exists $L>0$ such that

$|F_{i}(x,X)-F_{i}(y,X)|\leq L(\|X\|+1)|x-y|$ (1.9)

for all $X\in S^{N}$ and $x,y\in\bm{R}^{N}$. Here $\|X\|$ stands for the
operator norm of $X$.

  4. (iv)

$F_{i}$ is homogeneous of order one. Namely,

$F_{i}(x,\mu X)=\mu F_{i}(x,X)$ (1.10)

for $\mu\geq 0$, $x\in\bm{R}^{N}$, $X\in S^{N}$.

We shall give two examples of $F:\bm{R}^{N}\times S^{N}\to\bm{R}$ satisfying
above conditions.

  * •

Let $0<\gamma<1$. The operator

$F(D^{2}u)=\max\left\\{-\frac{\bigtriangleup
u}{1-\gamma},-\frac{\bigtriangleup u}{1+\gamma}\right\\}$

is nonlinear and convex. Equation $\partial_{t}u+F(D^{2}u)=0$ is called the
Barenblatt equation of Elasto-Plastic equation. See [9] and [11].

  * •

Let $N=2$. Then

$F(D^{2}u)=\min\left\\{\max\\{-\bigtriangleup u,-2\bigtriangleup
u\\},-u_{x_{1}x_{1}}-2u_{x_{2}x_{2}}\right\\}$

is a nonlinear and nonconvex operator.

We now state the comparison principle. The proof is given in Section 5.

######  Theorem 1.1 (Comparison principle).

Assume that $p,q\geq 1$ and let $T>0$. Let $(u_{1},u_{2})\in USC\cap
L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$ be a viscosity subsolution and
$(v_{1},v_{2})\in LSC\cap L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$ be a
viscosity supersolution of (1.1), respectively. If

$\displaystyle u_{i}(\cdot,0)\leq v_{i}(\cdot,0)\quad\mbox{in
}\bm{R}^{N}\mbox{ for }i=1,2,$

then

$\displaystyle u_{i}\leq v_{i}\quad\mbox{in }\bm{R}^{N}\times(0,T)\mbox{ for
}i=1,2.$

Existence of viscosity solutions to (1.1) and (1.2) is guaranteed by the
following:

######  Theorem 1.2.

Assume that $p,q\geq 1$ and $pq>1$. Let $u_{10},u_{20}\in BUC(\bm{R}^{N})$.
There exist $T>0$ and a unique viscosity solution $(u_{1},u_{2})$ of (1.1)
satisfying (1.2) in $\bm{R}^{N}\times[0,T]$. Furthermore, if $u_{i0}\geq 0$
for $i=1,2$, then $u_{i}\geq 0$ for $i=1,2$, as long as the solution exists.
Moreover, $u_{i}\in BUC(\bm{R}^{N}\times[0,T))$.

We temporarily go back to the results obtained by Meneses and Quaas [16, 17].
They treated (1.6) for the case $F=F(D^{2}u)$ is an $x$-independent operator
which satisfies (1.7)–(1.10) with the ellipticity constants
$0<\lambda\leq\Lambda$. Then there exists $\alpha=\alpha(F)>0$ such that if
$1<p\leq 1+1/\alpha$, then there exists no global-in-time solutions for any
$u_{0}\in BUC(\bm{R}^{N})$. While if $p>1+1/\alpha$, then there exists a
global-in-time solution for some $u_{0}\in BUC(\bm{R}^{N})$. These results
mean that $1+1/\alpha$ is the Fujita exponent for (1.6).

######  Remark 1.1.

We give several remarks about $\alpha=\alpha(F)$.

  1. (i)

If $F$ and $G$ are uniformly elliptic homogeneous such that $F\leq G$, then

$\alpha(F)\leq\alpha(G).$

Moreover, if $0<\lambda\leq\Lambda$ are the ellipticity constants of $F$, then
it holds that

$\frac{N\lambda}{2\Lambda}\leq\alpha(\mathcal{P}_{\lambda,\Lambda}^{-})\leq\alpha(F)\leq\alpha(\mathcal{P}_{\lambda,\Lambda}^{+})\leq\frac{N\Lambda}{2\lambda}.$

See [1, (3.21)] and [17, Lemma 2.2].

  2. (ii)

If $F=F(X)$ is convex, then $\alpha(F)\geq N/2$ and this inequality is strict
unless $F$ is linear (see [1, Example 3.12]).

  3. (iii)

Note that $\alpha$ coincides with the eigenvalue of

$F(D^{2}\psi)-\frac{1}{2}y\cdot\nabla\psi=\alpha\psi,\quad y\in\bm{R}^{N}.$
(1.11)

We next give a remark for the case $F=F(x,D^{2}u)$ depending on $x$. In [17],
it was shown that there exists $\tilde{\alpha}=\tilde{\alpha}(F)>0$ such that
for all solutions of

$\partial_{t}w+F(x,D^{2}w)=0,\quad x\in\bm{R}^{N},\ t>0,\quad
w(x,0)=w_{0}(x),\quad x\in\bm{R}^{N}$

satisfy

$\lim_{t\to\infty}t^{\tilde{\alpha}}\|w(\cdot,t)\|_{L^{\infty}}<\infty,$

whenever $w_{0}\in BUC(\bm{R}^{N})$ satisfies $0\leq w_{0}(x)\leq
A\exp(-B|x|^{2})$ for some $A,B>0$. While if $\beta>\tilde{\alpha}$, it holds
that

$\lim_{t\to\infty}t^{\beta}\|w(\cdot,t)\|_{L^{\infty}}=\infty.$

This is well-known for the case $F(D^{2}u)=-\bigtriangleup u$ (see e.g. [15]).
For $x$ depending case, the Fujita exponent is also given by
$1+1/\tilde{\alpha}$. However, the critical case $p=1+1/\tilde{\alpha}$ has
never been treated for $x$ depending case even for the single equation.

In this paper, we would like to prove the existence of global-in-time
solutions for the system of fully nonlinear parabolic equations (1.1). In our
setting, we have the choices of a combination of $F_{1}$ and $F_{2}$. Let
$\alpha_{i}=\alpha(F_{i})>0$ be the corresponding eigenvalue of (1.11)
replacing $F$ by $F_{i}$. As mentioned above, $\alpha_{i}>N/2$ if $F_{i}$ is
convex and nonlinear. Therefore, we can expect a different condition to
guarantee the existence of a global-in-time solution from (1.4).

Our main result is the following:

######  Theorem 1.3.

Let $F_{1}$, $F_{2}$ be independent of $x$. Suppoe that $F_{i}$’s satisfy
(1.7)–(1.10). Further assume that $p,q\geq 1$ satisfy $pq>1$ and

$p>\frac{\Lambda_{2}}{\lambda_{1}},\quad q>\frac{\Lambda_{1}}{\lambda_{2}}.$
(1.12)

There exist positive constants $\alpha_{1}$ and $\alpha_{2}$ such that, if

$\frac{p+1}{pq-1}<\alpha_{1}\quad\mathit{and}\quad\frac{q+1}{pq-1}<\alpha_{2},$
(1.13)

then there exists a global-in-time solution to (1.1) and (1.2) for some
$u_{10}$, $u_{20}\in BUC(\bm{R}^{N})$.

Our theorem gives a sufficient condition for the exsistence of global-in-time
solutions to (1.1) similar to the Fujita type result. We consider a solution
$\psi$ of the problem

$F(D^{2}\psi)-\frac{1}{2}y\cdot D\psi=\mu\psi,\quad
y\in\bm{R}^{N},\quad\lim_{|y|\to\infty}\psi(y)=0,$

where $F$ satisfies (1.7)–(1.10). To prove our main theorem, let us apply an
estimate for $\psi$ of the form

$c\exp(-\delta|y|^{2})\leq\psi(y)\leq C\exp(-\delta|y|^{2}).$

Using this estimate under the condtion (1.12), we can find a supersolution to
obtain global-in-time solutions. See Section 4. Note that (1.12) is not needed
for the single equation.

The rest of this paper is organized as follows. In Section 2, we give a
precise definition of viscosity solutions of the Cauchy problem and prepare
several notation. In Section 3, we give a proof of the existence of local-in-
time solutions of (1.1) and (1.2). In Section 4, we prove Theorem 1.3. We give
a proof of Theorem 1.1 in Section 5. In the appendix, for the convenience of
the reader, we give a detailed proof of Perron’s method.

##  2 Preliminaries

For a real valued function $f$ defined in $\bm{R}^{N}\times(0,T)$, define the
upper (resp. lower) semi-continuous envelope $f^{*}$ (resp. $f_{*}$) of $f$ by

$\displaystyle f^{*}(x,t):=\lim_{\varepsilon\to
0}\sup_{\begin{subarray}{c}y\in B(x,\varepsilon)\\\
|s-t|<\varepsilon\end{subarray}}f(y,s),\quad f_{*}(x,t):=\lim_{\varepsilon\to
0}\inf_{\begin{subarray}{c}y\in B(x,\varepsilon)\\\
|s-t|<\varepsilon\end{subarray}}f(y,s),$ (2.1)

for $x\in\bm{R}^{N}$, $t\in(0,T)$. It is well known that $f^{*}$ is upper
semi-continuous, $f_{*}$ is lower semi-continuous and $f_{*}\leq f\leq f^{*}$.
Furthermore, if $f$ is upper semi-continuous, then $f^{*}=f$. The same
property holds for $f_{*}$.

We prepare some notation. Let $A$ be a subset of $\bm{R}^{N}\times[0,\infty)$.
The sets $USC(A)$ and $LSC(A)$ stand for the set of upper semicontinuous
functions on $A$ and lower semicontinuous functions on $A$ respectively. Let
$\Omega\subset\bm{R}^{N}$ and let $BUC(\Omega)$ denote the set of bounded
uniformly continuous functions on $\Omega$. For $T>0$,
$C^{2,1}=C^{2,1}(\bm{R}^{n}\times(0,T))$ denotes the set of all functions
which is $C^{2}$ in the variable $x$ and $C^{1}$ in the variable $t$.

We recall the definition of viscosity solutions of general parabolic systems

$\displaystyle\partial_{t}u_{i}+G_{i}(x,t,u_{1},\ldots,u_{m},Du_{i},D^{2}u_{i})=0,\quad\mbox{in
}\Omega\times(0,T),\ \mbox{for }i=1,\ldots,m,$ (2.2)

where $T\in(0,\infty]$ and $\Omega$ is an open subset of $\bm{R}^{N}$.

######  Definition 2.1.

We call $u=(u_{1},\ldots,u_{m}):\Omega\times(0,T)\to\bm{R}^{m}$ a viscosity
subsolution (resp., supersolution) of (2.2) if for
$(i,x,t,\phi)\in\\{1,\ldots,m\\}\times\Omega\times(0,T)\times
C^{2,1}(\Omega\times(0,T))$,

$\displaystyle\partial_{t}\phi(x,t)+G_{i}(x,t,u_{1}^{*}(x,t),\ldots,u_{m}^{*}(x,t),D\phi(x,t),D^{2}\phi(x,t))$
$\displaystyle\leq 0,$ $\displaystyle(resp.\
\partial_{t}\phi(x,t)+G_{i}(x,t,{u_{1}}_{*}(x,t),\ldots,{u_{m}}_{*}(x,t),D\phi(x,t),D^{2}\phi(x,t))$
$\displaystyle\geq 0)$

provided that $u_{i}^{*}-\phi$ (resp. ${u_{i}}_{*}-\phi$) attains its local
maximum (resp., minimum) at $(x,t)\in\Omega\times(0,T)$. We call
$u:\Omega\times(0,T)\to\bm{R}^{m}$ a viscosity solution of (2.2) if $u$ is a
viscosity sub- and supersolution of (2.2).

