\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 54, pp. 1--33.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/54\hfil Nash-Moser techniques]
{Nash-Moser techniques for nonlinear boundary-value problems}

\author[Markus Poppenberg\hfil EJDE--2003/54\hfilneg]
{Markus Poppenberg}

\address{Markus Poppenberg \hfill\break
Fachbereich Mathematik\\
Universit\"at Dortmund\\
D-44221 Dortmund, Germany}
\email{poppenberg@math.uni-dortmund.de}


\date{}
\thanks{Submitted July 14, 2002. Published May 5, 2003.}
\subjclass[2000]{35K60, 58C15, 35K30}
\keywords{Nash-Moser, inverse function theorem,
boundary-value problem, \hfill\break\indent
parabolic, analytic semigroup, evolution system,
maximal regularity, trace theorem}


\begin{abstract}
 A new linearization method is introduced for smooth
 short-time solvability of initial boundary value problems for
 nonlinear evolution equations. The technique based on an inverse
 function theorem of Nash-Moser type is illustrated by an
 application in the parabolic case. The equation and the boundary
 conditions may depend fully nonlinearly on time and space
 variables. The necessary compatibility conditions are transformed
 using a Borel's theorem. A general trace theorem for normal
 boundary conditions is proved in spaces of smooth functions by
 applying tame splitting theory in Fr\'{e}chet spaces. The
 linearized parabolic problem is treated using maximal regularity
 in analytic semigroup theory, higher order elliptic a priori
 estimates and simultaneous continuity in trace theorems in Sobolev
 spaces.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definitiion}[theorem]{Definitiion}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
%\allowdisplaybreaks

\section{Introduction}\label{A}

The purpose of this paper is to introduce a new linearization
method for smooth short-time solvability of initial boundary
value problems for nonlinear evolution equations. The technique
based on an inverse function theorem of Nash-Moser type is
illustrated by an application in the parabolic case. The equation
and the boundary conditions may depend fully nonlinearly on time
and space variables. The general Theorem \ref{F3} applies to a
nonlinear evolutionary boundary value problem provided that the
linearized equation with linearized boundary conditions is well
posed; here a loss of derivatives is allowed in the estimates of
the linearized problem. An application in the parabolic case is
given in Theorem \ref{I2}.



We mention some points of the proof which might be of independent
interest. A Borel's theorem is applied to transform the
compatibility conditions. A trace theorem is proved for normal
boundary operators in spaces of smooth functions using tame
splitting theory in Fr\'{e}chet spaces. Some results on
simultaneous continuity in trace theorems in Sobolev spaces are
proved. In the application, higher order Sobolev norm estimates
including the dependence of the constants from the coefficients
are derived for the linearized parabolic problem using analytic
semigroup theory involving evolution operators and maximal
regularity.

Inverse function theorems of Nash-Moser type
\cite{Ha,Hoer4,LoZe,Po12,Po30} have been applied to partial
differential equations in several papers, for instance, concerning
small global solutions in \cite{Klai}, periodic solutions in
\cite{Qin,Kor,Fec}, and local solutions in \cite{Ha}, III.\ 2.2 or
\cite{LaPoTe,Po13,Po14,Po18,Po19}. Different from these articles
we consider initial boundary value problems including
compatibility conditions in this note. It seems that the technique
introduced in this paper is the first general linearization method
in the literature based on a Nash Moser type inverse function
theorem which applies to smooth initial boundary value problems
with loss of derivatives including compatibility conditions.

This paper continues and completes the work in \cite{Po14} where the
whole space case is considered. It turned out that the case
of boundary value problems treated in this note is completely different
from the whole space case and requires substantially other methods.

In the literature many results are known on linear and on
nonlinear parabolic boundary value problems. It is beyond the
scope of this paper to give a complete survey. We here only
mention some articles which contain also additional references.
For the classical linear theory of parabolic equations and systems
we refer to \cite{AgVi,Fried,Solo,LaSoUr,LioMa}. Early results on
short-time solvability of nonlinear second order equations can be
found in the references [1] through [8] of the survey paper
\cite{OlKr}. Since then nonlinear parabolic problems have been
studied in many papers, for instance in
\cite{Hud,Fried,LaSoUr,KrKaLo}, or, more recently, in
\cite{Lin,Lap,Skry}. Semigroup theory has been applied by many
authors to the solution of linear and of nonlinear parabolic
problems, we refer to \cite{Sobo,Tana,Paz,Ama,AcTe1,Luna,Ama1}.

This paper is organized as follows. Section \ref{B} contains
notations which are used throughout this article. In section
\ref{C} a smoothing property for Fr\'{e}chet spaces is recalled
from \cite{Po8} which is required as a formal assumption in the
inverse function theorem of Nash-Moser type \cite{Po12}. The
spaces $C^\infty_0([0,T], H^\infty(\Omega))$ are shown to enjoy
this property with uniform constants for all small $T>0$; here
$C_0^\infty$ denotes the subspace of $C^\infty$ containing
functions vanishing with all derivatives at the origin.

In section \ref{F} the inverse function theorem \cite{Po12} is
used to linearize the initial boundary value problem. Mainly due
to compatibility conditions this approach is completely different
from the whole space case \cite{Po14}. A transformation based on a
Borel's theorem  gives a reduction to zero compatibility
conditions. The smallness assumptions required by the inverse
function theorem can be achieved by choosing a small time interval
without supposing smallness assumptions on the initial values.
This is based on a uniformity argument and on  Borel's theorem.

Using results of section \ref{E} the linear
problem is reduced to a problem with homogeneous boundary conditions.
The results of section \ref{E} might be of independent interest.
A trace theorem including estimates is proved for normal boundary operators
in spaces of smooth functions by applying the tame splitting theorem \cite{PoV} in Fr\'{e}chet
spaces. Note that classical right inverses for trace operators in Sobolev
spaces constructed e.g.\ by the Fourier transform depend on the order of
the Sobolev space and do not induce right inverses in spaces of
smooth functions. In addition, results on simultaneous continuity are proved
for trace theorems in Sobolev spaces.

Sections \ref{K}, \ref{D}, \ref{I} contain an application in the
parabolic case.

In section \ref{K} the linearized parabolic initial boundary value problem
of arbitrary order is considered. Under suitable para\-bolicity assumptions
the necessary higher order Sobolev norm estimates are proved.
In order to derive the appropriate dependence of the constants from
the coefficients these estimates are formulated and proved by means of a
symbolic calculus involving the weighted multiseminorms $[~~]_{m,k}$
introduced in \cite{Po12}. The estimates are based on maximal regularity
in H\"older spaces and on the results of section \ref{E}
on simultaneous continuity in trace theorems.

In section \ref{D} we obtain sufficient conditions of elliptic type
for the para\-bolicity assumptions of section \ref{K}.
It is shown that the constants in the higher order elliptic a
priori estimates due to Agmon, Douglis, Nirenberg \cite{AgDoNi}
depend on the coefficients of the problem as required by the Nash-Moser
technique; we note that this means more than only uniformity as stated
in \cite{AgDoNi}. Furthermore, resolvent estimates due to Agmon \cite{Agm1}
are used to establish the assumptions of section \ref{K}.

Finally, in section \ref{I} the short-time solvability
of the nonlinear parabolic problem is proved in Theorem \ref{I2}
under general and natural assumptions. It is enough that the
linearized problem together with the linearized boundary conditions
is given by a regular elliptic problem in the usual sense
(cf.\ Definition \ref{G1} or \cite{LioMa}).

The technical Theorems \ref{F3}, \ref{F4}, \ref{E5} provide a general
framework for applications to evolutionary boundary value problems where
a loss of derivatives appears in the estimates of the linearized problem.
This might be interesting for further applications which are not
accessible to standard methods due to a loss of derivatives, for instance
to other evolution equations or to coupled systems
involving Navier Stokes system and heat equation
where a loss of derivatives appears due to the coupling.

\section{Preliminaries}
\label{B}

We shall consider Fr\'{e}chet spaces $E,F,\ldots$ equipped with a
fixed sequence $\,\|~~\|_0\le\|~~\|_1\le\|~~\|_2\le\ldots$ of
seminorms defining the topology. The product $E\times F$ is
endowed with the seminorms $\|(x,y)\|_k=\max\{\|x\|_k,\|y\|_k\}$.
A linear map $T:E \to F$ is called tame (cf. \cite{Ha}) if there
exist an integer $b $ and constants $c_k$ so that $\|Tx\|_k\le
c_k\|x\|_{k+b} $ for all $k$ and $x$. A linear bijection $\,T$ is
called a tame isomorphism if both $T$ and $T^{-1}$ are tame.

