\documentclass[reqno]{amsart}
%\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 51, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/51\hfil System of totally characteristic equations]
{Solvability of a system of totally characteristic equations
related to K\"ahler metrics}

\author[J. E. C. Lope, M. P. F. Ona \hfil EJDE-2017/51\hfilneg]
{Jose Ernie C. Lope, Mark Philip F. Ona}

\address{Jose Ernie C. Lope \newline
Institute of Mathematics, College of Science,
 University of the Philippines Diliman,
Quezon City, Philippines}
\email{ernie@math.upd.edu.ph}

\address{Mark Philip F. Ona \newline
Institute of Mathematics, College of Science,
 University of the Philippines Diliman,
Quezon City, Philippines}
\email{mpona@math.upd.edu.ph}

\thanks{Submitted October 31, 2016. Published February 21, 2017.}
\subjclass[2010]{35G50, 35A20}
\keywords{Totally characteristic equation;
 singular partial differential equation;
\hfill\break\indent  majorant function; formal Gevrey class}

\begin{abstract}
 We consider a system of equations composed of a higher order singular
 partial differential equation of totally characteristic type and several
 higher order non-Kowalevskian linear equations. This system is a higher
 order version of a system that arose in Bielawski's investigations
 on K\"ahler metrics. We first prove that this system has a unique
 holomorphic solution. We then show that if the coefficients of the
 system are in some formal Gevrey class, then the unique solution is
 also in the same formal Gevrey class.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In 2002, Bielawski considered the system
\begin{equation}
 \begin{gathered}
t\partial_{t}u  =f(t,x,u,\partial_xu,w_1,\ldots,w_{N})\\
\partial_{t}w_i  =\ell_i(x;\partial_x)u+h_i(x)\quad\text{ for }
i=1,\ldots,N,
\end{gathered}\label{eq:B}
\end{equation}
where the function $f$ is holomorphic with respect to all its variables
and each $\ell_i$ is a second order linear differential operator
whose coefficients are functions of $x$. He  showed that it has
a unique holomorphic solution $(u,w_1,\ldots,w_{N})$ that satisfies
$u(0,x)\equiv0$ and $w_i(0,x)\equiv0$ for all $i$. This unique
solvability result was necessary in showing that it is possible to
extend a K\"ahler metric on a complex manifold $X$ to a Ricci-flat
K\"ahler metric in a neighborhood of $X$ in a line bundle $L$,
under the condition that the canonical $S^{1}$-action on $L$ is
Hamiltonian \cite{biel}.

The first equation in this system is very similar to the one studied
by G\'erard and Tahara in \cite{gt1990} and in several succeeding
papers in the 1990s. In fact, in proving the unique existence of a
holomorphic solution to \eqref{eq:B}, Bielawski first converted it
into an integro-differential equation and then suitably modified the
method of G\'erard-Tahara to tackle the resulting equation.

Suppose that the function $g(t,x,u,v)$ is holomorphic in a neighborhood
of the origin
$(0,0,0,0)\in\mathbb{C}_{t}\times\mathbb{C}_x^{n}\times\mathbb{C}_{u}
\times\mathbb{C}_{v}^{n}$
and satisfies $g(0,x,0,0)\equiv0$. The equation of G\'erard and
Tahara is the nonlinear singular partial differential equation
\begin{equation}
\begin{aligned}
t\partial_{t}u & =g(t,x,u,\partial_xu) \\
 & =a(x)t+b(x)u+c(x)\partial_xu+g_2(t,x,u,\partial_xu),
\end{aligned} \label{GT}
\end{equation}
where we have denoted by $g_2(t,x,u,v)$ the collection of all nonlinear
terms of $g(t,x,u,v)$ with respect to the variables $t$, $u$ and
$v$. Under the assumptions that $c(x)\equiv0$ and $b(0)$ is not
a positive integer, G\'erard and Tahara showed that \eqref{GT} has
a unique solution $u(t,x)$ that satisfies $u(0,x)\equiv0$. This
is also known as the nonlinear Fuchsian case of \eqref{GT}.

In 1999, Chen and Tahara considered the \emph{totally characteristic}
case of \eqref{GT}, that is, instead of assuming that $c(x)\equiv0$,
they assumed that $c(x)$ vanishes at the origin but not identically
zero. More precisely, they assumed that $c(x)=x\widetilde{c}(x)$
with $\widetilde{c}(0)\neq0$. Under some Poincar\'e condition involving
$b(0)$ and $\widetilde{c}(0)$, they were able to show that the totally
characteristic case also has a unique solution $u(t,x)$ that satisfies
$u(0,x)\equiv0$.

In \cite{lo2016}, we extended the unique solvability result of Bielawski
to the following higher order version of \eqref{eq:B}:
\begin{equation}
\begin{gathered}
(t\partial_{t})^{m}u  =F\big(t,x,\{(t\partial_{t})^j
\partial_x^{\alpha}u\}_{j+|\alpha|\leq m,j<m},\{w_i\}_{i=1,\ldots,N}\big)\\
\partial_{t}^{q}w_i  =L_i(t,x;\partial_x)u+H_i(t,x),\quad i=1,\ldots,N,
\end{gathered}\label{eq:LO}
\end{equation}
where $m,q\geq1$ and each $L_i$ is a linear differential operator
of order $q+1$ having coefficients which are dependent on both $x$
and $t$. Note that the first equation of the system is an $m$th
order singular nonlinear Fuchsian equation. We used a family of majorant
functions used in \cite{lt2001} and \cite{pon} to show that the
formal solution is convergent.

In this article, we revisit Bielawski's system of equations, this time
under the assumption that the first equation is a singular equation
of totally characteristic type. We shall show that the higher order
version of the system possesses a unique holomorphic solution under
some Poincar\'e condition. Finally, we generalize our results to
the case when the coefficients of the partial Taylor expansion of
\eqref{eq:LO} are not holomorphic functions of $x$ but rather belong
in some formal Gevrey class. This generalization is inspired by the
work of Pong\'erard \cite{pon}.

\section{Holomorphic Solutions}

\subsection{Main results}

We denote the set of all nonnegative integers by $\mathbb{N}$, and set
$\mathbb{N}^{*}=\mathbb{N}\setminus\{0\}$. For any $v=(v_1,\ldots,v_{n})$
and $\alpha=(\alpha_1,\ldots,\alpha_{n})$, we define
$v^{\alpha}=v_1^{\alpha_1}\cdots v_{n}^{\alpha_{n}}$
and $|\alpha|=\alpha_1+\cdots+\alpha_{n}$.
Let $(t,x)\in\mathbb{C}_{t}\times\mathbb{C}_x$
and fix $m,q\in\mathbb{N}^{*}$. Consider the  system of
differential equations
\begin{equation} \label{hol-sys-e1}
\begin{gathered}
(t\partial_{t})^{m}u
 =F\bigl(t,x,\bigl\{(t\partial_{t})^j\partial_x^{\alpha}u
\bigr\}_{(j,\alpha)\in\Lambda},\{w_i\}_{i=1}^{N}\bigr),\\
\partial_{t}^{q}w_i  =\mathcal{L}_i(t,x;\partial_x)u+H_i(t,x)\quad
\text{for }i=1,\ldots,N,
\end{gathered}
\end{equation}
where the function $F(t,x,Y,Z)$ satisfies $F(0,x,0,0)\equiv0$ and
is holomorphic in some neighborhood containing
\[
\{(t,x,Y,Z)\in\mathbb{C}^{2+\sharp\Lambda+N}:
|t|\leq r_0,|x|\leq R_0,|Y_{j,\alpha}|\leq R_1,|Z_i|\leq R_1\},
\]
for some positive constants $r_0$, $R_0$ and $R_1$. The index
set $\Lambda$ is defined by
$\Lambda=\{(j,\alpha)\in\mathbb{N}^2:j+\alpha\leq m,\ j<m\}$
and its cardinality is denoted by $\sharp\Lambda$. The linear differential
operator $\mathcal{L}_i$ is of order $q+m$ and of the form
\begin{equation}
\mathcal{L}_i(t,x;\partial_x)
=\sum_{\gamma\leq q+m}\mathcal{L}_{i,\gamma}(t,x)\partial_x^{\gamma},\label{hol-L-exp}
\end{equation}
where for all $i=1,\ldots,N$ and $\gamma\leq q+m$, the functions
$\mathcal{L}_{i,\gamma}(t,x)$ and $H_i(t,x)$ are holomorphic in
some neighborhood of $\{(t,x):|t|\leq r_0,|x|\leq R_0\}$.