We also define a solution to the Cauchy problem.

######  Definition 2.2.

Let $u=(u_{1},\ldots,u_{m}):\Omega\times(0,T)\to\bm{R}^{m}$ be a viscosity
subsolution of (2.2). We call $u$ a viscosity subsolution of the Cauchy
problem (2.2) and

$u_{1}(\cdot,0)=u_{10},\dots,u_{m}(\cdot,0)=u_{m0}$

if $u$ satisfies

$u_{1}(\cdot,0)\leq u_{10},\dots,u_{m}(\cdot,0)\leq u_{m0}\quad\mathrm{in}\
\Omega.$

A viscosity supersolution is also defined in the same way.

######  Definition 2.3.

Define parabolic semi-jet $PJ^{2,+}u(x,t)$ of a function
$u:\bm{R}^{N}\times(0,\infty)\to\bm{R}$ at
$(x,t)\in\bm{R}^{N}\times[0,\infty)$ by

$\displaystyle PJ^{2,+}u(x,t)$ (2.3)
$\displaystyle:=\biggl{\\{}(a,z,X)\in\bm{R}\times\bm{R}^{N}\times S^{N}\
\biggl{|}\ u(y,s)\leq u(x,t)+\langle z,y-x\rangle$
$\displaystyle\quad+\frac{1}{2}\langle
X(y-x),y-x\rangle+a(s-t)+o(|y-x|^{2}+|s-t|)\quad\mathrm{as}\ y\to x,s\to
t.\biggr{\\}},$

where $\langle\cdot,\cdot\rangle$ denotes the standard innnar product on
$\bm{R}^{N}$. We also define $PJ^{2,-}u(x,t):=-PJ^{2,+}(-u(x,t))$. Moreover, a
sort of closure of semi-jet $\overline{PJ}^{2,\pm}u(x,t)$ is defined as
follows: $(a,z,X)\in\bm{R}\times\bm{R}^{N}\times S^{N}$ is a point of
$\overline{PJ}^{2,\pm}u(x,t)$ if there exist sequences
$(x_{k},t_{k})\in\bm{R}^{N}\times(0,\infty)$ and $(a_{k},z_{k},X_{k})\in
PJ^{2,\pm}u(x,t)$ such that

$x_{k}\to x,\quad t_{k}\to t,\quad u(x_{k},t_{k})\to u(x,t),\quad a_{k}\to
a,\quad z_{k}\to z,\quad X_{k}\to X$

as $k\to\infty$.

##  3 Existence of local-in-time solutions

In this section, we give proof of Theorem 1.2. To prove the local existence of
viscosity solution, we refer to important results from [2]. The following
Lemma is modified for the convenience of our argument.

######  Lemma 3.1.

Let $F:S^{N}\to\bm{R}$ be continuous and let satisfy the ellipticity condition

$F(Y)\leq F(X)\quad\mathrm{whenever}\ X\leq Y,\quad X,Y\in S^{N}.$ (3.1)

  1. (i)

If $u_{0}$ is uniformly continuous on $\bm{R}^{N}$, then the Cauchy problem

$\partial_{t}u+F(D^{2}u)=0\quad\mathrm{in}\ \bm{R}^{N},\quad
u(\cdot,0)=u_{0}\quad\mathrm{in}\ \bm{R}^{N}$ (3.2)

has a unique viscosity solution $u\in C(\bm{R}^{N}\times[0,\infty))$, which is
uniformly continuous in $x\in\bm{R}^{N}$. Moreover, if $u_{0}\in
BUC(\bm{R}^{N})$, then the unique solution $u$ of (3.2) is bounded and
uniformly continuous in $x\in\bm{R}^{N}$.

  2. (ii)

Assume $u_{0}\in BUC(\bm{R}^{N})$. Then the solution $u$ of (3.2) generates a
semigroup $\\{S(t)\\}_{t\geq 0}$ on $BUC(\bm{R}^{N})$, which satisfies the
following properties.

    1. (1)

For any $\varphi,\psi\in BUC(\bm{R}^{N})$,

$\|S(t)\varphi-S(t)\psi\|_{L^{\infty}}\leq\|\varphi-\psi\|_{L^{\infty}},\quad
t>0.$

    2. (2)

For any $\varphi\in BUC(\bm{R}^{N})$,

$\lim_{t\to+0}\|S(t)\varphi-\varphi\|_{L^{\infty}}=0.$

###### Proof of Theorem 1.2.

The proof is based on [16]. It can be seen that $\mathcal{P}_{i}^{-}$
satisfies (3.1) by the definition. Let $\\{S_{i}(t)\\}$ be an order preserving
semigroup generated by $\mathcal{P}_{i}^{-}$. Then, by Lemma 3.1 for each
$i=1,2$, $z_{i}(x,t)=[S_{i}(t)u_{i0}](x)$ is a viscosity solution to

$\partial_{t}z_{i}+\mathcal{P}_{i}^{-}(D^{2}z_{i})=0,\quad
x\in\bm{R}^{N},\,t>0,\quad z_{i}(x,0)=u_{i0}(x),\quad x\in\bm{R}^{N}.$

Furthermore, $S_{i}(t)$ satisfies

$\|S_{i}(t)\varphi-
S_{i}(t)\psi\|_{L^{\infty}}\leq\|\varphi-\psi\|_{L^{\infty}},\quad t>0$ (3.3)

for any $\varphi,\psi\in BUC(\bm{R}^{N})$ and

$\lim_{t\to+0}\|S_{i}(t)\varphi-\varphi\|_{L^{\infty}}=0.$

Let $T>0$. Define
$\Psi:(BUC(\bm{R}^{N}\times[0,T])^{2}\to(BUC(\bm{R}^{N}\times[0,T])^{2}$ by

$\Psi[v_{1},v_{2}](t):=(\Phi_{1}[v_{2}](t),\Phi[v_{1}](t)),\quad 0\leq t\leq
T,$

where

$\displaystyle\Phi_{1}[v_{2}](t)$
$\displaystyle:=S_{1}(t)u_{10}+\int_{0}^{t}S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))\,ds,$
$\displaystyle\Phi_{2}[v_{1}](t)$
$\displaystyle:=S_{2}(t)u_{20}+\int_{0}^{t}S_{2}(t-s)(|v_{1}(s)|^{q-1}v_{1}(s))\,ds.$

For the sake of convenience, we shall show that $\Psi$ is a contraction on a
closed subset of $(BUC(\bm{R}^{N}\times[0,T]))^{2}$. For $M>0$ and $T>0$, the
closed ball $B_{T,M}:=\\{v\in BUC(\bm{R}^{N}\times[0,T]):\sup_{0\leq t\leq
T}\|v(t)\|_{L^{\infty}(\bm{R}^{N})}\leq M\\}$ is a complete metric space.
Without loss of generality, we may assume $u_{10}\not\equiv 0$. Moreover, we
only need to consider $\Phi_{1}$ due to the symmetry. Set
$M:=2(\|u_{10}\|_{L^{\infty}(\bm{R}^{N})}+\|u_{20}\|_{L^{\infty}(\bm{R}^{N})})>0$.
Let $v_{2},\tilde{v}_{2}\in B_{T,M}$. Thanks to (3.3), we see that
$\|S_{i}(t)u_{10}\|_{L^{\infty}(\bm{R}^{N})}\leq\|u_{10}\|_{L^{\infty}(\bm{R}^{N})}$
for all $t\in[0,T]$ and so $S(t)u_{10}\in B_{M}$. Moreover, we have

$\left|\int_{0}^{t}S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))\,ds\right|\leq\int_{0}^{t}\||v_{2}(s)|^{p-1}v_{2}(s)\|_{L^{\infty}(\bm{R}^{N})}\leq
tM^{p}$

for $t\in[0,T]$. Thus,

$\|\Phi_{1}[v_{2}](t)\|_{L^{\infty}(\bm{R}^{N})}\leq\|u_{10}\|_{L^{\infty}(\bm{R}^{N})}+TM^{p}.$
(3.4)

We next use (3.3) to see

$\displaystyle|\Phi_{1}[v_{2}](t)-\Phi_{1}[\tilde{v}_{2}](t)|$
$\displaystyle\leq\int_{0}^{t}\|\\{(|v_{2}(s)|^{p-1}v_{2}(s))-(|\tilde{v}_{2}(s)|^{p-1}\tilde{v}_{2}(s))\\}\|_{L^{\infty}(\bm{R}^{N})}\,ds.$

By the mean value theorem, we see that there exists some $C>0$ such that

$|\Phi_{1}[v_{2}](t)-\Phi_{1}[\tilde{v}_{2}](t)|\leq CTM^{p-1}\sup_{0\leq
t\leq T}\|v_{2}(t)-\tilde{v}_{2}(t)\|_{L^{\infty}(\bm{R}^{N})}$

for $t\in[0,T]$. It follows that

$\sup_{0\leq t\leq
T}\|\Phi_{1}[v_{2}]-\Phi_{1}[\tilde{v}_{2}]\|_{L^{\infty}(\bm{R}^{N})}\leq
CTM^{p-1}\sup_{0\leq t\leq
T}\|v_{2}-\tilde{v}_{2}\|_{L^{\infty}(\bm{R}^{N})}.$ (3.5)

Therefore, taking $T>0$ small enough, we see that $\Psi$ is a contraction map
on $(B_{T,M})^{2}$. By (3.4) and (3.5), the Banach fixed point theorem is
applied and there exists a unique fixed point so that
$\Psi[v_{1},v_{2}]=(\Phi_{1}[v_{2}],\Phi_{2}[v_{1}])=(v_{1},v_{2})\in
B_{T,M}^{2}$. Namely, we have

$\displaystyle v_{1}(t)$
$\displaystyle=S_{1}(t)u_{10}+\int_{0}^{t}S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))\,ds,$
(3.6) $\displaystyle v_{2}(t)$
$\displaystyle=S_{2}(t)u_{20}+\int_{0}^{t}S_{2}(t-s)(|v_{1}(s)|^{q-1}v_{1}(s))\,ds$

in $\bm{R}^{N}\times[0,T]$. Furthermore, it follows from (3.6) that

$\|v_{1}(t)-S_{1}(t)u_{10}\|_{L^{\infty}(\bm{R}^{N})}\leq M^{p}t\to 0$

as $t\to+0$, hence

$\lim_{t\to+0}\|v_{1}(t)-u_{10}\|_{L^{\infty}(\bm{R}^{N})}=0.$

We have the same convergence of $v_{2}$ by the same argument.