A continuous nonlinear map $\Phi:(U\subset E)\to F$ between
Fr\'{e}chet spaces, $U$ open, is called a $C^1$-map if the
derivative $\Phi'(x)y= \lim\limits_{t\to
0}\,\frac{1}{t}(\Phi(x+ty)-\Phi(x))$ exists for all $x\in U,\,y\in
E$ and is continuous as a map $\Phi':U\times E\to F$. $\Phi$ is
called a $C^2$-map if it is $C^1$ and the second derivative
$\Phi''(x)\{y_1,y_2\}=\lim\limits_{t\to 0}
\frac{1}{t}(\Phi'(x+ty_2)y_1-\Phi'(x)y_1)$ exists and is
continuous as a map $\Phi'' : U\times E^2\to F$. Similar
definitions apply to higher derivatives $\Phi^{(n)}$; $\Phi$ is
called $C^\infty$ if it is $C^n$ for all $n$. Given a function of
two variables $\Phi=\Phi(x,y)$ we can also consider the partial
derivatives $\Phi_x$ and $\Phi_y$ where e.g. $\Phi_x(x,y)z=
\lim\limits_{t\to 0}\,\frac{1}{t}(\Phi(x+tz,y)-\Phi(x,y))$.
One-dimensional derivatives $\Phi_t,t\in\mathbb{R}$ are alternatively
considered as a map $\Phi_t:U\times \mathbb{R}\to F$ or as a map
$\Phi_t:U\to F$, respectively. For these notions we refer to {\bf
\cite{Ha}}, I.3.

Let $\Omega\subset \mathbb{R}^n$ be bounded and open with
$C^\infty$-boundary $\partial\Omega$. In this paper, we restrict
ourselves to the case of bounded domains $\Omega$; most results
are formulated in a way such that a generalization to uniformly
regular domains of class $C^\infty$ in the sense of \cite{Brow},
section 1 or \cite{Ama}, Ch. III, p. 642 is obvious (cf.
\cite{Po14}). For any integer $k\ge 0$ the Sobolev space
$H^k(\Omega)$ is equipped with its natural norms (where
$|\alpha|=\alpha_1+\ldots +\alpha_n$ for $\alpha\in\mathbb{N}_0^n$)
\begin{equation}
\label{B1eq}
\|u\|_k=(\sum_{|\alpha|\le k}\int_\Omega|\partial^\alpha
u(x)|^2\,dx)^{1/2},\quad u\in H^k(\Omega).
\end{equation}
The space $H^\infty(\Omega)=\bigcap_{k=0}^\infty\,H^k(\Omega)$ is
a Fr\'{e}chet space with the norms
$(\|~~\|_k)_{k=0}^\infty$. On the algebra $H^\infty(\Omega)$ we
can consider $\sup$-norms
\begin{equation}
\label{B2eq}
\|u\|_k^\infty=\sup_{|\alpha|\le k}\,\sup_{x\in\Omega}\;|\partial^\alpha
u(x)|, \quad u\in H^\infty(\Omega)
\end{equation}
since by Sobolev's imbedding theorem there
are constants $c_k>0$ such that
\begin{equation}
\label{B3eq}
\|u\|_k^\infty\le c_k\|u\|_{k+b},\quad u\in H^{k+b}(\Omega), \quad
b:=[n/2]+1>n/2.
\end{equation}

The Sobolev space $(H^s(\partial\Omega),\|~~\|_s)$ is defined as usual
for a real $s\ge 0$ using a partition of unity (cf. \cite{Wlo1}, I. 4.2.).
In particular, for an integer $k\ge 1$ the space $H^{k-1/2}(\partial\Omega)$
is the class of functions $\phi$ which are the boundary values of
functions $u\in H^{k}(\Omega)$; the space
$H^{k-1/2}(\partial\Omega)$ can be equipped with the equivalent norm
\begin{equation}
\label{B4eq}
\|\phi\|_{k-1/2}=\inf\,\{\|u\|_k:u\in H^k(\Omega),
u=\phi \;\mbox{\rm on}\;\partial \Omega\},
\phi\in H^{k-1/2}(\partial\Omega).
\end{equation}
The Fr\'{e}chet space
$H^\infty(\partial\Omega)=\bigcap_{k=0}^\infty H^{k}(\partial\Omega)$
is equipped with these norms.

The Fr\'{e}chet space $C^\infty(\overline{\Omega})$ of all
$C^\infty$-functions on $\Omega$ such that all partial
derivatives are uniformly continuous on $\Omega$ is equipped with
the norms $(\|~~\|_k^\infty)_{k=0}^\infty$. The Fr\'{e}chet space
$C^\infty(\partial\Omega)$ of all smooth functions on the manifold
$\partial\Omega$ is endowed with the norm system
$(\|~~\|_k^\infty)_{k=0}^\infty$ defined as usual using cutoff
functions and a partition of unity (cf. \cite{Po9}, 4.14.). It is
well known that there exists a linear continuous extension
operator $R_\Omega:C^\infty(\partial\Omega)\to
C^\infty(\overline{\Omega})$ such that $\|R_\Omega f\|_k\le
c_k\|f\|_k$ for all $k$ and constants $c_k>0$; this follows e.g.
from \cite{Se} using a partition of unity.

A vector valued function $u=(u_1,\ldots ,u_M)$ belongs to
$H^\infty(\Omega,\mathbb{R}^M)$ if each coordinate $u_j$ is in
$H^\infty(\Omega)$; the same applies to
$H^\infty(\partial\Omega,\mathbb{R}^M)$. We use of the following symbolic
calculus introduced in \cite{Po12}. Let $p,q\ge 0$ be integers,
$p+q\ge 1$, and let $E_1,\ldots , E_p,F_1,\ldots,F_q$ be linear
spaces each equipped with a sequence $|~~|_0\le |~~|_1\le |~~|_2
\le\ldots$ of seminorms. For any integer $m,k\ge 0$ and $x_1\in
E_1,\ldots, x_p\in E_p,\,y_1\in F_1,\ldots , y_q\in F_q$ we define
\[
\label{B5eq}
[x_1,...,x_p;y_1,...,y_q]_{m,k}=
\sup\limits\{|x_{k_1}|_{m+i_1}...|x_{k_r}|_{m+i_r}
|y_1|_{m+j_1}...|y_q|_{m+j_q}\}
\]
the 'sup' running over all $i_1,\ldots , i_r ,j_1, \ldots ,j_q\ge
0$ and $1\le k_1,\ldots ,k_r\le p$ with $0\le r\le k$ and
$i_1+\ldots+i_r+j_1+\ldots +j_q\le k$ (for $\,r=0$ the
$|x|$-terms are omitted). For $q=0$ we write $[x_1,\ldots
,x_p]_{m,k}$ (the $|y|$-terms are omitted) and for $p=0$ we write
$[;y_1,\ldots ,y_q]_{m,k}$. For $m=0$ we write
$[\ldots]_{k}=[\ldots]_{0,k}$. Observe that
$[x_1,\ldots,x_p;y_1,\ldots ,y_q]_{m,k}$ is a seminorm seperately
in each component $y_j$ while it is completely nonlinear in the
$x_i$-components. The weighted multiseminorms $[~~]_{m,k}$ are
increasing in $m$ and in $k$. For the purely nonlinear terms
(i.e., $q=0$) we have $[x_1,\ldots ,x_p]_{m,0}=1$ and $[x_1,\ldots
,x_p]_{m,k}\ge 1$ for all $m,k$. For properties of the terms
$[~~]_{m,k}$ we refer to \cite{Po12}, 1.7.; we shall often apply
rules like $[x]_{m,k}\cdot [x]_{m,i} \le [x]_{m,k+i}$ and
$[x]_{m,i+k}\le \max\{1,|x|_{m+i}^{i+k}\}[x]_{m+i,k}\le
C'[x]_{m+i,k}$ if $|x|_{m+i}\le C$. If Sobolev spaces
$H^\infty(\Omega)$ are involved then the following applies. The
expressions $[u]_{m,k}$ and $[u;v]_{m,k}$ are defined by the
corresponding Sobolev norms $\|u\|_i,\|v\|_j$. The terms
$\|u\|_{m,k}^\infty$ or $\|u,v\|_{m,k}^\infty$ (i.e., $p=2,q=0$)
are defined by $\sup$-norms $\|u\|_i^\infty,\|v\|_j^\infty$. The
expression $[u;v]_{m,k}^\infty$ (i.e., $p=q=1$) is defined by
means of the $\sup$-norms $\|u\|_i^\infty$ and Sobolev norms
$\|v\|_j$. For a real number $t$ let $[t]$ denote the largest
integer $j$ with $j\le t$.