We define the set $\Lambda_0=\{(j,\alpha)\in\Lambda:
\partial_{Y_{j,\alpha}}F(0,x,0,0)\not\equiv0\}$.
Under the above assumptions, we can expand $F(t,x,Y,Z)$ as
\begin{align*}
F(t,x,Y,Z)
&=a(x)t  +\sum_{(j,\alpha)\in\Lambda_0}b_{j,\alpha}(x)Y_{j,\alpha}
  +\sum_{1\leq i\leq N}d_i(x)Z_i \\
&\quad +\sum_{p+|\nu|+|\mu|\geq2}g_{p,\nu,\mu}(x)t^{p}Y^{\nu}Z^{\mu}.
\end{align*}
Now suppose that for all $(j,\alpha)\in\Lambda_0$, we have
$b_{j,\alpha}(x)=x^{\alpha}\lambda_{j,\alpha}(x)$
for some holomorphic function $\lambda_{j,\alpha}$ that satisfies
$\lambda_{j,\alpha}(0)\neq0$. In other words, we are assuming that
the first equation of \eqref{hol-sys-e1} is of \emph{totally characteristic
type} as defined by Chen-Tahara in \cite{ct1999}. Let
\begin{equation}
P(\tau,\xi)=\tau^{m}-\sum_{(j,\alpha)\in\Lambda}\lambda_{j,\alpha}(0)
\tau^j\xi(\xi-1)\cdots(\xi-\alpha+1).\label{hol-P}
\end{equation}
We have the following result on the existence and uniqueness of a
holomorphic solution.

\begin{theorem} \label{hol-thm}
If $P(\tau,\xi)\neq0$ for all $(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$,
then \eqref{hol-sys-e1} has a unique holomorphic solution $(u,w_1,\ldots,w_{N})$
that satisfies $u(0,x)\equiv0$ and $\partial_{t}^{k}w_i(0,x)\equiv0$
for $k=0,1,\ldots,q-1$ and $i=1,\ldots,N$.
\end{theorem}

\subsection{Existence of a unique formal solution}

Under the assumption that $b_{j,\alpha}=x^{\alpha}\lambda_{j,\alpha}$,
we can rewrite each $b_{j,\alpha}(t\partial_{t})^j\partial_x^{\alpha}u$
as $\lambda_{j,\alpha}(t\partial_{t})^j(x\partial_x)(x\partial_x-1)
\cdots(x\partial_x-\alpha+1)u$.
Thus if we let $x\lambda_{j,\alpha}^{*}(x)
=\lambda_{j,\alpha}(x)-\lambda_{j,\alpha}(0)$,
then the first equation of \eqref{hol-sys-e1} can be written as
\begin{equation} \label{hol-eqn1-exp}
\begin{aligned}
P(t\partial_{t},x\partial_x)u
& =a(x)t+\sum_{(j,\alpha)\in\Lambda_0}x\bigl(x^{\alpha}
\lambda_{j,\alpha}^{*}(x)\bigr)(t\partial_{t})^j\partial_x^{\alpha}u
+\sum_{1\leq i\leq N}d_i(x)w_i\\
 & \quad+\sum_{p+|\nu|+|\mu|\geq2}g_{p,\nu,\mu}(x)t^{p}
\prod_{(j,\alpha)\in\Lambda}\big((t\partial_{t}\big)^j
\partial_x^{\alpha}u)^{\nu_{j,\alpha}}\prod_{1\leq i\leq N}w_i^{\mu_i},
\end{aligned}
\end{equation}
where $P$ is the polynomial defined in \eqref{hol-P}.

We wish to find a formal solution $(u,w_1,\ldots,w_N)$ of the form
$u(t,x)=\sum_{j\geq 1, k\geq 0}$ $u_{j,k}t^jx^k$ and
$w_i(t,x)= \sum_{j\geq q, k\geq 0}w_{i,j,k}t^j x^k$ that satisfy
 \eqref{hol-sys-e1}.  Let us expand the coefficients as follows:
 $a(x)=\sum_{k\geq 0}a_kx^k$,
$\lambda_{\ell,\alpha}^*(x)=\sum_{k\geq 0}\lambda^*_{\ell,\alpha,k}x^k$,
$d_i(x)=\sum_{k\geq 0}d_{i,k}x^k$ and
$g_{p,\nu,\mu}(x)=\sum_{k\geq 0}g_{p,\nu,\mu,k}x^k$ for the first equation,
and $\mathcal{L}_{i,\gamma}(t,x)
=\sum_{j\geq 0, k\geq 0}\mathcal{L}_{i,\gamma,j,k}t^jx^k$ and $ H_i(t,x)
=\sum_{j\geq 0, k\geq 0}H_{i,j,k}t^jx^k$ for the second equation of the system.
Under the assumption that $P(\tau,\xi)\neq 0$, for any
$(\tau,\xi)\in\mathbb{N}^*\times\mathbb{N}$,  we see that
\begin{gather*}
u_{1,0}=\begin{cases}
P(1,0)^{-1}a_0  & \text{if } q>1\\
P(1,0)^{-1}(a_0 + \sum_{1\leq i\leq N}d_{i,0}w_{i,1,0})
 &\text{ if } q=1
\end{cases}\\
w_{i,q,0}=(q!)^{-1}H_{i,0,0}.
\end{gather*}
In addition, for $J\geq 1$ and $K\geq 0$, there exists functions
$\mathcal{F}_{J,K}$ and $\mathcal{G}_{J,K}$ such that
\begin{gather*}
\begin{aligned}
w_{i,J+q,K}
&= \frac{J!}{(J+q)!}\mathcal{F}_{J,K}
\Big(\{\mathcal{L}_{i,\gamma,j,k}\}_{i\leq N, j\leq J, k\leq K, \gamma\leq q+m}, \\
&\quad
\Big\{\frac{(k+\gamma)!}{k!}u_{j,k+\gamma}\Big\}_{j\leq J, k\leq K, \gamma\leq q+m},
  H_{J,K}\Big)
\end{aligned}\\
\begin{aligned}
u_{J,K} &= \frac{1}{P(J,K)}\mathcal{G}_{J,K}
\Big(a_K, \{\lambda^*_{\ell,\alpha,k}\}_{k< K},
\Big\{\frac{J^{\ell}k!}{(k-\alpha)!}u_{J,k}\Big\}_{k< K,
 (\ell,\alpha)\in\Lambda_0},\\
&\quad\{d_{i,k}\}_{i\leq N, k\leq K}, \{w_{i,J,k}\}_{i\leq N, k\leq K},
\{g_{p,\nu,\mu, k}\}_{k\leq K, p+|\nu|+|\mu|\geq 2},\\
&\quad \Big\{\frac{J^{\ell}(k+\alpha)!}{k!}u_{j,k+\alpha}
\Big\}_{j<J, k< K, (\ell,\alpha)\in\Lambda}, \{w_{i,j,k}\}_{j<J,k\leq K}\Big).
\end{aligned}
\end{gather*}
Observe that for any $k\geq 0$, the coefficients $w_{i,q,k}$ are uniquely
determined by the function $H_i$. These will enable us to solve for $u_{1,k}$.
We move forward by solving for $w_{i,q+1,k}$ for any $k\geq 0$, which will
allow us to solve for $u_{2,k}$. We follow these steps to recursively and uniquely
determine all the coefficients of the formal solution.

We have thus shown that there exists a unique formal solution to \eqref{hol-sys-e1}.
It remains to show that this formal solution is convergent.

\subsection{Convergence of the formal solution}

To show convergence, we  use the majorant method. For power series
$a(z)=\sum_{|\alpha|\geq0}a_{\alpha}z^{\alpha}$ and
$A(z)=\sum_{|\alpha|\geq0}A_{\alpha}z^{\alpha}$,
we say that $A$ majorizes $a$, written as $a\ll A$, if for all
$|\alpha|\geq0$, we have $|a_{\alpha}|\leq A_{\alpha}$. We construct
a system of majorant relations whose solution majorizes the formal
solution to \eqref{hol-sys-e1}.