By the regularity theory (see e.g. [13, Theorem 1.6, Chapter 13], [14, Theorem
14.10], [20, Theorem 4.13]), we know that $S_{i}(t)u_{i0}$ is a classical
solution of $\partial_{t}w+\mathcal{P}^{-}_{i}(D^{2}w)=0$. It follows from
(3.6) that $\partial_{t}v_{1},\partial_{t}v_{2}$ exist. Taking the derivative
of the right-hand side of (3.6), we see that

$\displaystyle\frac{\partial}{\partial
t}[S_{1}(t)u_{10}]=-\mathcal{P}_{1}^{-}(D^{2}[S_{1}(t)u_{10}]),$
$\displaystyle\frac{\partial}{\partial
t}\int_{0}^{t}S_{1}(t-s)(|v_{2}(s)|^{p-1}v(s))\,ds$
$\displaystyle=-\int_{0}^{t}\mathcal{P}^{-}_{1}(D^{2}[S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))])\,ds+|v_{2}(t)|^{p-1}v_{2}(t).$

The same estimate also allows $\partial_{t}v_{2}$ to exist. Thus, they satisfy

$\displaystyle\partial_{t}v_{1}+\mathcal{P}^{-}_{1}(D^{2}S_{1}(t)u_{10})$
$\displaystyle=-\int_{0}^{t}\mathcal{P}^{-}_{1}(D^{2}[S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))])\,ds+|v_{2}|^{p-1}v_{2}(t),$
$\displaystyle\partial_{t}v_{2}+\mathcal{P}^{-}_{2}(D^{2}S_{2}(t)u_{20})$
$\displaystyle=-\int_{0}^{t}\mathcal{P}^{-}_{2}(D^{2}[S_{2}(t-s)(|v_{1}(s)|^{q-1}v_{1}(s))])\,ds+|v_{1}|^{q-1}v_{1}(t).$

It follows from the property
$\mathcal{P}_{i}^{-}(X+Y)\geq\mathcal{P}_{i}^{-}(X)+\mathcal{P}_{i}^{-}(Y)$,
that

$\displaystyle-\int_{0}^{t}\mathcal{P}^{-}_{1}(D^{2}[S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))])\,ds$
$\displaystyle\geq-\mathcal{P}^{-}_{1}\left(\int_{0}^{t}D^{2}[S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))])\,ds\right).$

Furthermore, since $S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))$ is of $C^{2}$ as a
function of $x$, we have

$\int_{0}^{t}D^{2}[S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))]\,ds=D^{2}\int_{0}^{t}S_{1}(t-s)(|v_{2}(s)|^{p-1}v_{2}(s))\,ds,$

hence

$\displaystyle\partial_{t}v_{1}+\mathcal{P}_{1}^{-}\left(D^{2}\left[S_{1}(t)u_{10}+\int_{0}^{t}S_{1}(t-s)|v_{2}|^{p-1}v_{2}(s)\,ds\right]\right)$
$\displaystyle\geq|v_{2}|^{p-1}v_{2}(t),$ (3.7)
$\displaystyle\partial_{t}v_{2}+\mathcal{P}_{2}^{-}\left(D^{2}\left[S_{2}(t)u_{20}+\int_{0}^{t}S_{2}(t-s)|v_{1}|^{q-1}v_{1}(s)\,ds\right]\right)$
$\displaystyle\geq|v_{1}|^{q-1}v_{1}(t)\quad$

in $\bm{R}^{N}\times[0,T]$. Note that the integral preserve linear properties.
We note that for $i=1,2$, $F_{i}$ satisfies

$\mathcal{P}^{-}_{i}(X)\leq F_{i}(x,X),\quad x\in\bm{R}^{N},\,X\in S^{N}.$

Consequently, we then deduce from (3.7) that

$\displaystyle\partial_{t}v_{1}+F_{1}\left(x,D^{2}v_{1}\right)\geq|v_{2}|^{p-1}v_{2},\quad\partial_{t}v_{2}+F_{2}\left(x,D^{2}v_{2}\right)\geq|v_{1}|^{q-1}v_{1}$

for $x\in\bm{R}^{N}$ and $t>0$. Namely, $(v_{1},v_{2})$ is a viscosity
supersolution of (1.1) and (1.2). Replacing $\mathcal{P}^{-}$ by
$\mathcal{P}^{+}$, we can also obtain a viscosity subsolution of (1.1)
satisfying (1.2).

By the Perron method and the comparison principle, there exists a continuous
viscosity solution $(u_{1},u_{2})$ of (1.1) satisfying (1.2). Nonnegativity of
solutions follows from the comparison principle.

Finally, we shall show that $u_{i}\in BUC(\bm{R}^{N}\times[0,T))$ for $i=1,2$.
We refer to [8, Section 3.5] for the method. Let
$(\underline{u}_{1},\underline{u}_{2})$ and
$(\overline{u}_{1},\overline{u}_{2})$ be viscosity subsolution and viscosity
supersolution to (1.1) and (1.2) obtained above. We can see that

$\underline{u}_{i}(x,t)-u_{i0}(x)\leq
u_{i}(x,t)-u_{i0}(x)\leq\overline{u}_{i}(x,t)-u_{i0}(x)$

for $x,y\in\bm{R}^{N}$, $t>0$. There exists a modulus of continuity
$\omega:[0,\infty)\to[0,\infty)$, $\omega(0)=0$, since
$\underline{u}_{i},\overline{u}_{i}\in BUC(\bm{R}^{N}\times[0,T))$. Then we
have

$\underline{u}_{i}(x,t)-u_{i0}(x)\geq-\omega(t)$

and

$\overline{u}_{i}(x,t)-u_{i0}(x)\leq\omega(t),$

which implies that

$\sup_{\begin{subarray}{c}{i=1,2}\\\
x\in\bm{R}^{n}\end{subarray}}|u_{i}(x,t)-u_{i0}(x)|\leq\omega(t).$ (3.8)

For fixed $h>0$, set

$\displaystyle\overline{v}_{i}(x,t)$
$\displaystyle:=u_{i}(x,t+h)+\omega(h),\quad x\in\bm{R}^{N},\ t\geq 0,$
$\displaystyle\underline{v}_{i}(x,t)$
$\displaystyle:=u_{i}(x,t+h)-\omega(h),\quad x\in\bm{R}^{N},\ t\geq 0,$

for $i=1,2$. Then $(\overline{v}_{1},\overline{v}_{2})$ is a viscosity
supersolution and $(\underline{v}_{1},\underline{v}_{2})$ is a viscosity
subsolution to (1.1) and (1.2). We see from (3.8) that
$\underline{v}_{i}(x,0)\leq u_{i0}(x)\leq\overline{v}_{i}(x,0)$. By Theorem
1.1, we see that

$\underline{v}_{i}(x,t)\leq u_{i}(x,t)\leq\overline{v}_{i}(x,t)\quad
x\in\bm{R}^{N},\ t\in[0,T].$

Therefore, we obtain

$|u_{i}(x,t)-u_{i}(x,t+h)|\leq\omega(h)$

for all $x\in\bm{R}^{N}$. This shows that $u_{i}$ is uniformly continuous with
respect to variable $t$. Since $u_{i0}\in BUC(\bm{R}^{N})$, there exists
another modulus of continuity $\omega$ so that

$\sup_{i=1,2}|u_{i0}(x)-u_{i0}(y)|\leq\omega(|x-y|)$

for $x,y\in\bm{R}^{N}$. Similarly to the above discussion, set for
$h\in\bm{R}^{N}$,

$\displaystyle\overline{w}_{i}(x,t):=u_{i}(x+h,t)+\omega(|h|),\quad
x\in\bm{R}^{N},\ t\geq 0,$
$\displaystyle\underline{w}_{i}(x,t):=u_{i}(x+h,t)-\omega(|h|),\quad
x\in\bm{R}^{N},\ t\geq 0,$

Then $(\underline{w}_{1},\underline{w}_{2})$ is a viscosity subsolution and
$(\overline{w}_{1},\overline{w}_{2})$ is a supersolution to (1.1) and (1.2).
Since $\underline{w}_{i}(x,0)\leq u_{i}(x,0)\leq\overline{w}_{i}(x,0)$, by
Theorem 1.1, we obtain

$\sup_{i=1,2}|u_{i}(x,t)-u_{i}(x+h,t)|\leq\omega(|h|)$

for all $t\in[0,T)$. Summarizing, $u_{i}$’s are uniformly continuous in
$\bm{R}^{N}\times[0,T)$. ∎

##  4 Existence of global-in-time solutions (proof of Theorem 1.3)

In this section, we shall prove the existence of global-in-time solutions to
(1.1) and (1.2). We use the following Lemma.

######  Lemma 4.1 ([1, Lemma 3.10]).

Let $0<\lambda\leq\Lambda$. Assume that $F$ satisfies (1.8) and (1.10). For
each $\delta<(4\Lambda)^{-1}$, there exists $C>0$ such that

$\psi(y)\leq C\exp(-\delta|y|^{2}),\quad y\in\bm{R}^{N},$

where $\psi$ is the profile function of a unique positive self-similar
solution $\Phi$ of $\partial_{t}u+F(D^{2}u)=0$ appearing as
$\Phi(x,t)=t^{-\alpha(F)}\psi(x/\sqrt{t})$. Likewise, for each
$\delta>(4\lambda)^{-1}$, there exists $C>0$ such that

$C\exp(-\delta|y|^{2})\leq\psi(y),\quad y\in\bm{R}^{N}.$

###### Proof of Theorem 1.2.

For $i=1,2$, let $\psi_{i}$ be a positive solution of the eigenvalue problem

$F_{i}(D^{2}\psi_{i})-\frac{1}{2}y\cdot D\psi_{i}=\mu\psi_{i},\quad
y\in\bm{R}^{N},\quad\lim_{|y|\to\infty}\psi(y)=0.$ (4.1)

Let $(\alpha(F_{i}),\psi_{i})$ be the eigenpair of (4.1). Set
$\alpha_{i}=\alpha(F_{i})$. The existence of solution to (4.1) is obtained in
[1, Section 3]. See also [16]. Let us look forward a supersolution to (1.1) of
the form

$\overline{u_{1}}(x,t):=\varepsilon(t+1)^{a}\phi_{1}(x,t+1),\quad\overline{u_{2}}(x,t):=\tilde{\varepsilon}(t+1)^{b}\phi_{2}(x,t+1),$

where $\phi_{i}$ is defined by

$\phi_{i}(x,t):=t^{-\alpha_{i}}\psi_{i}(t^{-\frac{1}{2}}x),\quad
x\in\bm{R}^{N},\,t>0$

and $a,b$ will be defined. We refer to [19, Section 32] for the case of linear
diffusion. For each $i=1,2$, $\phi_{i}$ satisfies

$\partial_{t}\phi_{i}+F_{i}(D^{2}\phi_{i})=0\quad\mathrm{in}\
\bm{R}^{N}\times(0,\infty)$ (4.2)

in the sense of viscosity solution. In fact, by the argument used in [16,
Lemma 3.1], we can see that $\phi_{i}$ satisfies (4.2).

In what follows, we shall find a sufficient condition that
$(\overline{u}_{1},\overline{u}_{2})$ becomes a viscosity supersolution of
(1.1). We have

$\displaystyle\partial_{t}\overline{u_{1}}$
$\displaystyle=a\varepsilon(t+1)^{a-1}\phi_{1}+\varepsilon(t+1)^{a}\partial_{t}\phi_{1}(x,t+1),\quad
x\in\bm{R}^{N},\ t\geq 0.$ (4.3)

Assume that $\overline{u}_{1}-\Phi$ attains its minimum at $(x,t)$ and
satisfies that

$(\overline{u}_{1}-\Phi)(x,t)=0.$

Note that the function

$\phi_{1}(\cdot,\cdot+1)-\frac{1}{\varepsilon(t+1)^{a}}\Phi$

also attains minimum at $(x,t)$. Since $\phi_{1}(\cdot,\cdot+1)\in
C^{1,1}(\bm{R}^{N}\times(0,\infty))$ and $\phi_{1}$ is a viscosity solution of
(4.2), it holds that

$\displaystyle\partial_{t}\Phi(x,t)$
$\displaystyle=a\varepsilon(t+1)^{a-1}\phi_{1}+\varepsilon(t+1)^{a}\partial_{t}\phi_{1}$
$\displaystyle\geq
a\varepsilon(t+1)^{a-1}\phi_{1}-F_{1}\left(\frac{1}{\varepsilon(t+1)^{a}}D^{2}\Phi\right)$
$\displaystyle=\varepsilon(t+1)^{a-1}\phi_{1}-F_{1}(D^{2}\Phi).$

On the other hands, we have

$\displaystyle\overline{u_{2}}^{p}$
$\displaystyle=\tilde{\varepsilon}^{p}(t+1)^{bp}\phi_{2}^{p},\quad t\geq 0.$
(4.4)

Combining (4.3)–(4.4), we see that

$\partial_{t}\Phi+F_{1}(D^{2}\Phi)-\overline{u_{2}}^{p}\geq(t+1)^{bp-\alpha_{2}p}[\varepsilon
a(t+1)^{a-1-\alpha_{1}-bp+\alpha_{2}p}\psi_{1}-\tilde{\varepsilon}^{p}\psi_{2}^{p}]$
(4.5)

at $(x,t)$. In the same way, it holds that

$\partial_{t}\Phi+F_{2}(D^{2}\Phi)-\overline{u_{1}}^{q}\geq(t+1)^{aq-\alpha_{1}q}[\varepsilon
b(t+1)^{b-1-\alpha_{1}-aq+\alpha_{1}q}\psi_{2}-\tilde{\varepsilon}^{q}\psi_{1}^{q}]$
(4.6)

at $(x,t)$.