\section{A smoothing property for Fr\'{e}chet spaces}
\label{C}

In the inverse function theorem \ref{C5}
the Fr\'{e}chet spaces are assumed to satisfy smoothing property
(S) introduced in \cite{Po8}, 3.4 and property (DN) of Vogt \cite{V1}.
A Fr\'{e}chet space $E$ has property (DN) if there is $b$
such that for any $n$ there are $k_n$ and
$c_n>0$ such that for all $x\in E$ we have
\begin{equation}
\label{C1eq}
\|x\|_n^2\le c_n\|x\|_b\|x\|_{k_n}
\end{equation}
We say that $E$ has smoothing property {\rm (S)} if there exist $b,p\ge 0$ 
and constants $c_n>0$ such that for any $\theta\ge 1$ and any
$x\in E$ and for any sequence $(A_n)_n$ satisfying $\|x\|_n\le
A_n\le A_{n+1}$ and $A_n^2\le A_{n-1}A_{n+1}$ for all $n$ there
exists an element $S_\theta x\in E$ (which may depend on $x$ and
on the sequence $(A_n)$) such that
\begin{equation} \label{C2eq}
\begin{gathered}
\|S_\theta x\|_n  \le  c_n\theta^{n+p-k} A_k, \quad b\le k\le n+p\\
\|x-S_\theta x\|_n  \le  c_k \theta^{n+p-k}A_k, \quad k\ge n+p.
\end{gathered}
\end{equation}

Smoothing property (S) generalizes (cf. \cite{Po8})
the classical smoothing operators (cf.
\cite{Ha}, \cite{Hoer4}, \cite{LoZe}).
For a Fr\'{e}chet space $E$ and $T>0$ we put
\begin{equation}
\label{C3eq}
C_0^\infty([0,T],E)=\Big\{u\in C^\infty([0,T],E):u^{(j)}(0)=0
, j=0,1,2,\ldots\Big\}.
\end{equation}
In case $E$ is one-dimensional we write $C^\infty_0[0,T]$ instead
of $C_0^\infty([0,T],E)$.

\begin{lemma}
\label{C2} Let $T_1>0$. The spaces $C_0^\infty[0,T]$ have property
{\rm (S)} with $b=p=0$ where $c_n$ in {\rm (\ref{C2eq})} may be
chosen uniformly for all $0<T\le T_1$. \end{lemma}

\begin{proof}
The space $\mathcal{D} [0,2]$ of all smooth function with support in
$[0,2]$ has property (S) with $b=p=0$ (cf. \cite{Po8}, 5.1). The
space $C_0^\infty[0,1]$ is a quotient space of $\mathcal{D} [0,2]$ by means
of restriction and hence a direct summand of $\mathcal{D} [0,2]$ using an
extension operator (cf. Seeley \cite{Se} or \cite{Po9}, 4.8).
Therefore, $C_0^\infty[0,1]$ inherits property (S) from $\mathcal{D} [0,2]$
with $b=p=0$. To prove uniformity we assume that $T_1=1$. We have
\begin{equation}
\label{C4eq}
\|f\|_k^{[0,T]}=\sup\limits_{j=0}^k\sup\limits_{t\in [0,T]}|f^{(j)}(t)|
=\sup\limits_{t\in [0,T]}|f^{(k)}(t)|=:|f|_k^{[0,T]}
\end{equation}
for $f\in C_0^\infty[0,T]$ and $0<T\le 1$. Put
$\Gamma_T:C_0^\infty[0,1]\to C_0^\infty[0,T],\Gamma_Tf(x)=f(x/T)$.
Notice that $|\Gamma_T f|_k^{[0,T]}=T^{-k}|f|_k^{[0,1]}$. If
$S_\theta$ is induced by property (S) in $C_0^\infty[0,1]$ then
$\Gamma_T\circ S_{T\theta}\circ \Gamma_T^{-1}$ gives property (S)
for $C_0^\infty[0,T]$ with the same constants.
\end{proof}

The uniformity part of Lemma \ref{C2} does not work e.g.\ for
$C^\infty[0,T]$. For a Fr\'{e}chet space $E$ and a sequence
$0\le\alpha_0\le \alpha_1\le\ldots \nearrow +\infty$ we consider
the power series space of $E$-valued sequences
$x=(x_j)_{j=1}^\infty\subset E$ defined by \label{C5eq}
\[
\Lambda_\infty^\infty(\alpha;E)=\{(x_j)_j\subset E:
\|x\|_k=\sup_{i=0}^k\sup_j\|x_j\|_{k-i}e^{i\alpha_j}<\infty,
k=0,1,\ldots\}.
\]
In case ${\rm dim}\,E=1$ we write $\Lambda_\infty^\infty(\alpha)$
instead of $\Lambda_\infty^\infty(\alpha;E)$. The corresponding
space defined by $l^2$-norms instead of $\sup$-norms is denoted
by $\Lambda_\infty^2(\alpha)$.

\begin{lemma}
\label{C3}
If $E$ has property {\rm (S)} then $\Lambda_\infty^\infty(\alpha;E)$
has {\rm (S)} as well.
\end{lemma}

\begin{proof}
Let $0\not= x\in \Lambda_\infty^\infty(\alpha;E)$ and $\|x\|_k\le
A_k\le A_{k+1}, A_k^2\le A_{k-1}A_{k+1}$. We may assume that
$t\mapsto \log\,A_t$ is convex and increasing. We have
\[
\|x_j\|_i\le \inf\limits_{i\le k\in\mathbb{N}_0}e^{(i-k)\alpha_j}A_k=:B_i^j
\le D_{i+1}^j:=\inf\limits_{i+1\le t\in \mathbb{R}} e^{(i+1-t)\alpha_j}A_t\le A_{i+1}
\]
for any $i,j$. It is easy to see that $D_{i+1}^j\le D_{i+2}^j$ and
$(D_{i+1}^j)^2\le D_{i}^jD_{i+2}^j$ for all $i,j$. We hence may
choose $S_\theta x_j$ according to the sequence $(D_{i+1}^j)_i$
such that
\begin{equation} \label{C6aeq}
\begin{gathered}
\|S_\theta x_j\|_n  \le  c_n\theta^{n+p+1-k} D_k^j , \quad
b+1\le k\le n+p+1\\
\|x_j-S_\theta x_j\|_n  \le  c_k \theta^{n+p+1-k}D_k^j , \quad k\ge n+p+1.
\end{gathered}
\end{equation}
We define $T_\theta x$ for $\theta\ge 1$ by $(T_\theta x)_j=0$ if
$e^{\alpha_j}\ge \theta$ and $(T_\theta x)_j=S_\theta x_j$ if
$e^{\alpha_j} <\theta$. For $e^{\alpha_j}\ge \theta$ we get for
$k\ge n+p+1$ and $0\le i\le n$ the estimate
\begin{equation}
\label{C7eq}
\|x_j\|_{n-i}e^{i\alpha_j} \le
e^{(n-k)\alpha_j}A_k\le \theta^{n-k}A_k.
\end{equation}
For $e^{\alpha_j}<\theta$ we establish
for $k\ge n+p+1$ and $0\le i\le n$ the estimate
\begin{equation}
\label{C8eq}
\|x_j-S_\theta x_j\|_{n-i} e^{i\alpha_j} \le
c_k\theta^{n-i+p+1-k}e^{i\alpha_j}D_k^j\le
c_k \theta^{n+p+1-k}A_k.
\end{equation}
Let $e^{\alpha_j} <\theta$ and $b+1\le k\le n+p+1$. In the case
$0\le i\le k-b-1$ we get
\begin{equation}
\label{C9eq}
\|S_\theta x_j\|_{n-i}e^{i\alpha_j}\le
c_{n-i} D_{k-i}^j\theta^{n+p+1-k}e^{i\alpha_j}\le c_{n-i}\theta^{n+p+1-k}A_k
\end{equation}
and for $k-b-1\le i\le n$ we obtain (where we may assume that
$p\ge b$)
\begin{equation}
\label{C10eq}
\|S_\theta x_j\|_{n-i}e^{i\alpha_j}  \le
c_{n-i}\theta^{n+p-i-b}D_{b+1}^je^{i\alpha_j}\le
c_{n-i}\theta^{n+p+1-k}A_k
\end{equation}
since $D_{b+1}^je^{i\alpha_j}\le e^{(i+b+1-k)\alpha_j}A_k
\le \theta^{i+b+1-k} A_k$. This gives the result.\\
\end{proof}

\begin{proposition}
\label{C4} Let $\Omega\subset\mathbb{R}^n, n\ge 2$, be open and bounded
with $C^\infty$-boundary. Let $T_1>0$ and an integer $ m\ge 1$ be
fixed. Then the spaces $C_0^\infty([0,T],H^\infty(\Omega))$ and
$C_0^\infty([0,T],H^\infty(\partial\Omega))$ equipped with the
norms
\begin{equation}
\label{C11eq}
\|u\|_k=\sup\Big\{\|u^{(i)}(t)\|_{k-mi}:t\in[0,T],0\le i\le k/m \Big\}
\end{equation}
have properties {\rm (S), (DN)}. In addition, the constants
$c_n,k_n,b,p$ in the above definitions of {\rm (S), (DN)} can be
chosen uniformly for all $0<T\le T_1$. \end{proposition}

\begin{proof}
Clearly the spaces have (DN); the uniformity statement holds since
$C_0^\infty[0,T]$ is a subspace (by trivial extension) of
$C^\infty[-1+T,T]\cong C^\infty[0,1]$ if $T\le 1$. It is enough to
show property (S) for the spaces equipped with the new norm system
$(\|~~\|_{mk})_{k=0}^\infty$ (cf. \cite{Po4}, 4.3). There are tame
isomorphisms $H^\infty(\Omega)\cong \Lambda^\infty_\infty(\alpha)$
for $\alpha_j= (\log\,j)/n$ and $H^\infty(\partial\Omega)\cong
\Lambda_\infty^\infty(\beta)$ for $\beta_j=(\log\,j)/(n-1)$; this
is proved in \cite{Po9}, 4.10, 4.14. We put
$\tilde{\alpha}_j=m\alpha_j$ and obtain a tame isomorphism
\begin{equation}
\label{C12eq}
\Big(C_0^\infty([0,T],H^\infty(\Omega)),(\|~~\|_{mk})_{k=0}^\infty\Big)\cong
\Lambda^\infty_\infty(\tilde{\alpha};C_0^\infty[0,T]).
\end{equation}
The same argument applies to $H^\infty(\partial\Omega)$. Now
\ref{C2}, \ref{C3} give the
assertion.
\end{proof}