For simplicity suppose that constants $r_0,R_0$ and $R_1$
are all less than $1$. We choose $M$ large enough such that the functions
$F(t,x,Y,Z)$, $\mathcal{L}_{i,\gamma}(t,x)$ (for $i=1,\ldots,N,\gamma\leq q+m$),
$H_i(t,x)$ (for $i=1,\ldots,N$) and $\lambda_{j,\alpha}^{*}(x)$
(for $(j,\alpha)\in\Lambda_0)$
appearing in \eqref{hol-sys-e1}, \eqref{hol-L-exp} and \eqref{hol-eqn1-exp},
are bounded by $M$ for all $|x|\leq R_0$, $|t|\leq r_0$, $|Y_{j,\alpha}|\leq R_1$
and $|Z_i|\leq R_1$. In addition, we fix a sufficiently large
$A>0$ such that for all $(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$,
we have
\begin{equation}
\frac{1}{|P(\tau,\xi)|}\leq\frac{A}{1+\tau^{m}+\xi^{m}}.
\end{equation}
We can choose such a constant since $P(\tau,\xi)\neq0$ for all
$(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$
and since the polynomial $P(\tau,\xi)$ is of degree $m$ in $\tau$
and $\xi$.

\begin{proposition} \label{prop2}
Consider the  system of majorant relations
\begin{gather}
\begin{aligned}
 & \Big[(t\partial_{t})^{m}+(x\partial_x)^{m}+1\Big]U(t,x) \\
 & \gg\frac{AM}{1-x/R_0}\Big[\frac{t}{r_0}
 +x\sum_{(j,\alpha)\in\Lambda_0}x^{\alpha}(t\partial_{t})^j
 \partial_x^{\alpha}U+\sum_{1\leq i\leq N}\frac{W_i}{R_1}\Big] \\
 &\quad +\sum_{p+|\nu|+|\mu|\geq2}\frac{AM}{1-x/R_0}\frac{t^{p}}{r_0^{p}
 R_1^{|\nu|+|\mu|}}\prod_{(j,\alpha)\in\Lambda}\Big((t\partial_{t})^j
 \partial_x^{\alpha}U\Big)^{\nu_{j,\alpha}}
 \prod_{1\leq i\leq N}W_i^{\mu_i}\,,
\end{aligned}\label{hol-maj-sys-e1} \\
\partial_{t}^{q}W_i\gg\frac{M}{1-x/R_0-t/r_0}\Big[\sum_{\gamma\leq q+m}
 \partial_x^{\gamma}U+1\Big]\quad\text{for }i=1,\ldots,N\,, \label{hol-maj-sys-e2}
\\
  U(0,x)\gg0,\quad \partial_{t}^{k}W_i(0,x)\gg0\quad \text{for }i=1,\ldots,N,\;
k=0,\ldots,q-1\,. \label{hol-maj-sys-e3}
\end{gather}
Then for any $(U,W_1,\ldots,W_{N})$ that satisfies the above relations,
we have $u\ll U$ and $w_i\ll W_i$ for $i=1,\ldots,N$, where
$(u,w_1,\ldots,w_{N})$ is the unique formal solution of \eqref{hol-sys-e1}.
\end{proposition}

The above proposition  implies that the task of proving the convergence of
the formal solution is reduced to finding holomorphic functions $U(t,x)$
and $W_i(t,x)$ that satisfy the above relations. The proof of this
proposition is an easy calculation and will be omitted here.

\subsection{Majorant functions}

The same family of majorant functions found in \cite{lo2016} will
be used in the proof of our main result. Let $S=1+1/2^2+\cdots=\pi^2/6$.
For $i\in\mathbb{N}$ and $s\in\mathbb{N}^{*}$, define the following
family of functions:
\[
\varphi_i(x)=\frac{1}{4S}\sum_{n\geq0}\frac{x^{n}}{(n+1)^{2+i}}
\quad\text{and}\quad\Phi_i^{s}(t,x)
=\sum_{p\geq0}t^{p}\frac{D^{sp}\varphi_i(x)}{(sp)!}.
\]
Note that for any $i\in\mathbb{N}$ and $s\in\mathbb{N}^{*}$, $\varphi_i(x)$
converges when $|x|<1$, and $\Phi_i^{s}(t,x/R)$ converges when
$|t|^{1/s}+|x|<R$. We enumerate some useful properties of $\varphi_i$
and $\Phi_i^{s}$.

\begin{proposition} \label{prop3}
The following hold for all $i\in\mathbb{N}$, $s\in\mathbb{N}^{*}$:
\begin{enumerate}
\item $\varphi_i(x)\varphi_i(x)\ll2^{i}\varphi_i(x)$
\item $\varphi_{i+1}(x)\ll\varphi_i(x)$
\item $2^{-3-i}\varphi_i(x)\ll\varphi'_{i+1}(x)\ll\varphi_i(x)$ %\item For any $\epsilon\in(0,1)$, there exists a constant $B_{i,\epsilon}>0$%such that %\[%\frac{1}{1-\epsilon x}\,\varphi_i(x)\ll B_{i,\epsilon}\varphi_i(x).%\]
\item For any $k\in\mathbb{N}$, $xD^{k}\varphi_i(x)\ll2^{2+i}D^{k}\varphi_i(x)$
\item $\Phi_i^{s}(x)\Phi_i^{s}(x)\ll2^{i}\Phi_i^{s}(x)$
\item For any $\epsilon\in(0,1)$, there exists a constant $B_{i,\epsilon}>0$
such that for all $j\in\mathbb{N}$, $(1-\epsilon x)^{-1}D^j\varphi_i(x)\ll B_{i,\epsilon}D^j\varphi_i(x).$
\item For a sufficiently small $R(<2^{-2-i})$,
\begin{enumerate}
\item $(p+q)!D^{p}\varphi_i(x/R)\ll p! D^{p+q}\varphi_i(x/R)$
\item $\varphi_i(t+x/R)\ll\Phi_i^{s}(t,x/R)$
\item For any $\epsilon\in(0,1)$, there exists a constant $B_{i,\epsilon}>0$
such that
\[
\frac{\Phi_i^{s}(t,x/R)}{1-\epsilon(t+x/R)}\ll B_{i,\epsilon}\Phi_i^{s}(t,x/R).
\]
\end{enumerate}
\end{enumerate}
\end{proposition}

\begin{proof}
The first four assertions easily follow from the definition of $\varphi_i$.
The proof for (5) may be found in \cite{pon} but essentially rests
on (1) and the fact that
\[
\frac{D^{k}(\varphi_i^2)}{k!}=\sum_{j=0}^{k}\frac{D^{k-j}\varphi_i}{(k-j)!}\frac{D^j\varphi_i}{j!}.
\]
Item (6) follows from the estimates $4S\epsilon^{n}(n+1)^{2+i}\leq B_{i,\epsilon}$
for any $n\geq0$, and $D^j[(1-\epsilon x)^{-1}\varphi_i(x)]
\gg(1-\epsilon x)^{-1}D^j\varphi_i(x)$
for any $j\geq0$. The proof for (7a) is also found in \cite{pon}
where it was shown by induction that
$(p+1)D^{p}\varphi_i(x/R)\ll D^{p+1}\varphi_i(x/R)$.
This is equivalent in showing that for all $k,p,i\in\mathbb{N}$,
\[
R\Big(\frac{k+p+2}{k+p+1}\Big)^{2+i}\frac{p+1}{k+p+1}<1
\]
for some $R$. This is achieved when $R$ is chosen to be less than
$2^{-2-i}$. Item (7b) easily follows from (7a), and (7c) follows
from (6) and (7b).
\end{proof}

We now present holomorphic functions that satisfy all the relations
in Proposition \ref{prop2}.

\begin{proposition} \label{prop4}
Let $r\in(0,r_0)$ and $\eta=m+q$. Then there exist positive constants
$L_1$, $L_2$, $L_3$, $c$ and $R(<2^{-2-\eta}R_0)$ such that the
functions
\begin{gather}
U(t,x)=L_1t\Phi_{\eta}^{\eta}\Big(\frac{t}{cr},\frac{x}{R}\Big),\label{hol-U} \\
W_i(t,x)=L_2(cr)^{q+1}\sum_{k=1}^{\infty}(k+1)\big(\frac{t}{cr}\big)^{k+q}
\frac{D^{\eta k}\varphi_{\eta}(x/R)}{(\eta k)!}
+L_3t^{q}\Phi_{\eta}^{\eta}\Big(\frac{t}{cr},\frac{x}{R}\Big)\label{hol-W}
\end{gather}
for $i=1,\ldots,N$ satisfy \eqref{hol-maj-sys-e1}, \eqref{hol-maj-sys-e2}
and \eqref{hol-maj-sys-e3}.
\end{proposition}

\begin{proof}
For brevity, assume that the argument of $\Phi_i^{s}$ is always
$(t/cr,x/R)$ and the argument of $\varphi_i$ is always $x/R$,
and thus omit these from our notations. We also choose the constant
$K$ to be sufficiently large such that
$(1-x/R_0-t/r_0)^{-1}\Phi_i^{\eta}\ll K\Phi_i^{\eta}$
and $(1-x/R_0)^{-1}D^j\varphi_i\ll KD^j\varphi_i$ for all
$i\leq\eta$, $0\leq j$ and $R<2^{-2-\eta}R_0$.