To ensure that the right-hand sides of (4.5), (4.6) become nonnegative, it
suffices that

$\displaystyle a-1-\alpha_{1}-bp+\alpha_{2}p\geq 0,\quad\varepsilon
a\psi_{1}\geq\tilde{\varepsilon}^{p}\psi_{2}^{p},$ $\displaystyle
b-1-\alpha_{2}-aq+\alpha_{1}q\geq
0,\quad\tilde{\varepsilon}b\psi_{2}\geq\varepsilon^{q}\psi_{1}^{q}.$

Solving

$a-1-\alpha_{1}-bp+\alpha_{2}p=0,\quad b-1-\alpha_{2}-aq+\alpha_{1}q=0,$

we find the conditions

$a=\alpha_{1}-\frac{p+1}{pq-1},\quad b=\alpha_{2}-\frac{q+1}{pq-1}.$ (4.7)

Under the conditions (4.7), $a>0$ and $b>0$ are equivalent to (1.13), that is,

$\alpha_{1}>\frac{p+1}{pq-1},\quad\alpha_{2}>\frac{q+1}{pq-1}.$

If $\varepsilon=\tilde{\varepsilon}$, then we can find $\varepsilon>0$ so
small that $a\geq\varepsilon^{p-1}(\psi_{2}^{p}/\psi_{1})$ and
$b\geq\varepsilon^{q-1}(\psi_{1}^{q}/\psi_{2})$. Note that
$\psi_{2}^{p}/\psi_{1}$ and $\psi_{1}^{q}/\psi_{2}$ are bounded. Indeed,
applying Lemma 4.1 for $F_{i}$, for each $a_{i}<(4\Lambda_{i})^{-1}$, there
exists $C_{i}^{+}>0$ such that

$\psi_{i}(y)\leq C_{i}^{+}\exp(-a_{i}|y|^{2}),\quad y\in\bm{R}^{N},\quad
i=1,2.$

We also see that for each $b_{i}>(4\lambda)^{-1}$, there exists $C_{i}^{-}>0$
such that

$C_{i}^{-}\exp(-b_{i}|y|^{2})\leq\psi_{i}(y),\quad y\in\bm{R}^{N},$

hence

$C_{1}\frac{\exp(-b_{2}|y|^{2})^{p}}{\exp(-a_{1}|y|^{2})}\leq\frac{\psi_{2}^{p}}{\psi_{1}}\leq
C_{2}\frac{\exp(-a_{2}|y|^{2})^{p}}{\exp(-b_{1}|y|^{2})},$

where $C_{1}=(C_{2}^{-})^{p}/C_{1}^{+}$ and $C_{2}=(C_{2}^{+})^{p}/C_{1}^{-}$.
We also have a similar estimate for $\psi_{1}^{q}/\psi_{2}$. Finally, it
follows from (1.12) that $\psi_{1}^{q}/\psi_{2}$ and $\psi_{1}^{q}/\psi_{2}$
are bounded. Therefore, assuming (1.13) and choosing
$u_{i0}(x):=\bar{u}_{i}(x,0)$, by the Perron method, there exists a global-in-
time solution $(u_{1},u_{2})$ to (1.1) and (1.2). ∎

##  5 Proof of Theorem 1.1

In this section, we prove Theorem 1.1.

######  Lemma 5.1 ([12, Proposition 3.8 (2)]).

Asssume that $F$ satisfy (1.8) and (1.9). There exists a modulus of continuity
$\omega_{F}:[0,\infty)\to\bm{R}$ such that, if $X$, $Y\in S^{N}$, $\mu>1$
satisfy

$\displaystyle-3\mu\begin{pmatrix}I&O\\\
O&I\end{pmatrix}\leq\begin{pmatrix}X&O\\\ O&-Y\end{pmatrix}\leq
3\mu\begin{pmatrix}I&-I\\\ -I&I\end{pmatrix},$

then

$F(y,Y)-F(x,X)\leq\omega_{F}\left(|x-y|+\mu|x-y|^{2}\right)$

for all $x,y\in\bm{R}^{N}$.

######  Lemma 5.2.

Let $(u_{1},u_{2})\in USC\cap L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$ (resp.,
$LSC\cap L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$) be a viscosity subsolution
(resp., supersolution) of (1.1). We set for $i=1,2$,

$\displaystyle w_{i}:=e^{-\nu t}u_{i},$

where $\lambda>0$ is a constant. Then, $(w_{1},w_{2})$ is a viscosity
subsolution (resp., supersolution) of

$\left\\{\begin{aligned} \partial_{t}w_{1}+F_{1}(x,D^{2}w_{1})+\nu
w_{1}-e^{(p-1)\nu t}|w_{2}|^{p-1}w_{2}=0,\quad x\in\bm{R}^{N},\ t>0,\\\
\partial_{t}w_{2}+F_{2}(x,D^{2}w_{2})+\nu w_{2}-e^{(q-1)\nu
t}|w_{1}|^{q-1}w_{1}=0,\quad x\in\bm{R}^{N},\ t>0.\end{aligned}\right.$ (5.1)

###### Proof.

We shall argue only $w_{1}$. Let $\varphi\in C^{2}(\bm{R}^{N}\times[0,T))$ be
such that $w_{1}-\varphi$ achieve a maximum at $(x_{0},t_{0})$ and

$(w_{1}-\varphi)(x_{0},t_{0})=0.$

Then for all $(x,t)\in\bm{R}^{N}\times[0,T)$,

$e^{-\nu
t}u_{1}(x,t)-\varphi(x,t)=(w_{1}-\varphi)(x,t)\leq(w_{1}-\varphi)(x_{0},t_{0})=0.$

We have $u_{1}(x,t)-e^{-\nu t}\varphi(x,t)\leq 0$ for all
$(x,t)\in\bm{R}^{N}\times[0,T)$. On the other hand, $u_{1}(x_{0},t_{0})-e^{\nu
t_{0}}\varphi(x_{0},t_{0})=0$. Thus, $u_{1}-e^{\nu t}\varphi$ attains a
maximum at $(x_{0},t_{0})$. Since $(u_{1},u_{2})$ is a viscosity subsolution
of (1.1), we have, at $(x,t)=(x_{0},t_{0})$,

$\displaystyle 0$ $\displaystyle\geq\nu e^{\nu t}\varphi+e^{\nu
t}\partial_{t}\varphi+F_{1}(x,e^{\nu t}D^{2}\varphi)-|u_{2}|^{p-1}u_{2}$
$\displaystyle=\nu e^{\nu t}\varphi+e^{\nu t}\partial_{t}\varphi+e^{\nu
t}F_{1}(x,D^{2}\varphi)-e^{p\nu t}|w_{2}|^{p-1}w_{2}.$

We here used (1.10). Therefore, we obtain

$0\geq\nu w_{1}+\partial_{t}\varphi+F_{1}(x,D^{2}\varphi)-e^{(p-1)\nu
t}|w_{2}|^{p-1}w_{2}.$

By the same argument, we also obtain

$0\geq\nu w_{2}+\partial_{t}\varphi+F_{2}(x,D^{2}\varphi)-e^{(q-1)\nu
t}|w_{1}|^{q-1}w_{1}.$

Consequently, $(w_{1},w_{2})$ is a viscosity solution of (5.1). ∎

Theorem 1.1 is shown by proving the following proposition:

######  Proposition 5.1.

Let $p,q\geq 1$ and $T>0$. Let $(u_{1},u_{2})\in USC\cap
L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$ be a viscosity subsolution and
$(v_{1},v_{2})\in LSC\cap L^{\infty}(\bm{R}^{N}\times[0,T))^{2}$ be a
viscosity supersolution of (5.1), respectively. Assume that there exists a
constant $R>0$ such that for all $\lambda>0$,

$|e^{\nu t}u_{i}|,|e^{\nu t}v_{i}|\leq R\quad\mathrm{in}\
\bm{R}^{N}\times[0,T),i=1,2.$ (5.2)

If $u_{i0}\leq v_{i0}$ in $\bm{R}^{N}$ for $i=1,2$, then $u_{i}\leq v_{i}$ in
$\bm{R}^{N}\times[0,T)$ for $i=1,2$.

###### Proof.

For $\mu$, $\delta>0$, define

$\theta_{\mu,\delta}:=\sup_{i,x,t}\\{u_{i}(x,t)-v_{i}(x,t)-\frac{\mu}{T-t}-\delta|x|^{2}\\}.$

By the assumption (5.2), we see that $\theta_{\mu,\delta}\leq 2e^{-\nu t}R\leq
2R$. Put $\displaystyle\theta:=\limsup_{\mu,\delta\to 0}\theta_{\mu,\delta}$.

If $\theta\leq 0$, for all $x$, $t$, $i$, we have

$u_{i}(x,t)-v_{i}(x,t)\leq\frac{\mu}{T-t}+\delta|x|^{2}+\theta_{\mu,\delta}.$

Taking $\limsup$, we then have $u_{i}(x,t)-v_{i}(x,t)\leq 0$.

To obtain a contradiction, suppose that $\theta>0$. There exists a subsequence
(expressed by the same symbol) such that $\theta_{\mu,\delta}\geq\theta/2>0$.
In what follows fix $\mu$ and $\delta$ sufficiently small. Consider the
doubling of the variables

$(i,x,y,t,s)\mapsto
u_{i}(x,t)-v_{i}(y,s)-\frac{\mu}{T-t}-\delta|x|^{2}-\frac{|x-y|^{2}}{2\varepsilon}-\frac{|t-s|^{2}}{2\varepsilon},$

where $\varepsilon\in(0,1)$ is a parameter. Assume that the doubling map
attains a maximum at
$(i_{\varepsilon},x_{\varepsilon},y_{\varepsilon},t_{\varepsilon},s_{\varepsilon})$.
We may assume
$(i_{\varepsilon},x_{\varepsilon},y_{\varepsilon},t_{\varepsilon},s_{\varepsilon})\in\\{1,2\\}\times\overline{B_{r_{\delta}}}^{2}\times[0,T-\tau_{\mu}]^{2}$
for some $r_{\delta}>0$ and $\tau_{\mu}>0$, where $B_{r}$ stand for the ball
centered at the origin with radius $r>0$.