In section \ref{F} we shall apply the following inverse function theorem
of Nash-Moser type which is proved in \cite{Po12}, 4.1
(cf. \cite{Ha}, \cite{Hoer4}, \cite{LoZe}).

\begin{theorem}
\label{C5} Let $E,F$ be Fr\'{e}chet spaces with smoothing property
{\rm (S)} and {\rm (DN)}. Let $U_0=\{x\in E:|x|_b<\eta\}$ for some
$b\ge 0,\eta >0$. Let $\Phi:(U_0\subset E)\to F$ be a $C^2$-map
with $\Phi(0)=0$ such that $\Phi'(x):E\to F$ is bijective for all
$x\in U_0$. Assume that there are an integer $d\ge 0$ such that
\begin{equation} \label{C13eq}
\begin{gathered}
\|\Phi'(x)v\|_k  \le  c_k[x;v]_{d,k} \\
\|\Phi'(x)^{-1}y\|_k  \le  c_k[x;y]_{d,k}\\
\|\Phi''(x)\{v,v\}\|_k  \le  c_k[x;v,v]_{d,k}
\end{gathered}
\end{equation}
for all $x\in U_0,v\in E, y\in F$ and all $k=0,1,2,\ldots$ with
constants $c_k>0$. Then there exist open zero neighbourhoods
$V=\{y\in F:\|y\|_s<\delta\}\subset F$ and $U\subset E$ such that
$\Phi:U\to V$ is bijective and $\Phi^{-1}:(V\subset F)\to E$ is a
$C^2$-map. If $\Phi$ is $C^n$ then $\Phi^{-1}$ is $C^n$ as well,
$2\le n\le \infty$. Moreover, the numbers $s\ge 0$ and $\delta >0$
depend only on the constants in the assumption, i.e., on
$b,d,\eta,c_k$ and on the constants in properties {\rm (S), (DN)}.
\end{theorem}

\section{Linearization of boundary-value problems}
\label{F}

Let $\Omega\subset \mathbb{R}^n$ be bounded and open with
$C^\infty$-boundary. We fix a real number $T>0$ and integers
$M\ge 1,m\ge 2$. We write $H^\infty(\Omega)=H^\infty(\Omega,\mathbb{R}^M)$
and $H^\infty(\partial\Omega)=H^\infty(\partial\Omega,\mathbb{R}^M)$. We
assume that $m$ is even and put
$I(m)=\{\alpha\in\mathbb{N}_0^n:|\alpha|\le m\}$. Let
$A\subset(\mathbb{R}^M)^{I(m)}$ be open; then the set
\begin{equation}
\label{F1aeq}
U_0=\{u\in H^\infty(\Omega):\{\partial^\alpha u(x)\}_{|\alpha|\le m}
\in A, x\in\overline{\Omega}\}
\end{equation}
is open in $H^\infty(\Omega)$ as well. Let $F\in
C^\infty([0,T]\times\overline{\Omega}\times\overline{A},\mathbb{R}^M),
F=F(t,x,u)$. We consider $\mathcal{F}:[0,T]\times (U_0\subset
H^\infty(\Omega))\to H^\infty(\Omega)$ defined by
\begin{equation}
\label{F1beq}
\mathcal{F}(t,u)(x)=F(t,x,\{\partial^\alpha u(x)\}_{|\alpha|\le m}),
\quad u\in U_0,t\in [0,T],x\in \Omega.
\end{equation}

It is proved in \cite{Po14}, section 2 (cf. \cite{Hoer4})
that $\mathcal{F}$ is a nonlinear $C^\infty$-map between Fr\'{e}chet spaces
where $\mathcal{F}':([0,T]\times U_0)\times (\mathbb{R}\times H^\infty(\Omega))
\to H^\infty(\Omega)$ is given by
$\mathcal{F}'(t,u)(s,v)=\mathcal{F}_t(t,u)s+\mathcal{F}_u(t,u)v$ where
\begin{equation}
\label{F1ceq}
\mathcal{F}_u(t,u)v=
\sum\limits_{|\alpha|\le m}\,F_{\partial^\alpha u}(t,\cdot,
\{\partial^\beta u(\cdot)\}_{|\beta|\le m})\partial^\alpha v.
\end{equation}
If $A$ is bounded then
\cite{Po14}, 2.3, 2.4 give with $b=[n/2]+1$ the estimates
\begin{equation}\label{F1eq}
\begin{gathered}
\|\mathcal{F}'(t,u)(s,v)\|_k  \le  c_k [(t,u);(s,v)]_{m+b,k}\\
\|\mathcal{F}''(t,u)\{(s,v),(s,v)\}\|_k  \le  c_k [(t,u);(s,v),
(s,v)]_{m+b,k}
\end{gathered}
\end{equation}
for all $t\in [0,T], u\in U_0, s\in \mathbb{R}, v\in H^\infty(\Omega)$
where $c_k>0$ are constants.

We define nonlinear boundary operators $\mathcal{B}_j$ and put $\mathcal{B}=(\mathcal{B}_1,\ldots ,\mathcal{B}_{m/2})$. For that we fix integers
$m_j\ge 0$ and choose open sets $A_j\subset (\mathbb{R}^M)^{I(m_j)}$ and
mappings $B_j\in C^\infty([0,T]\times \partial \Omega\times
\overline{A}_j,\mathbb{R}^M), j=1,\ldots ,m/2$. Then the sets
\begin{equation}
\label{F1deq}
U_j=\{u\in H^\infty(\Omega):\{\partial^\beta u(x)\}_{|\beta|\le m_j}
\in A_j, x\in \partial\Omega\}
\end{equation}
are open in $H^\infty(\Omega)$. We define $\mathcal{B}_j: [0,T]\times
(U_j \subset H^\infty(\Omega))\to H^\infty(\partial\Omega )$ by
\begin{equation}
\label{F1eeq}
\mathcal{B}_j(t,u)(x)=B_j(t,x,\{\partial^\beta u(x)\}_{|\beta|\le m_j}),
\; u\in U_j,t\in [0,T],x\in\partial\Omega.
\end{equation}
The arguments used in \cite{Po14}, section 2 or \cite{Hoer4}
show that $\mathcal{B}_j$ is a $C^\infty$-map between Fr\'{e}chet spaces
where
$\mathcal{B}_j': ([0,T]\times U_j) \times(\mathbb{R}\times H^\infty(\Omega))
\to H^\infty(\partial\Omega)$ is given by
$\mathcal{B}_j'(t,u)(s,v)=(\mathcal{B}_j)_t(t,u)s+(\mathcal{B}_j)_u(t,u)v$
where
\begin{equation}\label{F1feq}
(\mathcal{B}_j)_u(t,u)v =
\sum\limits_{|\alpha|\le m_j}(B_j)_{\partial^\alpha u}
(t,\cdot,\{\partial^\beta u(\cdot)\})\partial^\alpha v.
\end{equation}
For a bounded set $A_j$ the proof of \cite{Po14}, 2.3, 2.4 yields
\begin{equation}\label{F2eq}
\begin{gathered}
\|\mathcal{B}_j'(t,u)(s,v)\|_{k-\frac{1}{2}}  \le c_k [(t,u);(s,v)]_{m_j+b,k}\\
\|\mathcal{B}_j''(t,u)\{(s,v),(s,v)\}\|_{k-\frac{1}{2}}
\le  c_k [(t,u);(s,v),(s,v)]_{m_j+b,k}
\end{gathered}
\end{equation}
for all $t\in [0,T], u\in U_j, s\in \mathbb{R}, v\in H^\infty(\Omega)$
with some $c_k>0$ and $b$ as above.
Our goal is to solve the nonlinear initial boundary-value problem
\begin{equation} \label{F3eq}
\begin{gathered}
u_t  =  \mathcal{F}(t,u) \quad \mbox{in }\Omega , \;
t\in[0,T_0]  \\
\mathcal{B}(t,u)  =  h(t) \quad \mbox{on }\partial\Omega , \;
t\in [0,T_0] \\
u(0)  =  \phi\,.
\end{gathered}
\end{equation}
More precisely, for a given initial value $\phi\in
U:=\bigcap_{j=0}^{m/2} U_j\subset H^\infty(\Omega)$ and a given
boundary value $h\in
C^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$ we are looking for
a solution $u$ of problem (\ref{F3eq}) for some suitable small
$T_0>0$; by a solution we mean a function $u\in
C^\infty([0,T_0],H^\infty(\Omega))$ such that $u(t)\in U$ for all
$t\in [0,T_0]$ and (\ref{F3eq}) is satisfied.