Let us begin with \eqref{hol-maj-sys-e2}. Using Proposition \ref{prop3}(3)
and the fact that $\partial_{t}^j(t^jV)\gg V$ for any $V\gg0$,
the left-hand side can be estimated as follows:
\begin{equation}
\begin{aligned}
\partial_{t}^{q}W_i
& \gg L_2cr\sum_{k=0}^{\infty}\frac{(k+2)(k+q+1)!}{(k+1)!}
 \big(\frac{t}{cr}\big)^{k+1}\frac{D^{\eta k+\eta}\varphi_{\eta}}{(\eta k+\eta)!}
 +L_3\Phi_{\eta}^{\eta} \\
& \gg\frac{L_2t}{2^{\eta(\eta+2)}\eta^{q+1}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k}\frac{D^{\eta k}\varphi_0}{(\eta k)!}
 +L_3\Phi_{\eta}^{\eta}=\frac{L_2t\Phi_0^{\eta}}{2^{\eta(\eta+2)}\eta^{q+1}}
 +L_3\Phi_{\eta}^{\eta}.
\end{aligned}\label{2ndLHS}
\end{equation}
As for the corresponding right-hand side, we again use Proposition \ref{prop3}(3) to obtain
\begin{align*}
\partial_x^{\gamma}U & \ll\frac{L_1t}{R^{\gamma}}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}\frac{D^{\eta k+\gamma}\varphi_{\eta}}{(\eta k)!}\ll\frac{L_1}{R^{\eta}}t\Phi_0^{\eta}.
\end{align*}
Here, we used the fact that $\gamma\leq\eta$. Since $1\ll4S\Phi_{\eta}^{\eta}$,
the right-hand side of relation \eqref{hol-maj-sys-e2} will be majorized
by
\begin{align}
KM\Big(\frac{L_1\eta}{R^{\eta}}t\Phi_0^{\eta}+4S\Phi_{\eta}^{\eta}\Big),\label{2ndRHS}
\end{align}
where we used Proposition \ref{prop3}(7c) and $K$ is the constant defined at
the beginning of the proof. Comparing \eqref{2ndLHS} and \eqref{2ndRHS},
we obtain the following conditions for the constants $L_1$, $L_2$,
$L_3$ and $R$:
\begin{equation}
\frac{KML_1\eta}{R^{\eta}}\leq\frac{L_2}{2^{\eta(\eta+2)}\eta^{q+1}}
\quad\text{and}\quad 4SKM\leq L_3.\label{cond1}
\end{equation}

Let us now turn our attention to \eqref{hol-maj-sys-e1}. Since $\eta\geq m$,
we have
\begin{equation}
\begin{aligned}
(t\partial_{t})^{m}U
& =L_1cr\sum_{k=0}^{\infty}(k+1)^{m}\big(\frac{t}{cr}\big)^{k+1}
 \frac{D^{\eta k}\varphi_{\eta}}{(\eta k)!} \\
& \gg\frac{L_1cr}{\eta^{m}}\sum_{k=1}^{\infty}\big(\frac{t}{cr}\big)^{k+1}
 \frac{D^{\eta k}\varphi_{\eta}}{(\eta k-m)!} \\
& \gg\frac{L_1cr}{2^{m(\eta+2)}\eta^{m}}\sum_{k=1}^{\infty}
 \big(\frac{t}{cr}\big)^{k+1}\frac{D^{\eta k-m}\varphi_{\eta-m}}{(\eta k-m)!} \\
& =\frac{L_1cr}{2^{m(\eta+2)}\eta^{m}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k+2}
 \frac{D^{\eta k-m+\eta}\varphi_{\eta-m}}{(\eta k-m+\eta)!} \\
 & \gg\frac{L_1cr}{2^{m(\eta+2)}\eta^{m}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k+2}\frac{D^{\eta k}\varphi_{\eta-m}}{(\eta k)!} \\
&=\frac{L_1t^2}{2^{m(\eta+2)}\eta^{m}cr}\Phi_{\eta-m}^{\eta}.
\end{aligned}\label{1.1}
\end{equation}
We used Proposition \ref{prop3}(3) in the third line, while in the last simplification,
we used Proposition \ref{prop3}(7a) and the fact that $\eta-m\geq0$. Now, consider
the other terms on the left-hand side. Since $m\geq1$ and
$(x\partial_x)^{m}V\gg x\partial_xV$
for any holomorphic function $V\gg0$, we have
\begin{equation}
\begin{aligned}
(x\partial_x)^{m}U
& \gg\frac{L_1t}{R}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}
 \frac{xD^{\eta k+1}\varphi_{\eta}}{(\eta k)!} \\
 & \gg\frac{L_1t}{2^{\eta+2}R}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}
 \frac{xD^{\eta k}\varphi_{\eta-1}}{(\eta k)!}\\
&\gg\frac{L_1tx}{2^{\eta+2}R}\Phi_{\eta-m}^{\eta}.
\end{aligned}\label{1.2}
\end{equation}
To majorize the third summation on the right-hand side of \eqref{hol-maj-sys-e1},
we will estimate the term $((t\partial_{t})^{m}+1)U$ in the following
manner:
\begin{equation}
\begin{aligned}
\big((t\partial_{t})^{m}+1\big)U
& \gg L_1cr\sum_{k=1}^{\infty}(k+1)^{m}\big(\frac{t}{cr}\big)^{k+1}
 \frac{D^{\eta k}\varphi_{\eta}}{(\eta k)!}+L_1t\Phi_{\eta}^{\eta} \\
 & \gg L_1cr\sum_{k=0}^{\infty}(k+2)\big(\frac{t}{cr}\big)^{k+2}
\frac{D^{\eta k+\eta}\varphi_{\eta}}{(\eta k+\eta)!}+L_1t\Phi_{\eta}^{\eta}.
\end{aligned}\label{1stLHS2}
\end{equation}
Therefore, using \eqref{1.1}, \eqref{1.2} and \eqref{1stLHS2},
the left-hand side of \eqref{hol-maj-sys-e1} will majorize
\begin{equation}
\begin{aligned}
 & \frac{1}{2}\Big(\frac{L_1t^2}{2^{m(\eta+2)}\eta^{m}cr}
 \Phi_{\eta-m}^{\eta}+L_1t\Phi_{\eta}^{\eta}\Big)
 +\frac{L_1tx}{2^{\eta+2}R}\Phi_{\eta-m}^{\eta} \\
 & +\frac{1}{2}\Bigl[L_1cr\sum_{k=0}^{\infty}(k+2)
\big(\frac{t}{cr}\big)^{k+2}\frac{D^{\eta k+\eta}\varphi_{\eta}}{(\eta k+\eta)!}
+L_1t\Phi_{\eta}^{\eta}\Bigr].
\end{aligned}\label{1stLHS1}
\end{equation}