It follows from

$\theta_{\mu,\delta}\leq
u_{i}(x,t)-v_{i}(y,s)-\frac{\mu}{T-t}-\delta|x|^{2}-\frac{|x-y|^{2}}{2\varepsilon}-\frac{|t-s|^{2}}{2\varepsilon}$

at
$(i_{\varepsilon},x_{\varepsilon},y_{\varepsilon},t_{\varepsilon},s_{\varepsilon})$
that

$\displaystyle\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{2\varepsilon}+\frac{|t_{\varepsilon}-s_{\varepsilon}|^{2}}{2\varepsilon}+\frac{\mu}{T-t_{\varepsilon}}+\delta|x_{\varepsilon}|^{2}$
$\displaystyle\leq
u_{i_{\varepsilon}}(x_{\varepsilon},t_{\varepsilon})-v_{i_{\varepsilon}}(y_{\varepsilon},s_{\varepsilon})-\theta_{\mu,\delta}$
(5.3) $\displaystyle\leq 4R.$

Taking a subsequence of $i_{\varepsilon}$ such that
$i_{\varepsilon}\equiv\hat{i}\in\\{1,2\\}$ for sufficiently $\varepsilon\ll
1$. Take a subsequence if necessary, we find
$(\hat{x},\hat{t},\hat{i})\in\overline{B_{r_{\delta}}}\times[0,T-\tau_{\mu})\times\\{1,2\\}$
such that

$x_{\varepsilon},y_{\varepsilon}\to\hat{x},\quad
t_{\varepsilon},s_{\varepsilon}\to\hat{t},\quad i_{\varepsilon}\to\hat{i}$

as $\varepsilon\to 0$. It follows from (5.3) that

$\displaystyle\limsup_{\varepsilon\to
0}\left(\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{2\varepsilon}+\frac{|t_{\varepsilon}-s_{\varepsilon}|^{2}}{2\varepsilon}\right)$
$\displaystyle\leq\limsup_{\varepsilon\to
0}\left(u_{\hat{i}}(x_{\varepsilon},t_{\varepsilon})-v_{\hat{i}}(y_{\varepsilon},_{\varepsilon})-\frac{\mu}{T-t_{\varepsilon}}-\delta|x_{\varepsilon}|^{2}\right)-\theta_{\mu,\delta}$
$\displaystyle\leq
u_{\hat{i}}(\hat{x},\hat{t})-v_{\hat{i}}(\hat{y},\hat{s})-\frac{\mu}{T-\hat{t}}-\delta|\hat{x}|^{2}-\theta_{\mu,\delta}$
$\displaystyle\leq 0.$

Thus, we obtain

$\lim_{\varepsilon\to
0}\frac{|x_{\varepsilon}-y_{\varepsilon}|^{2}}{2\varepsilon}=\lim_{\varepsilon\to
0}\frac{|t_{\varepsilon}-s_{\varepsilon}|^{2}}{2\varepsilon}=0.$

To obtain a contradiction, suppose that $\hat{t}=0$. Then

$\displaystyle 0$ $\displaystyle\leq\limsup_{\varepsilon\to
0}\left(u_{\hat{i}}(x_{\varepsilon},t_{\varepsilon})-v_{\hat{i}}(y_{\varepsilon},_{\varepsilon})-\frac{\mu}{T-t_{\varepsilon}}-\delta|x_{\varepsilon}|^{2}\right)-\theta_{\mu,\delta}$
$\displaystyle=u_{\hat{i}}(\hat{x},0)-v_{\hat{i}}(\hat{x},0)-\frac{\mu}{T}-\delta|\hat{x}|^{2}-\theta_{\mu,\delta}$
$\displaystyle\leq 0.$

Therefore, we obtain

$u_{\hat{i}}(\hat{x},0)-v_{\hat{i}}(\hat{x},0)-\frac{\mu}{T}-\delta|\hat{x}|^{2}=\theta_{\mu,\delta}.$

which is impossible. Therefore, $\hat{t}$ must be positive.

By the Ishii lemma (see [10, Lemma 2.3.23], [8, Chapter 3]), it holds that

$\displaystyle\left(\frac{t_{\varepsilon}-s_{\varepsilon}}{\varepsilon},\frac{x_{\varepsilon}-y_{\varepsilon}}{\varepsilon},X\right)$
$\displaystyle\in\overline{PJ}^{2,+}\left(u_{\hat{i}}(x_{\varepsilon},t_{\varepsilon})-\delta|x_{\varepsilon}|^{2}-\frac{\mu}{T-t_{\varepsilon}}\right),$
$\displaystyle\left(\frac{t_{\varepsilon}-s_{\varepsilon}}{\varepsilon},\frac{x_{\varepsilon}-y_{\varepsilon}}{\varepsilon},Y\right)$
$\displaystyle\in\overline{PJ}^{2,-}v_{\hat{i}}(y_{\varepsilon},s_{\varepsilon})$

and

$\displaystyle-\frac{3}{\varepsilon}\begin{pmatrix}I&O\\\
O&I\end{pmatrix}\leq\begin{pmatrix}X&O\\\
O&-Y\end{pmatrix}\leq\frac{3}{\varepsilon}\begin{pmatrix}I&-I\\\
-I&I\end{pmatrix},$

where $\overline{PJ}^{2,+}$ are defined in (2.3). In general,

$\displaystyle\left(\frac{t_{\varepsilon}-s_{\varepsilon}}{\varepsilon},\frac{x_{\varepsilon}-y_{\varepsilon}}{\varepsilon},X\right)$
$\displaystyle\in\overline{PJ}^{2,+}\left(u_{\hat{i}}(x_{\varepsilon},t_{\varepsilon})-\delta|x_{\varepsilon}|^{2}-\frac{\mu}{T-t_{\varepsilon}}\right)$

is equivalent to

$\displaystyle\left(\frac{t_{\varepsilon}-s_{\varepsilon}}{\varepsilon}+\frac{\mu}{(T-t_{\varepsilon})^{2}},\frac{x_{\varepsilon}-y_{\varepsilon}}{\varepsilon}+2\delta
x_{\varepsilon},X+2\delta I\right)$
$\displaystyle\in\overline{PJ}^{2,+}u_{\hat{i}}(x_{\varepsilon},t_{\varepsilon}).$

We now set $i:=\hat{i}$, $p_{1}=p$, $p_{2}=q$ and $j$ denotes $j\neq i$. In
what following ,we drop the subscription $\varepsilon$ for simplisity. Since
$(u_{1},u_{2})$ is a viscosity subsolution to (5.1) and $(v_{1},v_{2})$ is a
viscosity supersolution to (5.1), we have

$\displaystyle\frac{t-s}{\varepsilon}+\frac{\mu}{(T-t)^{2}}+F_{i}(x,X+2\delta
I)+\nu u_{i}(x,t)-e^{(p_{i}-1)\nu t}|u_{j}|^{p_{i}-1}u_{j}\leq 0,$
$\displaystyle\frac{t-s}{\varepsilon}+F_{i}(y,Y)+\nu
v_{i}(y,s)-e^{(p_{i}-1)\nu t}|v_{j}|^{p_{i}-1}v_{j}\geq 0.$

Combining these inequalities, we see that

$\displaystyle\frac{\mu}{(T-t)^{2}}+\nu(u_{i}(x,t)-v_{i}(y,s))-e^{(p_{i}-1)\nu
t}(|u_{j}|^{p_{i}-1}u_{j}-|v_{j}|^{p_{i}-1}v_{j})$ (5.4) $\displaystyle\leq
F_{i}(y,Y)-F_{i}(x,X+2\delta I).$

By Lemma 5.1, there exists a modulus of continuity
$\omega_{F_{i}}:[0,\infty)\to\bm{R}$ such that

$F_{i}(y,Y)-F_{i}(x,X)\leq\omega_{F_{i}}\left(|x-y|+\frac{|x-y|^{2}}{\varepsilon}\right)$

for all $x,y\in\bm{R}^{N}$. This together with (1.8) implies that

$\displaystyle F_{i}(y,Y)-F_{i}(x,X+2\delta I)+F_{i}(x,X)-F_{i}(x,X)$ (5.5)
$\displaystyle\leq\omega_{F_{i}}\left(|x-y|+\frac{|x-y|^{2}}{\varepsilon}\right)+\mathcal{P}_{i}^{+}(-2\delta
I)$
$\displaystyle=\omega_{F_{i}}\left(|x-y|+\frac{|x-y|^{2}}{\varepsilon}\right)+2\delta\Lambda_{i}N.$

If $u_{j}(x,t)\leq v_{j}(y,s)$, then

$\nu(u_{i}-v_{i})-e^{(p_{i}-1)\nu
t}(|u_{j}|^{p_{i}-1}u_{j}-|v_{j}|^{p_{i}-1}v_{j})\geq\lambda(u_{i}-v_{i})\geq\lambda\frac{\theta}{2}>0.$
(5.6)

On the other hand, if $u_{j}(x,t)>v_{j}(y,s)$, it follows from the mean value
theorem and (5.2) that

$\displaystyle e^{(p_{i}-1)\nu
t}(|u_{j}|^{p_{i}-1}u_{j}-|v_{j}|^{p_{i}-1}v_{j})$
$\displaystyle=e^{(p_{i}-1)\nu
t}\frac{|u_{j}|^{p_{i}-1}u_{j}-|v_{j}|^{p_{i}-1}v_{j}}{u_{j}-v_{j}}(u_{j}-v_{j})$
(5.7) $\displaystyle\leq R^{p_{i}-1}(u_{j}-v_{j}).$

It follows from (5.6) and (5.7) that

$\nu(u_{i}-v_{i})-e^{(p_{i}-1)\nu
t}(|u_{j}|^{p_{i}-1}u_{j}-|v_{j}|^{p_{i}-1}v_{j})\geq\nu(u_{i}-v_{i})-R^{p_{i}-1}(u_{j}-v_{j}).$

Choose $\nu>0$ so large that

$\nu>\max\left\\{R^{p_{1}-1},R^{p_{2}-1}\right\\}$

to get

$\nu(u_{i}-v_{i})-R^{p_{i}-1}(u_{j}-v_{j})\geq(\nu-R^{p_{i}-1})(u_{i}-v_{i})\geq(\nu-R^{p_{i}-1})\frac{\theta}{2}>0.$
(5.8)

Summarazing (5.4)–(5.6), (5.8), we obtain

$\frac{\mu}{(T-t)^{2}}+(\nu-R^{p_{i}-1})\frac{\theta}{2}\leq\omega_{F_{i}}\left(|x-y|+\frac{|x-y|^{2}}{\varepsilon}\right)+2\delta\Lambda_{i}N.$

Dropping the first term of the left-hand and then passing to the limit
$\varepsilon\to 0$, we finally obtain

$(\nu-R^{p_{i}-1})\frac{\theta}{2}\leq 2\delta\Lambda_{i}N.$

Since $\delta$ is sufficiently small, this is a contradiction. ∎

##  Appendix A Perron’s method

In this appendix, we state Perron’s method and give its proof. Let $T>0$ and
let $S$ be the set of all viscosity solutions of (1.1) in
$\bm{R}^{N}\times(0,T)$.

######  Lemma A.1.

Let $T>0$. Assume that $S\neq\emptyset$. For $i=1,2$, set

$u_{i}(x,t):=\sup\\{v_{i}(x,t)|\ v=(v_{1},v_{2})\in S\\},\quad
x\in\bm{R}^{N},t\in(0,T).$

If

$\sup_{K}|u_{i}|<\infty,\quad i=1,2$

for any compact subset $K\subset\bm{R}^{N}\times(0,T)$, then $u=(u_{1},u_{2})$
is a viscosity subsolution of (1.1).