There are some natural necessary constraints on the given data
$h,\phi$. In order such that (\ref{F3eq}) can admit a smooth
solution the data $h, \phi$ have to satisfy the following
necessary compatibility conditions which are obtained by computing
$h^{j}(0)$ as a differential operator acting on $ \phi$ on the
boundary of $\Omega$ by means of (\ref{F3eq}). For instance, we
get $h(0)=\mathcal{B}(0,\phi)=:\Gamma_0(\phi)$ and
\begin{equation}
\label{F3aeq}
h'(0)=\mathcal{B}_t(0,\phi)+\mathcal{B}_u(0,\phi)\mathcal{F}(0,\phi)
=:\Gamma_1(\phi).
\end{equation}
In a similar way we obtain from (\ref{F3eq}) the necessary compatibility
conditions
\begin{equation}
\label{F4eq}
h^{(j)}(0)=\partial_t^j\mathcal{B}(t,u(t))_{|t=0}=:\Gamma_j(\phi)
\quad \mbox{\rm on}~\partial\Omega,\quad j=0,1,2,\ldots
\end{equation}
where differential operators $\Gamma_j$ acting on $\phi$ on
$\partial\Omega$ are obtained by first computing
$\partial_t^j\mathcal{B}(t,u(t))$, then replacing all derivatives
$\partial_t^i u$ by terms involving $u$ using $u_t=\mathcal{F}(t,u(t))$ and finally evaluating at $t=0$ using $u(0)=\phi$.
Analogously, the values $u^{j}(0)$ are a priori determined in
$\Omega$ by (\ref{F3eq}). For instance, we get
$u(0)=\phi=:\Psi_0(\phi)$ and $u'(0)=\mathcal{F}(0,\phi)=:\Psi_1(\phi)$ and
\begin{equation}
\label{F4aeq}
u''(0)=\mathcal{F}_t(0,\phi)+\mathcal{F}_u(0,\phi)\mathcal{F}(0,\phi)
=:\Psi_2(\phi).
\end{equation}
Using the first and the third equation in (\ref{F3eq})
we see that solutions $u$ satisfy
\begin{equation}
\label{F5eq}
u^{(j)}(0)=\partial_t^{j-1}\mathcal{F}(t,u(t))_{|t=0}=:\Psi_{j}(\phi)
\quad \mbox{in }\Omega,\; j=1,2,3,\ldots
\end{equation}
with differential operators $\Psi_j$ acting on $\phi$ in $\Omega$
where $\Psi_j$ are defined using $u_t=\mathcal{F}(t,u)$ and
$u(0)=\phi$. We note that (\ref{F5eq}) are by no means
compatibility conditions like (\ref{F4eq}). However, the a priori
knowledge of $u^{(j)}(0)$ can be used to transform problem
(\ref{F3eq}) such that solutions $v$ of the transformed problem
satisfy $v^{j}(0)=0$ for all $j$. This simplifies the
compatibility conditions (\ref{F4eq}). We shall apply the
following version of a theorem of E. Borel's \cite{Borel}.

\begin{lemma}
\label{F1a} Let $E$ be a Fr\'{e}chet space. Let
$(a_j)_{j=0}^\infty\subset E$ be an arbitrary sequence. Then there
is $\psi\in C^\infty ([0,1],E)$ such that $\psi^{(j)}(0)=a_j$ for
all $j$. \end{lemma}

The proof of this lemma follows the standard proof in
(cf. \cite{Hoer}, 1.2.6 or \cite{Po3}, 1.3).\smallskip

We choose $\psi\in C^\infty([0,T],H^\infty(\Omega))$ such that
$\psi(t)\in U$ for $t\in [0,T]$ and
\begin{equation}\label{F6eq}
\psi^{(j)}(0)=\Psi_j(\phi), \quad j=0,1,2,\ldots
\end{equation}
We put $v=u-\psi$ and get from (\ref{F3eq}) the transformed problem
\begin{equation}\label{F7eq}
\begin{gathered}
v_t  =  \mathcal{F}(t,v+\psi)-\psi'(t)
\quad \mbox{in }\Omega , \; t\in[0,T_0]  \\
\mathcal{B}(t,v+\psi)-\mathcal{B}(t,\psi)  =  h(t)-\mathcal{B}(t,\psi)
\quad \mbox{on } \partial\Omega , \; t\in [0,T_0] \\
v(0)  =  0\,.
\end{gathered}
\end{equation}

\begin{remark} {\rm \label{F1}
(i) If $u$ solves (\ref{F3eq}) then $v=u-\psi$ solves (\ref{F7eq}).
On the other hand,
if $v$ solves (\ref{F7eq}) then $u=v+\psi$ solves (\ref{F3eq}).
(ii) Solutions $u$ of (\ref{F3eq}) satisfy
$u^{(j)}(0)=\psi^{(j)}(0)$ 
for all $j$. On the other hand,
solutions $v$ of (\ref{F7eq}) automatically satisfy $v^{(j)}(0)=0$
for all $j$. (iii) For $\gamma(t)=\mathcal{B}(t,\psi(t))$ we have
$\gamma^{(j)}(0) =\Gamma_j(\phi)$ for all $j$. (iv) If $h$
satisfys $h^{(j)}(0)=\Gamma_j(\phi)$ for all $j$ then the right
hand side $\tilde{h}(t)=h(t)-\mathcal{B}(t,\psi(t))$ in (\ref{F7eq})
satisfies $\tilde{h}^{(j)}(0)=0$ for all $j$. (v) The left hand
side $\tilde{\mathcal{B}}(t,v)= \mathcal{B}(t,v+\psi(t))-\mathcal{B}(t,\psi(t))$ considered in (\ref{F7eq}) as an operator in $v$
satisfies $(\partial_t^j\tilde{\mathcal{B}})(0,0)=0$ for all $j$.
Note that
\[
(\partial_t \tilde{\mathcal{B}})(t,v)=
\mathcal{B}_t(t,v+\psi)+\mathcal{B}_u(t,v+\psi)\psi'
-\mathcal{B}_t(t,\psi) -\mathcal{B}_u(t,\psi)\psi'.
\]
\indent (vi) In the case of linear boundary conditions we have
$\tilde{\mathcal{B}}(t,v)=\mathcal{B}(t,v)$. (vii) The right hand side
$\tilde{\mathcal{F}}(t,v)=\mathcal{F}(t,v+\psi(t)) -\psi'(t)$ in
(\ref{F7eq}) considered as a nonlinear differential operator in
$v$ satisfies $(\partial_t^j\tilde{\mathcal{F}})(0,0)=0$ for all $j$.
This follows since $\tilde{\mathcal{F}}(0,0)=\mathcal{F}(0,\phi)-\psi'(0)=0$ and
\[
(\partial_t^j\tilde{\mathcal{F}})(0,0)=
\partial_t^j\{\mathcal{F}(t,\psi(t))\}_{t=0}-\psi^{(j+1)}(0)=
\Psi_{j+1}(\phi)-\psi^{(j+1)}(0)=0.
\]
}%End of \rm
\end{remark}

Using the above notation we hence may consider the
normalized problem
\begin{equation}\label{F8eq}
\begin{gathered}
u_t  =  \tilde{\mathcal{F}}(t,u) \quad \mbox{ in }\Omega ,\;
t\in[0,T_0]  \\
\tilde{\mathcal{B}}(t,u)  =  \tilde{h}(t) \quad \mbox{on }\partial\Omega ,\;
t\in [0,T_0] \\
u(0)  =  0\,,
\end{gathered}
\end{equation}
where we may assume the normalized conditions
\begin{equation} \label{F9eq}
\begin{gathered}
(\partial_t^j\tilde{\mathcal{F}})(0,0)  =  0, \quad j=0,1,2,\ldots  \\
(\partial_t^j \tilde{\mathcal{B}})(0,0)  =  0, \quad j=0,1,2,\ldots \\
\tilde{h}^{(j)}(0) =  0 , \quad j=0,1,2,\ldots
\end{gathered}
\end{equation}
where $\tilde{h}^{(j)}(0)=0$ are the natural compatibility
conditions for (\ref{F8eq}) if we assume the first two conditions
in (\ref{F9eq}). Since solutions $u$ of (\ref{F8eq}), (\ref{F9eq})
satisfy $u^{(j)}(0)=0$ for all $j$ we have to look for solutions
$u$ in the space $C_0^\infty([0,T_0],H^\infty(\Omega))$. We
formulate problem (\ref{F8eq}) by a mapping. We fix $T>0$ and put
$J=[0,T]$. We get an open set $W\subset
C_0^\infty(J,H^\infty(\Omega))$ by
\begin{equation}\label{F9aeq}
W=\big\{u\in C_0^\infty(J,H^\infty(\Omega)):u(t)\in U,t\in J\big\}
\end{equation}
where we may assume that $0\in W$. We define the nonlinear map
\begin{equation} \label{F9beq}
\Phi:(W\subset C_0^\infty(J,H^\infty(\Omega)))\to
C_0^\infty(J,H^\infty(\Omega))\times C_0^\infty(J,H^\infty(\partial\Omega))
\end{equation}
by
\begin{equation}\label{F9ceq}
\Phi(u)=(\partial_t u -\tilde{\mathcal{F}}(t,u)+\tilde{\mathcal{F}}(t,0),
\tilde{\mathcal{B}}(t,u)-\tilde{\mathcal{B}}(t,0)).
\end{equation}