We will deal separately  with each summation on the right-hand side
of \eqref{hol-maj-sys-e1}. Using Proposition \ref{prop3} items (7a), (3) and
(4) in this order, we have for any $j+\alpha\leq m$:
\begin{align*}
x^{\alpha}(t\partial_{t})^j\partial_x^{\alpha}U
& =\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}(k+1)^j
 \big(\frac{t}{cr}\big)^{k+1}\frac{x^{\alpha}D^{\eta k+\alpha}
 \varphi_{\eta}}{(\eta k)!}\\
 & \ll\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}(k+1)^j
 \big(\frac{t}{cr}\big)^{k+1}\frac{x^{\alpha}D^{\eta k+\alpha+j}
 \varphi_{\eta}}{(\eta k+j)!}\\
& \ll\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k+1}\frac{x^{\alpha}D^{\eta k}\varphi_{\eta-\alpha-j}}
 {(\eta k)!}\\
& \ll2^{\alpha(2+\eta)}L_1t\Phi_{\eta-m}^{\eta}\ll2^{m(2+\eta)}L_1
 t\Phi_{\eta-m}^{\eta},
\end{align*}
and therefore by Proposition \ref{prop3}(7c),
\begin{align*}
\frac{AMx}{1-x/R_0}\sum_{(j,\alpha)\in\Lambda_0}x^{\alpha}(t\partial_{t})^j
(\partial_x)^{\alpha}U\ll(2^{m(2+\eta)}\sharp\Lambda_0AKM)L_1tx\Phi_{\eta-m}^{\eta},
\end{align*}
where $\sharp\Lambda_0$ is the cardinality of $\Lambda_0$. Comparing
this to the term with $tx\Phi_{\eta-m}^{\eta}$ in \eqref{1stLHS1},
we then see that we need to satisfy the  inequality
\begin{equation}
2^{m(2+\eta)}\sharp\Lambda_0AKM\leq\frac{1}{2^{\eta+2}R}.\label{cond2}
\end{equation}
Now, we consider the linear terms containing $W_i$. Using Proposition
\ref{prop3}(7a), we have
\begin{equation}
\begin{aligned}
W_i & \ll L_2(cr)^{q+1}\ \sum_{k=0}^{\infty}(k+2)
 \big(\frac{t}{cr}\big)^{k+q+1}\frac{D^{\eta(k+1)
 +\eta(q-1)}\varphi_{\eta}}{(\eta(k+1)+\eta(q-1))!}+L_3t^{q}\Phi_{\eta}^{\eta} \\
 & =L_2(cr)^{q+1}\ \sum_{k=q-1}^{\infty}(k-q+3)\big(\frac{t}{cr}\big)^{k+2}
\frac{D^{\eta(k+1)}\varphi_{\eta}}{(\eta(k+1))!}+L_3t^{q}\Phi_{\eta}^{\eta} \\
 & \ll L_2(cr)^{q+1}\ \sum_{k=0}^{\infty}(k+2)\big(\frac{t}{cr}\big)^{k+2}
\frac{D^{\eta k+\eta}\varphi_{\eta}}{(\eta k+\eta)!}+L_3t^{q}\Phi_{\eta}^{\eta}.
\end{aligned}\label{wlinear}
\end{equation}
Note that $t^{q-1}\ll(1-t/r_0)^{-1}\ll(1-t/r_0-x/R_0)^{-1}\ll4SK\Phi_{\eta}^{\eta}$
since $r_0<1$. Thus, the summation of the $W_i$'s on the right-hand
side is majorized by
\begin{align}
\frac{4AKMNS}{R_1}\Bigl[L_2(cr)^2\sum_{k=0}^{\infty}(k+2)
\big(\frac{t}{cr}\big)^{k+2}\frac{D^{\eta k+\eta}\varphi_{\eta}}{(\eta k+\eta)!}
+KL_3t\Phi_{\eta}^{\eta}\Bigr],\label{1stRHSw}
\end{align}
and by comparing this with \eqref{1stLHS1}, we obtain the inequalities:
\begin{equation}
\frac{4AKMNSL_2cr}{R_1}\leq\frac{L_1}{2}\quad\text{and}\quad\frac{4AK^2MNSL_3}{R_1}\leq\frac{L_1}{2}.\label{cond3}
\end{equation}
As for the nonlinear terms, note that for $j+\alpha\leq m$, $j<m$,
we have
\begin{align*}
(t\partial_{t})^j(\partial_x^{\alpha})U
& \ll\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}(k+1)^j\big(\frac{t}{cr}\big)^{k+1}
\frac{D^{\eta k+\alpha+j}\varphi_{\eta}}{(\eta k+j)!}\\
 & \ll\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}(k+1)^j\big(\frac{t}{cr}\big)^{k+1}
\frac{D^{\eta k}\varphi_{\eta-\alpha-j}}{(\eta k+j)!}\\
 & \ll\frac{L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k+1}
\frac{D^{\eta k}\varphi_{\eta-m}}{(\eta k)!}
=\frac{L_1t}{R^{m}}\Phi_{\eta-m}^{\eta}.
\end{align*}
Note that we again used (7a) and (3) of Proposition \ref{prop3}. Since $cr<1$
and $r_0<1$, we see that
\begin{equation}
\begin{aligned}
W_i & \ll L_2(cr)^{q+1}\sum_{k=1}^{\infty}\big(\frac{t}{cr}\big)^{k+q}
 \frac{D^{\eta k-1}\varphi_{\eta-1}}{(\eta k-1)!}+L_3t^{q}\Phi_{\eta}^{\eta} \\
 & \ll L_2crt^{q}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}\frac{D^{\eta k}
 \varphi_{\eta-m}}{(\eta k)!}+L_3t^{q}\Phi_{\eta-m}^{\eta}\\
&\ll 4KS(L_2+L_3)t\Phi_{\eta-m}^{\eta}.
\end{aligned} \label{1stRHSW}
\end{equation}
Using the above estimates for $(t\partial_{t})^j(\partial_x^{\alpha})U$
and $W_i$, we can now majorize the remaining terms on the right-hand
side of \eqref{hol-maj-sys-e1}. Setting $\kappa=2^{\eta-m}\max\{\sharp\Lambda,N\}$,
the remaining terms of the right-hand side will be majorized by
\begin{equation}
\begin{aligned}
& \frac{AM}{1-x/R_0}\Big\{\sum_{k=1}^{\infty}\big(\frac{t}{r_0}\big)^{k}
+\sum_{\substack{k+|\nu|+|\mu|\geq2\\ |\nu|+|\mu|\geq 1}}
\big(\frac{t}{r_0}\big)^{k}\Big(\frac{L_1t\Phi_{\eta-m}^{\eta}}{R^{m}R_1}
 \Big)^{|\nu|} \\
&\quad\times \Big(\frac{4KS(L_2+L_3)t\Phi_{\eta-m}^{\eta}}{R_1}\Big)^{|\mu|}\Big\} \\
& \ll\frac{AM}{1-x/R_0}\Big\{\frac{1}{1-t/r_0}\cdot\frac{t}{r_0}
 +\Phi_{\eta-m}^{\eta}\sum_{\substack{k+i+j\geq2\\ i+j\geq1}}
\big(\frac{t}{r_0}\big)^{k}\Big(\frac{\kappa L_1t}{R^{m}R_1}\Big)^{i} \\
&\quad\times  \Big(\frac{\kappa4KS(L_2+L_3)t}{R_1}\Big)^j\Big\} \\
&\ll\frac{AM}{1-x/R_0}\Big\{\frac{4St\Phi_{\eta}^{\eta}}{r_0(1-t/r_0)}
+\Big(\frac{1}{r_0}+\frac{\kappa L_1}{R^{m}R_1}
 +\frac{\kappa4KS(L_2+L_3)}{R_1}\Big)^2 \\
 & \quad \times\frac{t^2\Phi_{\eta-m}^{\eta}}{1-\frac{t}{r_0}
 -\frac{\kappa L_1t}{R^{m}R_1}-\frac{\kappa4KS(L_2+L_3)t}{R_1}}\Big\} \\
& \ll AMK\Big\{\frac{4S}{r_0}t\Phi_{\eta}^{\eta}+\Big(\frac{1}{r_0}
 +\frac{\kappa L_1}{R^{m}R_1}+\frac{\kappa4KS(L_2+L_3)}{R_1}\Big)^2
 t^2\Phi_{\eta-m}^{\eta}\Big\},
\end{aligned} \label{1stRHSnonlin}
\end{equation}
where the last simplification is possible if
\begin{equation}
\frac{1}{r_0}+\frac{\kappa L_1}{R^{m}R_1}+\frac{\kappa4KS(L_2+L_3)}{R_1}
\leq\frac{1}{cr_0}.\label{cond4}
\end{equation}
Comparing \eqref{1stLHS1} and \eqref{1stRHSnonlin}, we obtain the
conditions
\begin{gather}
  \frac{4SAMK}{r_0}\leq\frac{L_1}{2},\label{cond5}\\
  AMK\Big(\frac{1}{r_0}+\frac{\kappa L_1}{R^{m}R_1}
+\frac{\kappa4KS(L_2+L_3)}{R_1}\Big)^2\leq\frac{L_1}{2^{m(\eta+2)+1}\eta^{m}cr}\,.
\label{cond6}
\end{gather}
By choosing a small enough $R$ and fixing it, and then choosing sufficiently
large $L_3$, $L_1$ and $L_2$ in that order, and lastly choosing
$c$ small enough, we can satisfy conditions \eqref{cond1}, \eqref{cond2},
\eqref{cond3}, \eqref{cond4}, \eqref{cond5} and \eqref{cond6}
so that $U$ and $W_i$ satisfy \eqref{hol-maj-sys-e1}, \eqref{hol-maj-sys-e2}
and \eqref{hol-maj-sys-e3}.
\end{proof}

In view of Proposition \ref{prop2}, the functions $U$ and $W_i$ defined
in \eqref{hol-U} and \eqref{hol-W} are majorants of $u$ and $w_i$,
respectively. Since $W_i\ll4KS(L_2+L_3)t\Phi_{\eta-m}^{\eta}$
from \eqref{1stRHSW}, we know that the formal solution $(u,w_1,\ldots,w_{n})$
converges on
\[
\{(t,x):|t/(cr)|^{1/\eta}+|x|\leq R\},
\]
and this completes our proof for Theorem \ref{hol-thm}.