###### Proof.

For $i=1,2$ and $\varphi\in C^{2,1}(\bm{R}^{N}\times(0,T))$, we assume that
$u_{i}^{*}-\varphi$ attains strict maximum at
$(x_{0},t_{0})\in\bm{R}^{N}\times(0,T)$. Choose $r>0$ so that
$[t_{0}-r,t_{0}+r]\subset(0,T)$. Then for any $\varepsilon>0$, we have

$u_{i}^{*}(x_{0},t_{0})=\lim_{\varepsilon\to 0}\sup_{\begin{subarray}{c}x\in
B(x_{0},\varepsilon)\\\
|t-t_{0}|<\varepsilon\end{subarray}}u_{i}(x,t)\leq\sup_{\begin{subarray}{c}x\in
B(x_{0},\varepsilon)\\\ |t-t_{0}|<\varepsilon\end{subarray}}u_{i}(x,t),$

where $u_{i}^{*}$ us defined in (2.1). For all $\tau>0$, there exist sequences
$x_{\tau,\varepsilon}\in B(x_{0},\varepsilon)$ and
$t_{\tau,\varepsilon}\in(t_{0}-r,t_{0}+r)$ such that

$\sup_{\begin{subarray}{c}x\in B(x_{0},\varepsilon)\\\
|t-t_{0}|<\varepsilon\end{subarray}}u_{i}(x,t)\leq
u_{i}(x_{\tau,\varepsilon},t_{\tau,\varepsilon})+\tau.$

For any $\tau>0$ there exists $\delta_{\tau}>0$ such that for all
$x\in\bm{R}^{N}$, $t\in(0,T)$, if $|x-x_{0}|+|t-t_{0}|<\delta_{\tau}$, then
$|\varphi(x,t)-\varphi(x_{0},t_{0})|<\tau$. For each $k=1,2,\dots$ set

$\varepsilon=\min\left\\{\frac{\delta_{1/k}}{2},\frac{1}{k},r\right\\}.$

There exist $x_{k}\in B_{r}(x_{0})$, $t_{k}\in(t_{0}-r,t+r)$ such that

$\displaystyle x_{k}\to x_{0},\quad t_{k}\to t_{0}\quad\mathrm{as}\
k\to\infty,$ $\displaystyle
u_{i}^{*}(x_{0},t_{0})<u_{i}(x_{k},t_{k})+\frac{1}{k},\quad|\varphi(x_{k},t_{k})-\varphi(x_{0},t_{0})|<\frac{1}{k}.$

Moreover, by the definition of $u_{i}$, there exists $(u_{1}^{k},u_{2}^{k})\in
S$ such that

$u_{i}(x_{k},t_{k})<u_{i}^{k}(x_{k},t_{k})+\frac{1}{k}.$

Choose $(y_{k},s_{k})\in\overline{B}_{r}(x_{0})\times[t_{0}-r,t_{0}+r]$ so
that $(u_{i}^{k})^{*}-\varphi$ attains its maximum at $(y_{k},s_{k})$. Taking
a subsequence (still denoted by the same symbol), we see that as $k\to\infty$,
$y_{k}\to\hat{y}$, $t_{k}\to\hat{t}$ for some
$\hat{y}\in\overline{B}_{r}(x_{0})$, $t_{k}\to\hat{t}\in[t_{0}-r,t_{0}+r]$. We
have

$\displaystyle(u_{i}^{*}-\varphi)(x_{0},t_{0})$
$\displaystyle<(u_{i}^{*}-\varphi)(x_{k},t_{k})+\frac{3}{k}$ (A.1)
$\displaystyle\leq((u_{i}^{k})^{*}-\varphi)(x_{k},t_{k})+\frac{3}{k}$
$\displaystyle\leq((u_{i}^{k})^{*}-\varphi)(y_{k},s_{k})+\frac{3}{k}$
$\displaystyle\leq(u_{i}^{*}-\varphi)(y_{k},s_{k})+\frac{3}{k}.$

Since $u_{i}^{*}$ is lower semicontinuous, we see that

$(u_{i}^{*}-\varphi)(x_{0},t_{0})\leq(u_{i}^{*}-\varphi)(\hat{y},\hat{t}).$

On the other hand, $u_{i}^{*}-\varphi$ has a strict maximum at
$(x_{0},t_{0})$, hence $x_{0}=\hat{x}$ and $t_{0}=\hat{t}$. Therefore,
$y_{k}\in\overline{B}_{r}(x_{0})$, $t_{k}\in[t_{0}-r,t_{0}+r]$ for
sufficiently large $k$. In addition, using (A.1) again, we have

$\lim_{k\to\infty}(u_{i}^{k})^{*}(y_{k},s_{k})=u_{i}^{*}(x_{0},t_{0})$

and

$\displaystyle\limsup_{k\to\infty}(u_{i}^{k})^{*}(y_{k},s_{k})\leq\limsup_{k\to\infty}u_{j}^{*}(y_{k},s_{k})\leq
u_{j}^{*}(x_{0},t_{0}),$

where $j\neq i$. Consequently, we obtain

$\partial_{t}\varphi(y_{k},s_{k})+F_{i}(y_{k},D^{2}\varphi(y_{k},s_{k}))\leq|u_{j}^{*}(y_{k},s_{k})|^{p_{i}-1}u_{j}^{*}(y_{k},s_{k}),$

hence

$\partial_{t}\varphi(x_{0},t_{0})+F_{i}(x_{0},D^{2}\varphi(x_{0},t_{0}))\leq|u_{j}^{*}(x_{0},t_{0})|^{p_{i}-1}u_{j}^{*}(x_{0},t_{0}),$

which completes the proof. ∎

######  Proposition A.1.

Assume that
$\xi=(\xi_{1},\xi_{2})\in(L^{\infty}_{\mathrm{loc}}(\bm{R}^{N}\times(0,T)))^{2}$
is a viscosity subsolution of (1.1) and
$\eta=(\eta_{1},\eta_{2})\in(L^{\infty}_{\mathrm{loc}}(\bm{R}^{N}\times(0,T)))^{2}$
is a viscosity supersolution of (1.1) for some $T>0$. If $\xi$ and $\eta$
satisfy

$\xi_{1}\leq\eta_{1},\quad\xi_{2}\leq\eta_{2}\quad\mathrm{in}\
\bm{R}^{N}\times(0,T),$

then

$u_{i}(x,t):=\sup\\{v_{i}(x,t)|v=(v_{1},v_{2})\in S,\ \xi\leq v\leq\eta\\}$

is a viscosity solution of (1.1) in $\bm{R}^{N}\times(0,T)$. Here $\xi\leq
v\leq\eta$ means that $\xi_{i}\leq v_{i}\leq\eta_{i}$ in
$\bm{R}^{N}\times(0,T)$ for $i=1,2$.

###### Proof.

By Lemma A.1, $u=(u_{1},u_{2})$ is a viscosity solution to (1.1). Suppose,
contrary to our claim, that there exist $\varphi\in C^{2,1}$ and
$(x_{0},t_{0})\in\bm{R}^{N}\times(0,T)$, $u_{i*}-\varphi$ attains a strict
minimum at $(x_{0},t_{0})$, $(u_{i*}-\varphi)(x_{0},t_{0})=0$ for some $i=1,2$
and exists $\theta>0$ such that

$\partial_{t}\varphi+F(x_{0},D^{2}\varphi)-|u_{i*}|^{p_{i}-1}u_{i*}<-\theta$
(A.2)

at $(x_{0},t_{0})$, where $i\neq j$ and $p_{1}=p$, $p_{2}=q$.

We firstly show that $\varphi(x_{0},t_{0})<\eta_{i*}(x_{0},t_{0})$. In fact,
we see that $\varphi\leq u_{i*}\leq\eta_{i*}$ and $u_{j*}\leq\eta_{j*}$ and
$\eta_{i*}-\varphi$ attains a minimum at $(x_{0},t_{0})$ if
$\varphi(x_{0},t_{0})=\eta_{i*}(x_{0},t_{0})$, thus by the definition of the
viscosity supersolution, we obtain

$\partial_{t}\varphi(x_{0},t_{0})+F(x_{0},D^{2}\varphi(x_{0},t_{0}))\geq|\eta_{j*}|^{p_{j}-1}\eta_{j*}(x_{0},t_{0})\geq|u_{j*}|^{p_{j}-1}u_{j*}(x_{0},t_{0}).$

This contradicts the assumption.

For any $\rho>0$, there exists $\varepsilon_{\rho}>0$ such that

$\displaystyle u_{j*}(x_{0},t_{0})-\rho$
$\displaystyle=\sup_{\varepsilon>0}\inf_{\begin{subarray}{c}|x-x_{0}|<\varepsilon_{\rho}\\\
|t-t_{0}|<\varepsilon_{\rho}\end{subarray}}u_{j}(x,t)-\rho$
$\displaystyle<\inf_{\begin{subarray}{c}|x-x_{0}|<\varepsilon_{\rho}\\\
|t-t_{0}|<\varepsilon_{\rho}\end{subarray}}u_{j}(x,t)$ $\displaystyle\leq
u_{j}(x,t)$

for $|x-x_{0}|<\varepsilon_{\rho}$ and $|t-t_{0}|<\varepsilon_{\rho}$. By
using the mean value theorem, there exists $\hat{\theta}$ such that

$\displaystyle|u_{j*}^{p_{i}-1}u_{j}$
$\displaystyle\geq|u_{j*}(x_{0},t_{0})-\rho|^{p_{i}-1}(u_{j*}(x_{0},t_{0})-\rho)$
$\displaystyle=|u_{j*}(x_{0},t_{0})|^{p_{i}-1}u_{j*}(x_{0},t_{0})$
$\displaystyle-\rho
p_{i}|\hat{\theta}u_{j*}(x_{0},t_{0})(x_{0},t_{0})+(1-\hat{\theta})(u_{j*}(x_{0},t_{0})-\rho)|^{p_{i}-1}.$

We can find $s_{0}>0$ so small that

$\rho
p_{i}|\hat{\theta}u_{j*}(x_{0},t_{0})(x_{0},t_{0})+(1-\hat{\theta})(u_{j*}(x_{0},t_{0})-\rho)|^{p_{i}-1}<\frac{\theta}{4}$

and

$|\partial_{t}\varphi(x_{0},t_{0})+F_{i}(x_{0},D^{2}\varphi(x_{0},t_{0}))-\partial_{t}\varphi(x,t)-F_{i}(x,D^{2}\varphi(x,t))|<\frac{\theta}{4}$

for $|x-x_{0}|<s_{0}$ and $|t-t_{0}|<s_{0}$. This together with (A.2) and the
continuity of $\partial_{t}\varphi$ and $F_{i}(\cdot,D^{2}\varphi)$ implies
that

$\displaystyle-\theta$
$\displaystyle>\partial_{t}\varphi(x,t)+F_{i}(x,D^{2}\varphi(x,t))-\frac{\theta}{4}-|u_{j*}|^{p_{i}-1}u_{j*}(x,t)-\frac{\theta}{4}$

for $|x-x_{0}|<s_{0}$ and $|t-t_{0}|<s_{0}$. Therefore,

$\partial_{t}\varphi(x,t)+F_{i}(x,D^{2}\varphi(x,t))-|u_{j*}|^{p_{i}-1}u_{j*}(x,t)<-\frac{\theta}{2}.$
(A.3)

It is already shown that
$u_{i*}(x_{0},t_{0})=\varphi(x_{0},t_{0})<\eta_{i*}(x_{0},t_{0})$. Set

$3\hat{\tau}:=\eta_{i*}(x_{0},t_{0})-u_{i*}(x_{0},t_{0})>0.$

Since $\eta_{i*}$ is lower semicontinuous and $\varphi$ is continuous, we can
find $s_{1}\in(0,s_{0})$ such that for all $|x-x_{0}|<s_{1}$ and
$t\in(t_{0}-s_{1},t_{0}+s_{1})$,