We note that $\Phi$ is well defined since
$\partial_t^j\{u_t-\tilde{\mathcal{F}}(t,u(t))+\tilde{\mathcal{F}}
(t,0)\}(0)=0$ and
$\partial_t^j \{\tilde{\mathcal{B}}(t,u(t))-\tilde{\mathcal{B}}
(t,0)\}(0)=0$ for all $j$ and every $u\in W$ in view of (\ref{F9eq}).
The map $\Phi$ is a $C^2$-map satisfying $\Phi(0,0)=0$
where the first derivative is
\begin{equation}\label{F9deq}
\Phi'(u)v=(\partial_tv -\tilde{\mathcal{F}}_u(t,u)v,
\tilde{\mathcal{B}}_u(t,u)v)
\end{equation}
For a fixed $T_1>0$ the first and third estimate
in (\ref{C13eq}) hold with uniform constants for $0<T\le T_1$
where the norms are defined by (\ref{C11eq}).
This follows from the proof of \cite{Po14}, 4.3
using (\ref{F1eq}), (\ref{F2eq}).
We consider the equation
\begin{equation}\label{F9eeq}
\Phi(u)=(\tilde{\mathcal{F}}(t,0),\tilde{h}(t)-\tilde{\mathcal{B}}(t,0)).
\end{equation}
The inverse function theorem \ref{C5} requires the smallness condition
\begin{equation}\label{F9feq}
\|\tilde{\mathcal{F}}(t,0)\|_s+\|\tilde{\mathcal{B}}(t,0)\|_s+
\|\tilde{h(t)}\|_s <\delta.
\end{equation}
By (\ref{F9eq}) condition (\ref{F9feq}) holds if $T>0$ is chosen
sufficiently small. We here shall have to observe that $s,\delta$
in Theorem \ref{C5} can be chosen uniformly for all $0<T\le T_1$.
We consider the nonlinear problem (\ref{F3eq}) for some given
initial value $\phi\in U$ and $h\in
C^\infty([0,T_1],H^\infty(\partial\Omega))$. We assume that the
compatibility conditions (\ref{F4eq}) hold.

\begin{theorem}\label{F3}
Let $T_1>0,\phi\in H^\infty(\Omega)$ and
$h\in C^\infty([0,T_1]),H^\infty(\partial\Omega))$
satisfy {\rm (\ref{F4eq})}.
Assume that there are $b\ge 0$ and $c_k>0$
and an open neighbourhood $U$ of $\phi$ in $H^\infty(\Omega)$
so that for any $0<T\le T_1$ and
$u\in $ $W= \{w\in C^\infty([0,T],H^\infty(\Omega)):w(t)\in U,t\in [0,T]\}$
the linear problem
\begin{equation}\label{F10eq}
\begin{gathered}
z_t(t)  =  \mathcal{F}_u(t,u(t))z(t) + f(t)
\quad \mbox{in }\Omega , \; t\in [0,T]  \\
\mathcal{B}_u(t, u(t))z(t)  =  g(t) \quad \mbox{on }\partial\Omega
, \; t\in [0,T]\\
z(0)  =  0
\end{gathered}
\end{equation}
admits for any $f\in C^\infty_0([0,T],H^\infty(\Omega))$ and
$g\in C_0^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$ a
unique solution $z\in C_0^\infty([0,T],H^\infty(\Omega))$
satisfying the estimates
\begin{equation}\label{F11eq}
\|z\|_k \le c_k[u;(f,g)]_{b,k},\quad k=0,1,2,\ldots
\end{equation}
Then {\rm (\ref{F3eq})} has a unique solution $u\in
C^\infty([0,T_0],H^\infty(\Omega))$ for some $T_0>0$.
\end{theorem}

\begin{proof}
We choose $\psi\in C^\infty([0,T_1],H^\infty(\Omega))$ satisfying
(\ref{F6eq}) such that $\psi(t)\in U$ for all $t\in[0,T_1]$. By
remark \ref{F1} (i) it is enough to solve problem (\ref{F7eq}) for
$v=u-\psi$. For that we define $\Phi$ by (\ref{F9eeq}) where
$\Phi(0,0)=0$ and $\tilde{\mathcal{F}},\tilde{\mathcal{B}}, \tilde{h}$
are defined as in Remark \ref{F1} satisfying (\ref{F9eq}). We have
to solve equation (\ref{F9eeq}). By our assumption on the linear
problem (\ref{F10eq}) the operator $\Phi'(u)$ is bijective for all
$u$ in some zero neighbourhood in
$C_0^\infty([0,T],H^\infty(\Omega)),0<T\le T_1$. The inequalities
(\ref{F11eq}) yield the second estimate in (\ref{C13eq}) while the
first and third estimate in (\ref{C13eq}) hold as observed above.
The assumptions of Theorem \ref{C5} on the spaces are satisfied by
Proposition \ref{C4}. Hence there exist numbers $s\ge 0$ and
$\delta>0$ as in Theorem \ref{C5} which can be chosen uniformly
for all $0<T\le T_1$. We can choose $T_0>0$ so small such that the
smallness condition (\ref{F9feq}) holds in $[0,T_0]$. Theorem
\ref{C5} gives a solution $v\in C_0^\infty([0,T_0],
H^\infty(\Omega))$ of problem (\ref{F7eq}) and thus a solution
$u=v+\psi$ of problem (\ref{F3eq}). The uniqueness can be shown
using Theorem \ref{C5} and a standard argument as in \cite{Po14},
Theorem 4.4.
This gives the result.
\end{proof}

Problem (\ref{F10eq}) can be reduced to a
problem with homogeneous boundary conditions provided that the
boundary conditions can be solved. Let $T_1>0$ and choose $U,W$
as in Theorem \ref{F3}. We assume there exist $b\ge 0$ and
$c_k>0$ such that for any $u\in W$ and $0<T\le T_1$
there exists a map
\begin{equation}
\label{F11aeq}
R_u:C_0^\infty([0,T],H^\infty(\partial\Omega)^{\frac{m}{2}})\to
C_0^\infty([0,T],H^\infty(\Omega))
, B_u(\cdot, u) R_u = {\rm Id}~
\end{equation}
which satisfies for all
$g\in C_0^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$ the estimates
\begin{equation}
\label{F11beq}
\|R_u g\|_k\le c_k [u;g]_{b,k}, \quad k=0,1,2,\ldots
\end{equation}
In section \ref{E} we show that such $R_u$ exist
for normal boundary conditions.

\begin{theorem}
\label{F4}
Let $T_1>0,\phi\in H^\infty(\Omega),h\in C^\infty([0,T_1]),
H^\infty(\partial\Omega))$ satisfy {\rm (\ref{F4eq})}.
Assume that there are $b\ge 0$ and $c_k>0$
and open sets $U, W$ as in Theorem {\rm \ref{F3}}
so that for any $0<T\le T_1$ and $u\in W$
there exist $R_u$ satisfying {\rm (\ref{F11aeq}), (\ref{F11beq})}
such that for any $f_1\in C_0^\infty([0,T],H^\infty(\Omega))$ the problem
\begin{equation}\label{F12eq}
\begin{gathered}
w_t(t)  =  \mathcal{F}_u(t,u(t))w(t) + f_1(t)
\quad \mbox{in }\Omega , \; t\in [0,T]  \\
\mathcal{B}_u(t, u(t))w(t)  =  0 \quad \mbox{on }\partial\Omega ,
\; t\in [0,T]\\
w(0)  =  0
\end{gathered}
\end{equation}
admits a unique solution $w\in C_0^\infty([0,T],H^\infty(\Omega))$
satisfying the estimates
\begin{equation}\label{F13eq}
\|w\|_k \le c_k[u;f_1]_{b,k},\quad k=0,1,2,\ldots
\end{equation}
Then {\rm (\ref{F3eq})} has a unique solution $u\in
C^\infty([0,T_0],H^\infty(\Omega))$ for some $T_0>0$.
\end{theorem}

\begin{proof}
Let $f\in C^\infty_0([0,T],H^\infty(\Omega))$ and $g\in
C_0^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$. We choose
$v=R_u g$ satisfying (\ref{F11beq}). We put $f_1(t)=f(t)
-v_t(t)+\mathcal{F}_u(t,u(t))v(t)$. Then $f_1\in
C_0^\infty([0,T],H^\infty(\Omega))$. By assumption we find a
solution $w\in C_0^\infty([0,T], H^\infty(\Omega))$ of
(\ref{F12eq}) satisfying (\ref{F13eq}). Then $z=v+w$ is a solution
of (\ref{F10eq}) satisfying (\ref{F11eq}). The solution is unique
by means of the unique solvability of (\ref{F12eq}). Hence Theorem
\ref{F3} gives the result.
\end{proof}