\section{Gevrey class solutions}

\subsection{Formulation and result}

In this section, we consider the case when the coefficients of \eqref{hol-sys-e1}
are in some formal Gevrey class. This space is defined as follows:
for $d\geq1$, the formal Gevrey class $G_x^d$ of index $d$
is defined to be the space of all formal series
$u(x)=\sum_{\alpha\geq0}u_{\alpha}x^{\alpha}$ such that
\[
\sum_{\alpha\geq0}\frac{u_{\alpha}x^{\alpha}}{(\alpha!)^{d-1}}
\]
is a convergent power series. Similarly, we define the formal Gevrey
class $G_x^d[Z]$ to be the space of all formal expansions
$u(x,Z)=\sum_{\alpha\geq0}u_{\alpha}(Z)x^{\alpha}$ such that
\[
\sum_{\alpha\geq0}\frac{u_{\alpha}(Z)x^{\alpha}}{(\alpha!)^{d-1}}
\]
is a convergent power series. The function $u(x,Z)$ is said to be
(formal) Gevrey of index $d$ in the variable $x$ and holomorphic
in the variable $Z$.

We now state the problem. Fix $d\geq1$ and consider the system
\begin{equation} \label{gev-sys-e1}
\begin{gathered}
(t\partial_{t})^{m}u  =F\bigl(t,x,\bigl\{(t\partial_{t})^j
 \partial_x^{\alpha}u\bigr\}_{(j,\alpha)\in\Lambda^d},
\{w_i\}_{i=1}^{N}\bigr),\\
\partial_{t}^{q}w_i  =\mathcal{L}_i(t,x;\partial_x)u+H_i(t,x)\quad\text{for }
i=1,\ldots,N,
\end{gathered}
\end{equation}
where $\Lambda^d=\left\{ (j,\alpha)\in\mathbb{N}^2:j+|d\alpha|\leq m,\; j<m\right\}$
and $\mathcal{L}_i$ is a linear operator given by
\begin{equation}
\mathcal{L}_i(t,x;\partial_x)=\sum_{|d\gamma|\leq q+m}\mathcal{L}_{i,\gamma}(t,x)\partial_x^{\gamma}.\label{gev-L}
\end{equation}
In addition, assume that
\begin{gather*}
F(t,x,Y,Z)  \in G_x^d[t,Y,Z]\\
\mathcal{L}_{i,\gamma}(t,x),H_i(t,x)  \in G_x^d[t].
\end{gather*}

Note that $\Lambda^d$ is a subset of the index set $\Lambda$ defined
in Section 2 and is equal to $\Lambda$ if $d=1$. In addition, the
linear operator $\mathcal{L}_i$ is of degree at most $\lfloor(q+m)/d\rfloor$,
where $\lfloor x\rfloor$ is the floor function of $x$.

Define $\Lambda_0^d=\{(j,\alpha)\in\Lambda^d;
\partial_{Y_{j,\alpha}}F(0,x,0,0)\not\equiv0\}$.
Assume that for all $(j,\alpha)\in\Lambda_0^d$, there exists
$\lambda_{j,\alpha}(x)\in G_x^d$ such that
$b_{j,\alpha}(x)=x^{\alpha}\lambda_{j,\alpha}(x)$
and $\lambda_{j,\alpha}(0)\neq0$. Define the polynomial
\begin{align*}
P^d(\tau,\xi)=\tau^{m}-\sum_{(j,\alpha)\in\Lambda_0^d}
\lambda_{j,\alpha}(0)\tau^j\xi(\xi-1)\cdots(\xi-\alpha+1).
\end{align*}
We have the following result.

\begin{theorem}  \label{gev-thm}  
If $P^d(\tau,\xi)\neq0$ for all $(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$,
then \eqref{gev-sys-e1} has a unique formal solution $(u,w_1,\ldots,w_{N})$
of class $G_x^d[t]$ that satisfies $u(0,x)\equiv0$ and
$\partial_{t}^{k}w_i(0,x)\equiv0$
for $k=0,1,\ldots,q-1$ and $i=1,\ldots,N$. 
\end{theorem}

Under the assumptions on $F$ and $\mathcal{L}_i$, \eqref{gev-sys-e1}
can be rewritten as
\begin{equation} \label{gev-sys-e2}
\begin{gathered}
\begin{aligned}
(t\partial_{t})^{m}u
& =a(x)t+\sum_{(j,\alpha)\in\Lambda_0^d}b_{j,\alpha}(x)
 (t\partial_{t})^j\partial_x^{\alpha}u+\sum_{1\leq i\leq N}d_i(x)w_i \\
& \quad+\sum_{p+|\nu|+|\mu|\geq2}g_{p,\nu,\mu}(x)t^{p}
 \prod_{(j,\alpha)\in\Lambda^d}\big((t\partial_{t}\big)^j
 \partial_x^{\alpha}u)^{\nu_{j,\alpha}}\prod_{1\leq i\leq N}w_i^{\mu_i}
\end{aligned}\\
\partial_{t}^{q}w_i  =\sum_{|d\gamma|\leq q+m}
\mathcal{L}_{i,\gamma}(t,x)\partial_x^{\gamma}u+H_i(t,x)\quad
\text{for }i=1,\ldots,N.
\end{gathered}
\end{equation}
As in the holomorphic case, the existence of a formal solution directly
follows from the assumption that $P^d(\tau,\xi)\neq0$ for all
$(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$.

\subsection{Preparatory lemmas}

To study the formal series of class $G_x^d[Z]$, we use the following
notation: given $u(x,Z)=\sum_{j\geq0}u_{j}(Z)x^j$, define
\[
u^{(d)}(x,Z)=\sum_{j\geq0}u_{j}(Z)(j!)^{d-1}x^j.
\]
We have a similar definition for formal series of class $G_x^d$.
In view of this notation, observe that if $u(x,Z)$ is of class $G_x^d[Z]$,
then there is a unique holomorphic function $w(x,Z)$ such that $w^{(d)}=u$.
We shall denote this $w$ by $\hat{u}$. We shall do the same for
formal series of class $G_x^d$.

It is easy to show that for formal series $u$ and $w$, 
$u\ll w\iff u^{(d)}\ll w^{(d)}$
and $u^{(d)}w^{(d)}\ll(uw)^{(d)}$. These two results imply that by
replacing $\varphi_i$ and $\Phi_i^{s}$ by $(\varphi_i)^{(d)}$
and $(\Phi_i^{s})^{(d)}$ respectively, the results for Proposition
\ref{prop3} will all hold except for (4), (6), (7a) and (7c). The next lemma
states the modified results.

\begin{lemma} \label{lem6}
The following hold for all $i\in\mathbb{N}$:
\begin{enumerate}
\item $x(D^{k}\varphi_i(x))^{(d)}\ll2^{2+i}(D^{k}\varphi_i(x))^{(d)}$;

\item For any $\epsilon\in(0,1)$, there exists a constant $B_{i,\epsilon}>0$
such that for all $j\in\mathbb{N}$, 
$((1-\epsilon x)^{-1})^{(d)}(D^j\varphi_i(x))^{(d)}
\ll B_{i,\epsilon}(D^j\varphi_i(x))^{(d)}$;

\item For a sufficiently small $R<2^{-2-i}$,
\[
(p+q)!(D^{p}\varphi_i(x/R))^{(d)}\ll(p!)(D^{p+q}\varphi_i(x/R))^{(d)};
\]

\item Let $p,q\in\mathbb{N}$. There exists a constant $C_{d}$ (dependent
on $d$ but not on $p$ and $q$) such that
\[
(D^{p+q}\varphi_i)^{(d)}\ll D^{p}(D^{q}\varphi_i)^{(d)}
\ll C_{d}(D^{\lceil dp\rceil +q}\varphi_i)^{(d)}.
\]
Here, $\lceil x\rceil $ is the ceiling of $x$;

\item For any $\epsilon\in(0,1)$ and $R<2^{-2-i}$, there exists a constant
$B_{i,\epsilon}>0$ such that
\[
\Big(\frac{1}{1-\epsilon(t+x/R)}\Big)^{(d)}(\Phi_i^{s})^{(d)}(t,x/R)
\ll B_{i,\epsilon}(\Phi_i^{s})^{(d)}(t,x/R).
\]
\end{enumerate}
\end{lemma}

\begin{proof}
The proofs for (2), (3) and (5) follow from our two previous assertions
and the proof for (1) uses the fact that $x=(x)^{(d)}$. To prove
(4), we note that $(D^{p+q}\varphi_i)^{(d)}\ll D^{p}(D^{q}\varphi_i)^{(d)}$
by inspection. For the second majorant relation, it is sufficient
to show that for all $j$, $p$, $q\geq0$, the quantity
\[
\frac{(j+p+q)!}{(j+\lceil dp\rceil+q)!}
\Big(\frac{j+\lceil dp\rceil +q+1}{j+p+q+1}\Big)^{2+i}
\Big(\frac{(j+p)!}{j!}\Big)^{d-1}
\]
is bounded, which easily follows from the definition of $\lceil x\rceil$.
\end{proof}

\subsection*{Proof of Theorem \ref{gev-thm}}

Let $\hat{F}(t,x,Y,Z)$, $\hat{\mathcal{L}}_{i,\gamma}(t,x)$ and
$\hat{H}_i(t,x)$ be the holomorphic functions derived from $F(t,x,Y,Z)$,
$\mathcal{L}_{i,\gamma}(t,x)$ and $H_i(t,x)$ in \eqref{gev-sys-e1}
and \eqref{gev-L}. Define $\lambda_{j,\alpha}^{*}$ by 
$x\lambda_{j,\alpha}^{*}(x)=\lambda_{j,\alpha}(x)-\lambda_{j,\alpha}(0)$.
Let $\hat{\lambda}_{j,\alpha}^{*}(x)$ be the holomorphic function
derived from $\lambda_{j,\alpha}^{*}$.