$\eta_{i*}(x,t)-\varphi(x,t)>\eta_{i*}(x_{0},t_{0})-\varphi(x_{0},t_{0})-\hat{\tau}=2\hat{\tau}.$

Therefore $\varphi(x,t)+2\hat{\tau}<\eta_{i*}(x,t))$ in $D$, where

$D:=B_{s_{1}}(x_{0})\times(t_{0}-s_{1},t_{0}+s_{1}).$

On the other hand, since $u_{i*}-\varphi$ attains a strict minimum at
$(x_{0},t_{0})$ and $(u_{i*}-\varphi)(x_{0},t_{0})=0$, there exist
$\varepsilon\in(0,s_{1}/2)$ and $\tau_{0}\in(0,\hat{\tau})$ such that

$(u_{i*}-\varphi)(x,t)\geq\min_{(x,t)\in
A}\\{(u_{i*}-\varphi)(x,t)\\}>\tau_{0}.$

Here we have set

$A:=\left(\overline{B_{s_{1}/2+\varepsilon}(x_{0})}\setminus
B_{s_{1}/2-\varepsilon}(x_{0})\right)\times\left\\{t\mid\frac{s_{1}}{2}-\varepsilon\leq|t-t_{0}|\leq\frac{s_{1}}{2}+\varepsilon\right\\}.$

We now define $(w_{1},w_{2})$ by

$\displaystyle w_{i}(x,t)$
$\displaystyle:=\begin{cases}\max\\{u_{i}(x,t),\varphi(x,t)+\tau_{0}\\}\quad&\mathrm{in}\
D/2,\\\ u_{i}(x,t)\quad\mathrm{in}\
(\bm{R}^{N}\times(0,T))\setminus(D/2),\end{cases}$ $\displaystyle w_{j}(x,t)$
$\displaystyle:=u_{j}(x,t)\quad\mathrm{in}\ \bm{R}^{N}\times(0,T),$

where

$D/2:=B_{s_{1}/2}(x_{0})\times\left(t_{0}-\frac{s_{1}}{2},t_{0}+\frac{s_{1}}{2}\right).$

In what follows, we shall show that $(w_{1},w_{2})$ is a viscosity subsolution
to (1.1) in $\bm{R}^{N}\times(0,T)$ satisfying $\xi_{k}\leq w_{k}\leq\eta_{k}$
for $k=1,2$. It follows from the definition of $w_{i}$ that we have
$\xi_{i}\leq u_{i}\leq w_{i}$ in $\bm{R}^{N}\times(0,T)$. Since
$\varphi(x,t)+\tau_{0}\leq\eta_{i*}$ in $D$, we see that
$\varphi+\tau_{0}\leq\eta_{i}$ in $D$, hence $w_{k}\leq\eta_{k}$ in
$\bm{R}^{N}\times(0,T)$ for $k=1,2$. Consequently, for $k=1,2$, we obtain

$\xi_{k}\leq w_{k}\leq\eta_{k}\quad\mathrm{in}\ \bm{R}^{N}\times(0,T).$

We can find $n\in\\{1,2,\dots\\}$ sufficiently large so that

$\frac{1}{n}<\frac{s_{1}}{2}-\varepsilon\quad\mathrm{and}\quad\frac{1}{n}<\frac{\tau_{0}}{2}$

and there exist $x_{n}\in B_{1/n}(x_{0})$ and $t_{n}\in\bm{R}$ with
$|t_{0}-t_{n}|<1/n$ such that

$u_{i}(x_{n},t_{n})<u_{i*}(x_{0},t_{0})+\frac{1}{n}.$

Moreover, it follows that

$u_{i*}(x_{0},t_{0})+\frac{1}{n}<u_{i*}(x_{0},t_{0})+\frac{\tau_{0}}{2}=\varphi(x_{0},t_{0})+\frac{\tau_{0}}{2}<\varphi(x_{0},t_{0})+\tau_{0}.$

Note that $(x_{n},t_{n})\in D/2$.

In what follows, we shall prove that $(w_{1},w_{2})$ is a viscosity
subsolution to (1.1) in $\bm{R}^{n}\times(0,T)$. Let us take
$\bm{R}^{N}\times(0,T)$ and $\psi\in C^{2,1}(\bm{R}^{N}\times(0,T))$
arbitrarily.

We firstly assume that $w_{i}^{*}-\psi$ attains a local maximum at
$(\hat{x},\hat{t})$. Consider the first case
$w_{i}^{*}(\hat{x},\hat{t})=u_{i}^{*}(\hat{x},\hat{t})$. Then

$\displaystyle u_{i}^{*}(\hat{x},\hat{t})-\psi(\hat{x},\hat{t})$
$\displaystyle=w_{i}^{*}(\hat{x},\hat{t})-\psi(\hat{x},\hat{t})$
$\displaystyle\geq w_{i}^{*}(x,t)-\psi(x,t)$ $\displaystyle\geq
u_{i}^{*}(x,t)-\psi(x,t)$

in $\bm{R}^{N}\times(0,T)$. Thus, $u_{i}^{*}-\psi$ attains its maximum at
$(\hat{x},\hat{t})$. Moreover, since $(u_{1},u_{2})$ is a subsolution to (1.1)
in $\bm{R}^{N}\times(0,T)$ and $u_{j}\equiv w_{j}$, we have

$\partial_{t}\psi+F_{i}(\cdot,D^{2}\psi)\leq|u_{j}^{*}|^{p_{i}-1}u_{j}^{*}=|w_{j}^{*}|^{p_{i}-1}w_{j}^{*}$

at $(\hat{x},\hat{t})$.

We next consider the second case
$w_{i}^{*}(\hat{x},\hat{t})=(\varphi+\tau_{0})^{*}(\hat{x},\hat{t})=\varphi(\hat{x},\hat{t})+\tau_{0}$.
Note that $(\hat{x},\hat{t})\in D/2$. The same argument above implies that
$\varphi+\tau_{0}-\psi$ attains its maximum at $(\hat{x},\hat{t})$. Thus, we
see that

$\partial_{t}\varphi(\hat{x},\hat{t})=\partial_{t}\psi(\hat{x},\hat{t}),\quad
D\varphi(\hat{x},\hat{t})=D\psi(\hat{x},\hat{t}),\quad
D^{2}\varphi(\hat{x},\hat{t})\leq D^{2}\psi(\hat{x},\hat{t}).$

It follows from (1.8) and (A.3) that

$\displaystyle\partial_{t}\psi+F_{i}(\cdot,D^{2}\psi)$
$\displaystyle\leq\partial_{t}\varphi+F_{i}(\cdot,D^{2}\varphi)$
$\displaystyle\leq|{u_{j}}_{*}|^{p_{i}-1}{u_{j}}_{*}$
$\displaystyle\leq|{u_{j}}^{*}|^{p_{i}-1}{u_{j}}^{*}$
$\displaystyle=|{w_{j}}^{*}|^{p_{i}-1}{w_{j}}^{*}$

at $(\hat{x},\hat{t})$.

We secondly assume that $w_{j}^{*}-\psi$ attains a local maximum at
$(\hat{x},\hat{t})$. Since $w_{j}=u_{j}$ in $\bm{R}^{N}\times(0,T)$,
$u_{j}^{*}-\psi$ also attains its maximum at $(\hat{x},\hat{t})$. Therefore,
we obtain

$\displaystyle\partial_{t}\psi+F_{j}(\cdot,D^{2}\psi)\leq|u_{i}^{*}|^{p_{j}-1}u_{i}^{*}\leq|w_{i}^{*}|^{p_{j}-1}w_{i}^{*}$

at $(\hat{x},\hat{t})$.

Consequently, $(w_{1},w_{2})$ is a viscosity subsotlution to (1.1) in
$\bm{R}^{N}\times(0,T)$. This contradicts the definition of $(u_{1},u_{2})$. ∎

######  Remark A.1.

Let $f_{1}$ and $f_{2}$ be a real valued function defined in a subset of
$\bm{R}^{M}$ with $M\in\\{1,2,\dots\\}$. Then
$\max\\{f_{1},f_{2}\\}^{*}=\max\\{f_{1}^{*},f_{2}^{*}\\}$. This fact allows us
to divide the cases $u_{i}^{*}=w_{i}^{*}$ or not.

## Acknoledgement

TK was partially supported by Grant-in-Aid for Early-Career Scientists JSPS
KAKENHI Grant Number 18K13436 and Tottori University of Environmental Studies
Grant-in-Aid for Special Research. RS was partially supported by Grant-in-Aid
for Early-Career Scientists JSPS KAKENHI Grant Number 18K13435 and funding
from Fukuoka University (Grant No. 205004).

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# Memory Efficient Patch-based Training for INR-based GANs

Namwoo Lee12 Hyunsu Kim1 Gayoung Lee1 Sungjoo Yoo2 Yunjey Choi1

1NAVER AI Lab 2Seoul National University
This work was done during an internship at NAVER AI Lab.

###### Abstract

Recent studies have shown remarkable progress in GANs based on implicit neural
representation (INR) - an MLP that produces an RGB value given its (x, y)
coordinate. They represent an image as a continuous version of the underlying
2D signal instead of a 2D array of pixels, which opens new horizons for GAN
applications (e.g., zero-shot super-resolution, image outpainting). However,
training existing approaches require a heavy computational cost proportional
to the image resolution, since they compute an MLP operation for every (x, y)
coordinate. To alleviate this issue, we propose a multi-stage patch-based
training, a novel and scalable approach that can train INR-based GANs with a
flexible computational cost regardless of the image resolution. Specifically,
our method allows to generate and discriminate by patch to learn the local
details of the image and learn global structural information by a novel
reconstruction loss to enable efficient GAN training. We conduct experiments
on several benchmark datasets to demonstrate that our approach enhances
baseline models in GPU memory while maintaining FIDs at a reasonable level.

##  1 Introduction

Recent advances in Generative Adversarial Networks (GANs) [8, 11, 12] enable
realistic image synthesis and show practical and diverse applicability such as
image-to-image translation [10, 5, 6, 16, 14], 3d-aware image generation [4,
19, 20, 9], real image editing [1, 26, 13], etc. Typical GANs view images as
2D pixel arrays and build them using convolutional filters. However, thanks to
the success of NeRF in 3D modeling, it is also getting popular to view images
as a continuous function in GANs. Implicit Neural Representations (INR) [22,
7, 21, 3, 18, 25] is a popular method that use a neural network to approximate
the continuous function. A number of recent studies including CIPS [2] and
INR-GAN [23] have proposed a model that combines the INR concept and GANs.
These INR-based GANs can naturally and easily do what was difficult in
convolutional GANs, such as partial patch generation, zero-shot super-
resolution, and image extrapolation.

Despite the advantages of INR-based GANs, it is difficult to train them
because they are hardware intensive due to a lot of network inference
proportional to the image size. Unlike convolutional GANs [12, 17] which use
upsampling and convolutional filters, pure INR-based GANs need to infer each
coordinate of an image, so it consumes much more GPU memory. For example, CIPS
requires 4 times more GPU memory than StyleGAN2. Therefore, reducing
computation costs is an important research topic to practically use INR-based
GANs. INR-GAN reduces the costs in the generator by factorizing the parameters
and progressively growing the feature maps similar to StyleGAN2. However,
their method is still computationally expensive because it starts with a
feature map of large size $(64^{2})$ and requires generating the entire image
for the discriminator.