\section{Normal boundary conditions}
\label{E}

In this section we are concerned with normal boundary conditions.
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with $C^\infty$-boundary.
Let $\{B_j\}_{j=1}^p$ be a set of differential operators
$B_j=B_j(x,\partial)$ of order $m_j$ given by
\begin{equation}
\label{E1aeq}
B_j=B_j(x,\partial)=\sum\limits_{|\beta|\le m_j}\, b_{j,\beta}(x)
\partial^\beta,\quad\quad j=1,\ldots ,p
\end{equation}
with $b_{j,\beta}\in C^\infty(\partial\Omega)$. There is a linear
extension operator $S: C^\infty(\partial\Omega)\to
C^\infty(\overline{\Omega})$ satisfying $\|Sf\|_k\le c_k\|f\|_k$
for all $k,f$ with constants $c_k>0$; this follows from Seeley
\cite{Se} using a partition of unity (cf. \cite{Po9}). We hence
may assume that $b_{j,\beta}\in C^\infty(\overline{\Omega})$. The
set $\{B_j\}_{j=1}^p$ is called normal (cf. \cite{LioMa},
\cite{Sche3}, \cite{Wlo1}) if $m_j\not=m_i$ for $j\not= i$ and if
for any $x\in\partial\Omega$ we have
$B_j^P(x,\nu)\not=0,j=1,\ldots ,p$ where $\nu=\nu(x)$ denotes the
inward normal vector to $\partial\Omega$ at $x$ and $B_j^P$
denotes the principal part of $B_j$. A normal set
$\{B_j\}_{j=1}^p$ is called a Dirichlet system if
$m_j=j-1,j=1,\ldots ,p$. We can consider the Dirichlet boundary
conditions $u\mapsto\frac{\partial^{j-1}u}{\partial
\nu^{j-1}}_{|\partial\Omega}, j=1,\ldots ,p$, which give for any
$k\ge p$ a trace operator
\begin{equation}\label{E1ceq}
T_k^p:H^k(\Omega)\to \prod\limits_{i=1}^{p}H^{k-i+1/2}(\partial\Omega),
\quad T_k^pu=\Big\{\Big(\frac{\partial^{j-1}u}{\partial \nu^{j-1}}
\Big)_{|\partial\Omega}\Big\}_{j=1}^p.
\end{equation}
The trace operators $T_k^p$ are surjective admitting
a continuous linear right inverse $Z_k^p$ which depends on $k$
(cf. \cite{LioMa}, \cite{Wlo1}).
To construct a tame linear right inverse for the induced
trace operator $T^p:H^\infty(\Omega)\to H^\infty(\partial\Omega)^p$
we apply tame splitting theory in Fr\'{e}chet spaces
developed by Vogt (cf. \cite{T}).

Let $(F_k)_{k},(G_k)_k$ be families of Hilbert spaces with
injective linear continuous imbeddings $F_{k+1}\hookrightarrow
F_k,G_{k+1}\hookrightarrow G_k$ for all $k$. Let $T_k:F_k\to G_k$
be surjective continuous linear maps such that
$(T_k)_{|F_{k+1}}=T_{k+1}$ for all $k$. Let $E_k=N(T_k)\subset
F_k$ denote the kernel of $T_k$; we have $E_{k+1}\hookrightarrow
E_k$ and
\begin{equation}
\label{E1deq}
0\quad \longrightarrow \quad E_k\quad\hookrightarrow\quad F_k\quad
\stackrel{T_k}{\longrightarrow}\quad G_k\quad\longrightarrow \quad 0
\end{equation}
are exact sequences of Hilbert spaces.
We equip the Fr\'{e}chet spaces $E=\bigcap_{k}E_k, F=\bigcap_{k}F_k,
G=\bigcap_{k}G_k$ with the induced norms. We then have a mapping
$T:F\to G$ defined by $Tx=T_k x, x\in F$ where $N(T)=E$. The following
splitting theorem is a simplified version of \cite{PoV}, 6.1, 6.2.

\begin{lemma}
\label{E1} Let $E_k,F_k,G_k,T_k$ and $E,F,G,T$ be as above where
{\rm (\ref{E1deq})} is an exact sequence of Hilbert spaces for
every $k$. Assume that there are tame isomorphisms
$E\cong\Lambda_\infty^2(\alpha)$ and
$G\cong\Lambda_\infty^2(\beta)$ for some $\alpha, \beta$. Then
\begin{equation}
\label{E1eeq}
0\quad \longrightarrow \quad E\quad\hookrightarrow\quad F\quad
\stackrel{T}{\longrightarrow}\quad G\quad\longrightarrow \quad 0
\end{equation}
is an exact sequence of Fr\'{e}chet spaces which splits tamely,
i.e., there is a tame linear map $Z:G\to F$ such that $T\circ Z
={\rm Id}_G$. \end{lemma}

\begin{lemma}
\label{E2} Let $p\ge 1$. The trace operator
$T^p:H^\infty(\Omega)\to H^\infty(\partial\Omega)^p$ admits a tame
linear right inverse $Z^p:H^\infty(\partial\Omega)^p\to
H^\infty(\Omega)$, $T^p\circ Z^p={\rm Id}$. \end{lemma}

\begin{proof}
The trace operator $T_k^p$ induces for $k\ge p$
an exact sequences
\begin{equation}
\label{E1feq}
0~\longrightarrow ~ N(T_k^p)\quad\hookrightarrow ~
H^k(\Omega)\quad\stackrel{T_k^p}{\longrightarrow} ~
\prod\limits_{i=1}^{p}H^{k-i+1/2}(\partial\Omega)~\longrightarrow~ 0
\end{equation}
of Hilbert spaces. We show using Lemma \ref{E1} that the sequence
\begin{equation}
\label{E1geq}
0\quad\longrightarrow\quad N(T^p)\quad\hookrightarrow
\quad H^\infty(\Omega)\quad\stackrel{T^p}{\longrightarrow}
\quad H^\infty(\partial\Omega)^p\quad\longrightarrow\quad 0
\end{equation}
of Fr\'{e}chet spaces splits tamely. Note that there are tame
isomorphisms $H^\infty(\Omega)\cong \Lambda_\infty^2(\alpha)$ and
$H^\infty(\partial \Omega)^p\cong \Lambda_\infty^2(\beta)$ (cf.
the proof of Proposition \ref{C4}). Let $\Delta$ denote the
Laplacian. We consider $\Delta^p$ as an unbounded operator in
$L^2(\Omega)$ under null Dirichlet boundary conditions, the domain
given by $D_p=N(T_{2p}^p)=\{u\in H^{2p}(\Omega):T_{2p}^pu=0\}$. It
is well known that the spectrum of $\Delta^p$ is discrete (cf.
\cite{Brow}, Theorem 17, \cite{Agm1}, Theorem 2.1). We thus can
choose $\lambda$ such that $\Delta^p-\lambda $ is an isomorphism
$D_p\to L^2(\Omega)$. Therefore, $\Delta^p-\lambda : N(T^p)\to
H^\infty(\Omega)$ is an isomorphism (cf. \cite{Wlo1}) which is a
tame isomorphism by means of classical elliptic a priori estimates
(cf. \cite{AgDoNi}, Theorem 15.2). Hence $N(T^p)\cong
H^\infty(\Omega) \cong \Lambda_\infty^2(\alpha)$ tamely
isomorphic. By Lemma \ref{E1} the sequence (\ref{E1geq})
splits tamely. This gives the result.
\end{proof}

For a differential operator $P=\sum_{|\alpha|\le m}
a_\alpha(x)\partial^\alpha$ with $a_\alpha\in C^\infty$ we put
\begin{equation}
\label{E1heq}
{\textstyle
\|P\|_i=\sum_{|\alpha|\le m}\|a_\alpha\|_i^\infty,
\quad i=0,1,\ldots
}
\end{equation}
For $P$ as above and $Q=\sum_{|\beta|\le n}b_\beta(x)\partial^\beta$ we get
\begin{equation}
\label{E1eq}
\|PQ\|_i\le C_i\sum_{j=0}^i\|P\|_{i-j}\|Q\|_{m+j}
\end{equation}
with constants $C_i>0$. For smooth nonvanishing functions $f$ we
get with $C_i>0$ depending only on $i,m,n$ and on
$\|1/f\|_0^\infty$ the estimates
\begin{equation}
\label{E2eq}
\|1/f\|_i^\infty\le C_i[f]_i,\quad\|P/f\|_i\le C_i [f;P]_i
\end{equation}
for all $i$. Here the expressions $[f]_i$ and $[f;P]_i$ are
defined by the norms $\|f\|_j^\infty$ and (\ref{E1heq}). To prove
a generalization of Lemma \ref{E2} to normal boundary conditions
we first consider the case of a half space. We consider the cubes
\begin{gather}
\label{E2aeq}
\Sigma=\{x\in\mathbb{R}^n:|x_i|< 1\; (i=1,\ldots ,n), \; x_n> 0\}.\\
\label{E2aaeq}
\sigma=\{x\in\mathbb{R}^n:|x_i|< 1\; (i=1,\ldots ,n), \; x_n=0\}.
\end{gather}
The following lemma is well known and is due to \cite{ArMi} (see
\cite{LioMa,Sche1,Sche3,Tana1,Wlo1}); we prove additional estimates which are important
for our purposes. In the following lemma we consider smooth
function on $\overline{\Sigma}$.