Suppose that the functions $\hat{F}(t,x,Y,Z)$,
 $\hat{\mathcal{L}}_{i,\gamma}(t,x)$,
$\hat{H}_i(t,x)$, and $\hat{\lambda}_{j,\alpha}^{*}(x)$ are bounded
by some constant $M>0$ for $|x|\leq R_0$, $|t|\leq r_0$, $|Y_{j,\alpha}|\leq R_1$
and $|Z_i|\leq R_1$. In addition, fix a constant $A_{d}>0$ such
that
\begin{equation}
\frac{1}{|P^d(\tau,\xi)|}\leq\frac{A_{d}}{1+\tau^{m}+\xi^{\lfloor m/d\rfloor}}
\end{equation}
for all $(\tau,\xi)\in\mathbb{N}^{*}\times\mathbb{N}$. We present
this proposition as a counterpart of Proposition \ref{prop2}.

\begin{proposition} \label{prop7}
Consider the following majorant system:
\begin{gather}
\begin{aligned}
 & \Big[(t\partial_{t})^{m}+(x\partial_x)^{\lfloor m/d\rfloor}+1\Big]U(t,x) \\
 &\gg\Big(\frac{AM}{1-x/R_0}\Big)^{(d)}\Big[\frac{t}{r_0}+x\sum_{(j,\alpha)
 \in\Lambda_0}x^{\alpha}(t\partial_{t})^j\partial_x^{\alpha}U+\sum_{1\leq1\leq N}
 \frac{W_i}{R_1}\Big] \\
 & \quad+\sum_{p+|\nu|+|\mu|\geq2}\Big(\frac{AM}{1-x/R_0}\Big)^{(d)}
\frac{t^{p}}{r_0^{p}R_1^{|\nu|+|\mu|}}\prod_{(j,\alpha)\in\Lambda}
\Big((t\partial_{t})^j\partial_x^{\alpha}U\Big)^{\nu_{j,\alpha}}
\prod_{1\leq i\leq N}W_i^{\mu_i}
\end{aligned} \label{gev-maj-sys-e1} \\
 \partial_{t}^{q}W_i\gg\Big(\frac{M}{1-x/R_0-t/r_0}\Big)^{(d)}
\Big[\sum_{|d\gamma|\leq q+m}\partial_x^{\gamma}U+1\Big] \label{gev-maj-sys-e2}\\
U(0,x)\gg0\quad\text{and}\quad\partial_{t}^{k}W_i(0,x)\gg0\quad \text{for }
i=1,\ldots,N,\; k=1,\ldots,q.\label{gev-maj-sys-e3}
\end{gather}
If the formal series $U(t,x)$ and $W_i(t,x)\ (i=1,\ldots,N)$ satisfy
the above relations, then $u\ll U$ and $w_i\ll W_i$ for $i=1,\ldots,N$,
where $(u,w_1,\ldots,w_{N})$ is the formal solution to \eqref{gev-sys-e1}.
\end{proposition}

The task of proving the Gevrey regularity of the formal solution is
now reduced to finding a formal solution $(U,W_1,\ldots,W_{N})$
of class $G_x^d$ that satisfies the above relations.

\begin{proposition} \label{prop8}
Let $r\in(0,r_0)$ and $\eta=m+q$. Then there exist positive constants
$L_1$, $L_2$, $L_3$, $c$ and $R(<2^{-2-\eta}R_0)$ such that
\begin{equation}
U(t,x)=L_1t(\Phi_{\eta}^{\eta})^{(d)}\Big(\frac{t}{cr},
\frac{x}{R}\Big)\label{gev-U}
\end{equation}
and
\begin{equation}
W_i(t,x)=L_2(cr)^{q+1}\sum_{k=1}^{\infty}(k+1)\big(\frac{t}{cr}\big)^{k+q}
\frac{(D^{\eta k}\varphi_{\eta}(x/R))^{(d)}}{(\eta k)!}
+L_3t^{q}(\Phi_{\eta}^{\eta})^{(d)}\Big(\frac{t}{cr},\frac{x}{R}\Big)\label{gev-W}
\end{equation}
for $i=1,\ldots,N$, satisfy \eqref{gev-maj-sys-e1}, \eqref{gev-maj-sys-e2}
and \eqref{gev-maj-sys-e3}.
\end{proposition}

The proof of this proposition is very similar to the proof of the
holomorphic case. The only notable difference is when we deal with
derivatives of $(\varphi_i)^{(d)}$ and $(\Phi_i^{s})^{(d)}$.
For brevity, we shall omit the arguments of $\Phi_i^{s}$ and $\varphi_i^{s}$
in our notations. Choose the constant $K$ to be sufficiently large
such that
\begin{gather*}
\Big(\frac{1}{1-x/R_0-t/r_0}\Big)^{(d)}(\Phi_i^{\eta})^{(d)} 
 \ll K(\Phi_i^{\eta})^{(d)},\\
\Big(\frac{1}{1-x/R_0}\Big)^{(d)}(D^j\varphi_i^{\eta})^{(d)} 
 \ll K(D^j\varphi_i^{\eta})^{(d)},
\end{gather*}
for any $i\leq\eta$, $0\leq j$ and $R<2^{-2-\eta}R_0$. Choose also a constant
$C_{d}$ to be sufficiently large to satisfy Lemma \ref{lem6}(4).

As in the result for the holomorphic case, we have the following estimate
for the left-hand side of \eqref{gev-maj-sys-e2}:
\begin{equation}
\partial_{t}^{q}W_i  \gg\frac{L_2t(\Phi_0^{\eta})^{(d)}}{2^{\eta(\eta+2)}
\eta^{q+1}}+L_3(\Phi_{\eta}^{\eta})^{(d)}.\label{2ndLHS-gev}
\end{equation}
For the corresponding right-hand side, note that if $|d\gamma|\leq q+m=\eta$
then $\lceil d\gamma\rceil \leq\eta$. By Lemma \ref{lem6}(4), we
have
\begin{align*}
\partial_x^{\gamma}U
& \ll\frac{C_{d}L_1t}{R^{\gamma}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k}\frac{(D^{\eta k+\lceil d\gamma\rceil }
 \varphi_{\eta})^{(d)}}{(\eta k)!}\\
& \ll\frac{C_{d}L_1t}{R^{\eta}}\sum_{k=0}^{\infty}
 \big(\frac{t}{cr}\big)^{k}\frac{(D^{\eta k}\varphi_0)^{(d)}}{(\eta k)!}\\
&\ll\frac{C_{d}L_1}{R^{\eta}}t(\Phi_0^{\eta})^{(d)}.
\end{align*}
Therefore by Lemma \ref{lem6}(5), the right-hand side of \eqref{gev-maj-sys-e2}
will be majorized by
\begin{align}
KM\Big(\frac{C_{d}L_1\lfloor\eta/d\rfloor}{R^{\eta}}t(\Phi_0^{\eta})^{(d)}
+4S(\Phi_{\eta}^{\eta})^{(d)}\Big).\label{2ndRHS-gev}
\end{align}
Comparing \eqref{2ndLHS-gev} and \eqref{2ndRHS-gev}, we obtain the
conditions
\begin{equation}
\frac{C_{d}KML_1\lfloor\eta/d\rfloor}{R^{\eta}}
\leq\frac{L_2}{2^{\eta(\eta+2)}\eta^{q+1}}\quad\text{and}\quad
4SKM\leq L_3.\label{cond1-gev}
\end{equation}