(a) Traditional INR-based generator (b) Multi-stage patch-based training
(Ours)

Figure 1: Traditional vs. Multi-stage patch-based training. (a) Training
existing INR-based GANs [23, 2] is computationally expensive as they require
performing an MLP operation $G$ on all (x, y) coordinates for full resolution
($16^{2}$ in the example). (b) Our proposed multi-stage patch-based training
enables efficient training of INR-based GANs by performing $G$ only on a
predetermined small number of (x, y) coordinates ($4^{2}$ in the example)
regardless of resolution. In the early stage (Stage 1), a coarse global image
is generated from the sparse grid, and in the later stages (Stage 2, 3), local
patches with fine details are generated from the dense grids. The local patch
generated in each later stage is regularized to match the corresponding region
in the image generated in the previous stage. In multi-stage patch-based
training, we omit the mapping network $F$ for brevity.

In this paper, we propose a method that can dramatically reduce the training
costs for INR-based GAN using multi-stage patch-based training. During
training, our method generates small patches ($32^{2}$) instead of entire
images, and the generated patches are fed to the discriminator. This patch-
wise training can save a lot of GPU memory, but since the discriminator only
sees the patches, it cannot give feedback on global structures. To solve this
problem, we propose a novel multi-stage training method and progressively
reduce the receptive field of each stage patch. Specifically, in the initial
stage, the target patch is coarsely and globally sampled from an image,
whereas in the final stage, the patch of equal size is densely and locally
sampled. Then, in order to transfer the knowledge about the global structure
of the previous stage to the current stage, we apply the consistency loss
between the current generated patches and the patches cropped from the
previously generated patches. By doing this, the final generator can generate
a globally consistent image while it is trained using only local patches. We
conduct extensive experiments with various datasets and show that our method
reduces the required size of GPU memory and training time effectively while
maintaining the quality of generated images comparable to the existing
methods.

##  2 Multi-stage patch-based training

We propose multi-stage patch-based training, which reduces the computational
cost for training INR-based GANs. We build upon the INR-based GAN [2] and keep
every other component except the training strategy and patch regularization,
including the adversarial loss, and hyperparameters. Overall framework can be
shown in Figure 1.

For efficient training, we aim to generate local patches instead of full
images (e.g. generating $64^{2}$ patches instead of creating $256^{2}$ images
can reduce the computational cost such as GPU memory by $\tfrac{1}{4}$).
However, it is known that the generator $G$ cannot learn the global structure
of an image by providing only small patches to the discriminator [17]. To
alleviate this problem, we adopt multi-stage training in which the generator
learns to produce a coarse full image in the early stage of training (Stage 1
in Figure 1b) and learns to generate local patches with fine details in the
later stages (Stage 2, 3 in Figure 1b).

Sparse-to-dense coordinate sampling. During training, we sample $(x,y)$
coordinates in a sparse-to-dense manner. We first define a set of integer
pixel coordinates grid as:

$\texttt{grid}\left(H,W,N\right)=\\{\left(\tfrac{H}{N}k,\tfrac{W}{N}k\right)\mid
0\leq k<N\\}$ (1)

where $H,W$ are the height and width of training image resolution,
respectively (e.g. $256^{2}$), and $k$ is an integer value. A small $N$ gives
sparsely sampled coordinates, while a large $N$ gives densely sampled ones. In
the first stage of training, we set $N$ to $\tfrac{H}{4}$ to reduce the size
of the coordinate grid to $\tfrac{1}{16}$ of its full resolution (sparse
sampling). In the second and third stages of training, we set $N$ to
$\tfrac{H}{2}$ and $H$, respectively (dense sampling). We apply appropriate
random cropping to reduce the computational cost in the later stages.

Coarse-to-fine patch generation. We train the generators to produce coarse
global images in the early stage of training and local patches with fine
details in the later stages. Here, we denote the generator for each stage
$i\in\left\\{1,2,3\right\\}$ as $G_{i}$ for clarity. Our generator $G_{i}$
takes as input a random Gaussian vector $\textbf{z}\in\mathds{R}^{128}$ shared
across all pixels and pixel coordinates $\left(x,y\right)\in\left\\{0\dots
W-1\right\\}\times\left\\{0\dots H-1\right\\}$.

(a) Image-based (86GB) | (b) Patch-based (30GB) | (c) Ours (31GB)
---|---|---
|  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |
Figure 2:  Qualitative comparison with the baselines and our method. The
first/second/third row shows samples from FFHQ/LSUN Church/AFHQ, respectively.
The image-based model offers the best quality but requires much GPU memory
(86GB), whereas the patch-based model needs much less GPU memory (30GB) but
generates globally inconsistent images. Our method uses the comparable amount
of GPU memory (31GB) to the patch-based model, while producing much better
image quality.

The first stage generator $G_{1}$ produces a coarse global image $I_{1}$ by
performing an MLP operation for each $\left(x,y\right)$ coordinates, while
keeping random vector $\mathbf{z}$ fixed:

$\displaystyle
I_{1}=\left\\{G_{1}\left(x,y;\mathbf{z}\right)\mid\left(x,y\right)\in\texttt{grid}\left(H,W,\tfrac{H}{4}\right)\right\\}.$
(2)

We train $G_{1}$ with an adversarial loss [8] to generate images that are
indistinguishable from real images of low resolution. Note that unlike
traditional INR-based GANs [2, 23], our method sets $N$ to $\tfrac{H}{4}$
instead of $H$, which efficiently reduces GPU memory.

Unlike $G_{1}$, we train the generators $G_{2}$, $G_{3}$ to produce local
patches instead of full images. We use the generator trained in the previous
stage to initialize the generator in the later stage (i.e. initialize $G_{2}$
with the weights of $G_{1}$). This helps to distill the global representation
learned in the previous stage. The equation is similar to that of $G_{1}$, but
the (x, y) coordinates are densely sampled and randomly selected:

$\displaystyle
I_{i}=\left\\{G_{i}\left(x,y;\mathbf{z}\right)\mid\left(x,y\right)\in\texttt{rcrop}(\texttt{grid}\left(H,W,N_{i}\right))\right\\},$
(3)

where rcrop indicates a random crop operation. We set $N_{2}$ to
$\tfrac{H}{2}$ and $N_{3}$ to $H$ for $G_{2}$ and $G_{3}$, respectively. We
obtain a coordinate grid of $\tfrac{1}{4}$ size compared to full resolution
through the rcrop operation, and use the small grid to efficiently train the
generators to produce local patches.

Patch regularization. In order to maintain consistency between the currently
generated patch $I_{i}$ and the region cropped from the previously generated
image (or patch) $I_{i-1}$, we apply patch regularization:

$\mathcal{L}_{patch}=\mathbb{E}\left[{\lVert\texttt{resize}(I_{i},\tfrac{1}{2})-\texttt{crop}(I_{i-1})\lVert}_{2}\right],$
(4)

where $\texttt{resize}(\cdot,\tfrac{1}{2})$ reduces the size of image in half.
The proposed patch regularization is simple and helps to distill the global
structure learned from the previous stage to the current stage.

| FID Scores$\downarrow$ | Computation Costs
---|---|---
Method |  FFHQ (5 days) |  Church (6 days) |  AFHQ (4 days) |  GPU mem.$\downarrow$ |  sec/iter$\downarrow$
Image-based |  8.51 |  6.42 |  10.00 |  86GB |  3.04
Patch-based |  41.65 |  18.48 |  39.39 |  30GB |  0.82
Ours |  24.38 |  10.08 |  17.13 |  31GB |  0.71
Table 1: Comparison on FID score and computational costs for each method.
While patch-based method is memory-efficient than the original image-based
method, it produces worse quality images in terms of the FID score. Our method
requires the same amount of GPU memory as the patch-based model, but produces
higher quality images. We also report the running time for each training
iteration.

##  3 Experiments

Our multi-stage patch-based method effectively reduces the required size of
GPU memory ($2.8\times$ lower) in training. In this section, we conduct
experiments on various benchmark datasets (FFHQ, LSUN Church, and AFHQ) to
verify the effectiveness of our method. All experiments are conducted at
$256\times 256$ scale with the $G_{3}$ generator, and we use the Fréchet
inception distance (FID) metric to show that our method still retains
comparable performance in image generation.

###  3.1 Baseline Models

Since CIPS [2] is one of the state-of-the-art INR-based GANs, we demonstrates
the applicability of our method to the CIPS model. To show the effectiveness
of our method, we compare our method with three baselines.

Image-based method is the original version of CIPS network. We do not change
any configurations from its paper.

Patch-based method is the patch-based training without our multi-stage
training and patch regularization. The network is trained with $4\times$
smaller patches and only adversarial loss term.

Gradient Accumulation is the same as Image-based method except for the batch
size. To avoid the GPU memory limitation, some recent works [11, 12] may use
small batch size and accumulate gradients. The network weights are updated
once every multiple batches, whose summation is equal to that of the original
batch size.

###  3.2 Main results

Figure 2 and Table 1 show the qualitative and quantitative results. For a fair
comparison, we trained all baselines with the same training time; 4, 5, 6 days
for AFHQ, FFHQ, and LSUN Church, respectively. We set the training time in
proportion to the size of the data. Gradient Accumulation method is excluded
from Table 1 because it needs $n\times$ more time if we want to use $n\times$
smaller batch size. Ours shows visually comparable quality compared to the
original CIPS network while it needs $2.8\times$ less memory of GPU. Without
our multi-stage training, image quality deteriorates significantly in the
patch-based method. Our method needs only $3\%$ additional GPU memory but
shows significantly better image generation quality than patch-based method
according to the FID score; FIDs increase by 17.27, 8.40, and 22.26 in FFHQ,
LSUN Church, and AFHQ, respectively. Since our method and the patch-based
model generate only part of an image, each training iteration takes
significantly less time than the image-based model, and we can run more
training iterations in the same amount of time. Note that our method is
slightly faster than the patch-based model because we can skip random cropping
for the first stage.

|  |
---|---|---
|  |
Stage 1 | Stage 2 | stage 3
Figure 3: Samples of each training phase in our multi-stage training method.
In the first stage, coarse and global contours are generated, and in the later
stage, more and more details are added. The ability to produce globally
consistent images is transferred by our patch regularization loss.

###  3.3 Effect of patch regularization

In multi-stage patch-based training, we propose patch regularization which
matches the generated patches of different training phases as we’ve discussed
in Section 2. Figure 3 shows our regularization makes the network produce
consistent structure in all stages. Stage 1 shows blurry but structurally
meaningful images, and stage 3 shows high-fidelity images while maintaining
the structure of the early stages. Without this loss term, our network cannot
fully exploit the advantage of the multi-stage training.

|
---|---
Figure 4: Extrapolation on LSUN Church using our method. The pixels in out-of-
boundary locations are properly generated.

###  3.4 Extrapolation Results

In Figure 4, we show the results of extrapolation on LSUN Church using our
method. Thanks to the advantages of INR-based model, our method can generate
an image of a size not seen during training by simply feeding the targeted
coordinates.

##  4 Conclusion and Discussion

In this paper, we propose multi-stage patch-based training, a novel and
scalable approach that can train INR-based GANs with a flexible computational
cost regardless of the image resolution. We conducted experiments on several
benchmark datasets and demonstrated that our method contributes to reducing
the required size of GPU memory in training INR-based GAN models.

Our method also has some limitations. The proposed patch regularization might
be too restrictive as it forces the patch generated in the current stage to be
strongly match the image in the previous step. Also, the performance of multi-
stage training for a specific dataset (i.e. FFHQ) could be more improved.
Improving the performance and devising more flexible regularization to extract
global structures would be one of the meaningful future work.

Acknowledgements. The authors thank NAVER AI Lab researchers for constructive
discussion. All experiments were conducted on NAVER Smart Machine Learning
(NSML) platform [15, 24].

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