Now for left-hand side of \eqref{gev-maj-sys-e1}, we have a similar
result as in the holomorphic case given by
\begin{equation}
(t\partial_{t})^{m}U\gg\frac{L_1t^2}{2^{m(\eta+2)}\eta^{m}cr}
(\Phi_{\eta-m}^{\eta})^{(d)}.\label{1.1-gev}
\end{equation}
For the term with $(x\partial_x)^{\lfloor m/d\rfloor}$, we use
the same technique as in the previous section to obtain
\begin{equation}
\begin{aligned}
(x\partial_x)^{\lfloor m/d\rfloor}U
& \gg L_1t\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}
 \frac{(x\partial_x)(D^{\eta k}\varphi_{\eta})^{(d)}}{(\eta k)!} \\
& \gg\frac{L_1t}{R}\sum_{k=0}^{\infty}\big(\frac{t}{cr}\big)^{k}
\frac{x(D^{\eta k+1}\varphi_{\eta})^{(d)}}{(\eta k)!} \\
&\gg\frac{L_1tx}{2^{\eta+2}R}(\Phi_{\eta-m}^{\eta})^{(d)}.
\end{aligned} \label{1.2-gev}
\end{equation}
Thus by using \eqref{1.1-gev} and \eqref{1.2-gev}, the left-hand
side of \eqref{gev-maj-sys-e1} will majorize
\begin{equation}
\frac{L_1t^2}{2^{m(\eta+2)}\eta^{m}cr}
(\Phi_{\eta-m}^{\eta})^{(d)}
+\frac{L_1tx}{2^{\eta+2}R}(\Phi_{\eta-m}^{\eta})^{(d)}
+L_1t(\Phi_{\eta}^{\eta})^{(d)}.\label{1stLHS1-gev}
\end{equation}
Similarly, we will majorize the summation with $W_i$'s separately
using a stronger majorant derived from $((t\partial_{t})^{m}+1)U$
which is given by
\begin{equation}
\big((t\partial_{t})^{m}+1\big)U
\gg L_1cr\sum_{k=0}^{\infty}(k+2)\big(\frac{t}{cr}\big)^{k+2}
\frac{(D^{\eta k+\eta}\varphi_{\eta})^{(d)}}{(\eta k+\eta)!}
+L_1t(\Phi_{\eta}^{\eta})^{(d)}.\label{1stLHS2b}
\end{equation}
For the corresponding right-hand side, note that for all
$j+\lceil d\alpha\rceil \leq m$,
\begin{align*}
x^{\alpha}(t\partial_{t})^j\partial_x^{\alpha}U
& \ll \frac{C_{d}L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}(k+1)^j
 \big(\frac{t}{cr}\big)^{k+1}\frac{x^{\alpha}(D^{\eta k+\lceil d\alpha\rceil }
 \varphi_{\eta})^{(d)}}{(\eta k)!}\\
 & \ll \frac{C_{d}L_1cr}{R^{\alpha}}\sum_{k=0}^{\infty}\big(\frac{t}{cr}
 \big)^{k+1}\frac{x^{\alpha}(D^{\eta k}\varphi_{\eta-\lceil d\alpha\rceil -j}
 )^{(d)}}{(\eta k)!}\\
 & \ll2^{\alpha(2+\eta)}C_{d}L_1t(\Phi_{\eta-m}^{\eta})^{(d)}
\ll 2^{m(2+\eta)}C_{d}L_1t(\Phi_{\eta-m}^{\eta})^{(d)},
\end{align*}
where we applied Lemma \ref{lem6}(1) to obtain the last line. Thus the linear
terms with $x^{\alpha}$ will have the  estimate
\[
\Big(\frac{AM}{1-x/R_0}\Big)^{(d)}x\sum_{(j,\alpha)\in\Lambda_0}x^{\alpha}
(t\partial_{t})^j(\partial_x)^{\alpha}U\ll(2^{m(2+\eta)}
\sharp\Lambda_0AC_{d}KM)L_1tx(\Phi_{\eta-m}^{\eta})^{(d)}.
\]
Comparing this to the term with $tx(\Phi_{\eta-m}^{\eta})^{(d)}$
in \eqref{1stLHS1-gev}, we have the necessary condition:
\begin{equation}
2^{m(2+\eta)}\sharp\Lambda_0AC_{d}KM\leq\frac{1}{2^{\eta+2}R}.\label{cond2-gev}
\end{equation}
For the linear terms containing $W_i$, we follow a similar process
as in the previous section by using $t^{q-1}\ll4SK(\Phi_{\eta}^{\eta})^{(d)}$
to obtain
\begin{equation}
\frac{4AKMNS}{R_1}\Big(L_2(cr)^2\sum_{k=0}^{\infty}(k+2)
\big(\frac{t}{cr}\big)^{k+2}\frac{(D^{\eta k+\eta}\varphi_{\eta})^{(d)}}
{(\eta k+\eta)!}+KL_3t(\Phi_{\eta}^{\eta})^{(d)}\Big).\label{1stRHSwb}
\end{equation}
By comparing this to \eqref{1stLHS2}, we obtain the following requirements:
\begin{equation}
\frac{4AKMNSL_2cr}{R_1}\leq \frac{L_1}{2}\quad\text{and}\quad
\frac{4AK^2MNSL_3}{R_1}\leq \frac{L_1}{2}.\label{cond3-gev}
\end{equation}
For the nonlinear terms, we have similar results given by
\begin{gather*}
(t\partial_{t})^j(\partial_x^{\alpha})U
\ll\frac{C_{d}L_1t}{R^{m}}(\Phi_{\eta-m}^{\eta})^{(d)}\\
W_i  \ll4SK(L_2+L_3)t(\Phi_{\eta-m}^{\eta})^{(d)}.
\end{gather*}
Therefore by again setting $\kappa=2^{\eta-m}\max\{\sharp\Lambda,N\}$,
the remaining terms of the right-hand side of \eqref{gev-maj-sys-e1}
will be majorized by
\begin{equation}
\begin{aligned}
 & \Big(\frac{AM}{1-x/R_0}\Big)^{(d)}\Big\{\sum_{k=1}^{\infty}
\big(\frac{t}{r_0}\big)^{k}+\sum_{\substack{k+|\nu|+|\mu|\geq2\\
|\nu|+|\mu|\geq1}}
\big(\frac{t}{r_0}\big)^{k}\Big(\frac{C_{d}L_1t(\Phi_{\eta-m}^{\eta})^{(d)}}
 {R^{m}R_1}\Big)^{|\nu|}\\
& \quad \times\Big(\frac{4SK(L_2+L_3)t(\Phi_{\eta-m}^{\eta})^{(d)}}
 {R_1}\Big)^{|\beta|}\Big\} \\
 &\ll AMK\Big\{\frac{4S}{r_0}t(\Phi_{\eta}^{\eta})^{(d)}
+\Big(\frac{1}{r_0}+\frac{\kappa C_{d}L_1}{R^{m}R_1}
+\frac{\kappa4SK(L_2+L_3)}{R_1}\Big)^2t^2(\Phi_{\eta-m}^{\eta})^{(d)}\Big\},
\end{aligned}\label{1stRHSnonlin-gev}
\end{equation}
where the last simplification is possible if
\begin{equation}
\frac{1}{r_0}+\frac{\kappa C_{d}L_1}{R^{m}R_1}
+\frac{\kappa4SK(L_2+L_3)}{R_1}\leq\frac{1}{cr_0}.\label{cond4-gev}
\end{equation}
Therefore, by \eqref{1stLHS1-gev} and \eqref{1stRHSnonlin-gev}, we
obtain the conditions
\begin{gather*}
  \frac{4SAMK}{r_0}\leq \frac{L_1}{2}\label{cond5-gev}\\
  AMK\Big(\frac{1}{r_0}+\frac{\kappa C_{d}L_1}{R^{m}R_1}
+\frac{\kappa4KS(L_2+L_3)}{R_1}\Big)^2\leq\frac{L_1}{2^{m(\eta+2)+1}
\eta^{m}cr}\label{cond6-gev}
\end{gather*}

Finally, similar to the previous section, we may choose constants
$L_1$, $L_2$, $L_3$, $c$ and $R$ that will satisfy conditions
\eqref{cond1-gev}, \eqref{cond2-gev}, \eqref{cond3-gev}, \eqref{cond4-gev},
\eqref{cond5-gev} and \eqref{cond6-gev} so that $U$ and $W_i$
satisfy the majorant system in Proposition \ref{prop7}.

\subsection*{Acknowledgements} 
This research was supported by a grant from
the Office of the Vice Chancellor for Research and Development of
the University of the Philippines Diliman.

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\end{document}