\begin{lemma}
\label{E3} Let $\{B_j\}_{j=1}^p$ and $\{B_j'\}_{j=1}^p$ be two
Dirichlet systems on $\overline{\sigma}$. Then there exist smooth
differential operators $\Lambda_{kj},1\le j\le k\le p$, of order
$k-j$ containing only tangential derivatives $\partial_1,\ldots,
\partial_{n-1}$ such that
\begin{equation}
\label{E2beq}
B_k'=\sum\limits_{j=1}^k\Lambda_{kj}B_j,
\quad k=1,\ldots ,p, \quad \mbox{on} \quad \overline{\sigma}
\end{equation}
where $\Lambda_{kk}$ is a function which vanishes nowhere on
$\overline{\sigma}$. In addition, we have
\begin{equation}
\label{E3eq}
\|\Lambda_{kj}\|_i\le C [B_j,\ldots,B_k;B_k']_{i+k-j}, \quad
1\le j\le k, \; i=0,1,\ldots
\end{equation}
with some constant $C>0$ depending only on $i,n,k$ and on
$\|1/\sigma_k\|_0^\infty$ where $\sigma_k$ is the nonvanishing
coefficient of the term $\partial_n^{k-1}$ in $B_k$.
\end{lemma}

\begin{proof}
Let first $B_k'= \partial_n^{k-1}$. We assume that
$B_k=\sum_{j=1}^k\Gamma_{kj}\partial_n^{j-1}$ where $\Gamma_{kj}$
has order $k-j$ and $\Gamma_{kk}$ is a function not vanishing on
$\overline{\sigma}$. We have
\begin{equation}
\label{E3aeq}
\partial_n^{k-1} = \Gamma_{kk}^{-1}B_k - \Gamma_{kk}^{-1}
\sum_{j=1}^{k-1}\Gamma_{kj}\partial_n^{j-1}=\sum\limits_{j=1}^k
\Lambda_{kj}B_j
\end{equation}
where $\Lambda_{kk}=\Gamma_{kk}^{-1} $ and
$\Lambda_{kj}=-\Gamma_{kk}^{-1}\sum_{l=j}^{k-1}\Gamma_{kl}\Lambda_{lj},
j<k$. For $j<k$ we get
\begin{equation}
\label{E3ceq}
\|\Lambda_{kj}\|_i \le
C\sum_{l=j}^{k-1}\sum_{m=0}^i[B_k]_{i-m}\|\Gamma_{kl}\Lambda_{lj}\|_m
\le C \sum\limits_{l=j}^{k-1}[B_k;\Lambda_{lj}]_{i+k-l}
\end{equation}
from (\ref{E1eq}), (\ref{E2eq}) and $\|\Lambda_{kk}\|_i\le
C[B_k]_i$. By induction we see that
\begin{equation}
\label{E3deq}
\|\Lambda_{kj}\|_i\le C[B_j,\ldots ,B_k]_{i+k-j}
\end{equation}
for all $k$. In the general case we may write
\begin{equation}
B_l'=\sum_{k=1}^l\Psi_{lk}\partial_n^{k-1}=\sum_{k=1}^l\sum_{j=1}^k\Psi_{lk}
\Lambda_{kj}B_j=\sum_{j=1}^l\Phi_{lj}B_j, \quad l=1,\ldots, p
\end{equation}
where $\Psi_{lk}$ are tangential operators of order $l-k$ and
$\Psi_{ll}$ does not vanish on $\overline{\sigma}$; here
$\Lambda_{kj}$ as above and
$\Phi_{lj}=\sum_{k=j}^l\Psi_{lk}\Lambda_{kj}$. From (\ref{E1eq}),
(\ref{E3deq}) we get
\begin{equation}
\label{E3eeq}
\|\Phi_{lj}\|_i \le
C\sum_{k=j}^l\sum_{m=0}^i\|\Psi_{lk}\|_{i-m}\|\Lambda_{kj}\|_{l-k+m}
\le C[B_j,\ldots ,B_l;B_l']_{i+l-j}
\end{equation}
which proves the result.
\end{proof}

The assertion of Lemma \ref{E3} is invariant w.r.t.\
normal coordinate transformations (cf. \cite{Wlo1}).
In Theorem \ref{E4} we follow \cite{Wlo1}, Theorem 14.1.

\begin{theorem}
\label{E4} Let $\{B_j\}_{j=1}^p$ be a smooth normal system. Then
there exists a linear map $R:H^\infty(\partial\Omega)^p\to
H^\infty({\Omega})$ such that $B_jRg=g_j$ for $j=1,\ldots,p$ and
any $g=\{g_j\}_{j=1}^p\in H^\infty(\partial\Omega)^p$. There is
$b\ge 0$ such that
\begin{equation}\label{E4eq}
\|Rg\|_k\le C \sum_{j=1}^p [B_j,\ldots , B_p;g_j]_{b,k},
\quad k=0,1,2,\ldots
\end{equation}
for all $g\in H^\infty(\partial\Omega)^p$
where $C$ depends only on $k,p,n,\Omega$ and on
\begin{equation}
\label{E4aeq}
\sup\{|B_j^P(x,\nu(x))|+|B_j^P(x,\nu(x))|^{-1}:x\in\partial\Omega,
\;j=1,\ldots ,p\}.
\end{equation}
\end{theorem}

\begin{proof}
We may assume that $\{B_j\}_{j=1}^p$ is a Dirichlet system. We
choose for $x\in \partial\Omega$ an open neighbourhood $U_x$ in
$\mathbb{R}^n$ and a normal diffeomorphism $U_x\leftrightarrow \{x\in
\mathbb{R}^n:|x_i|<1,i=1,\ldots, n\}$ where $U_x\cap \Omega\leftrightarrow
\Sigma, \, U_x\cap \partial\Omega\leftrightarrow \sigma, \,
U_x\cap \overline{\Omega}^c\leftrightarrow -\Sigma$. We cover
$\partial\Omega$ by finitely many open sets $U_i=U_{x_i}$ and
choose a subordinate partition of unity $\alpha_i$. We write
$D_j=\partial^{j-1}/\partial \nu^{j-1}$. By Lemma \ref{E3} we get
on $\overline{U_i}\cap\partial\Omega$ for $j=1,\ldots,p$ a
representation
\begin{equation}
\label{E4ceq}
B_j=\sum_{l=1}^j \Lambda_{jl}^i D_l,\quad
D_j=\sum_{l=1}^j\Phi_{jl}^i B_l, \quad
\sum_{l=m}^{j}\Lambda_{jl}^{i}\Phi_{lm}^{i}=\delta_{jm}
\end{equation}
where $\Lambda_{jl}^i,\Phi_{jl}^i$ are tangential differential
operators of order $j-l$ and $\Lambda_{jj}^i, \Phi_{jj}^i$ do not
vanish on $\overline{U_i}\cap\partial\Omega$. Note that
(\ref{E3deq}) holds for $\Phi_{jl}$. We choose smooth functions
$\beta_i$ such that $\beta_i=1$ on an open neighbourhood $V_i$ of
${\rm supp}\, \alpha_i$ and ${\rm supp}\, \beta_i\subset U_i$. Let
$Z^p$ be the extension operator from Lemma \ref{E2}. We define
\begin{equation}
\label{E5eq}
Rg=\sum_i \beta_i Z^p
\{\sum_{l=1}^j\Phi_{jl}^i(\alpha_ig_l)\}_{j=1}^p.
\end{equation}
Let $v_i=\beta_i Z^p(w_j^i)$ and
$w_j^{i}=\sum\Phi_{jl}^{i}(\alpha_ig_l)$ as in (\ref{E5eq}). We
claim that $D_j v_i=w_j^{i}$ on
$\overline{U_i}\cap\partial\Omega$. This holds on $V_i\cap\partial
\Omega$ by Lemma \ref{E2}. On $(\overline{U_i}\setminus
V_i)\cap\partial\Omega$ all derivatives of order $\le p-1$ of
$Z^p(w_j^{i})$ vanish; the normal derivatives are $w_j^{i}=0$ and
the tangential derivatives vanish since $Z^p(w_j^{i})=0$ on
$(\overline{U_i}\setminus V_i)\cap \partial\Omega$. Thus $D_j
v_i=0=w_j^{i}$ on this set. We obtain
\[
B_jRg=\sum_i B_j(v_i)=
\sum_i \sum_{l=1}^j\Lambda_{jl}^{i}w_l^{i}
=\sum_i \sum_{m=1}^j\sum_{l=m}^j\Lambda_{jl}^{i}\Phi_{lm}^{i}(\alpha_ig_m)
=g_j.
\]
By Lemma \ref{E2} we have $\|Z^p\{g_j\}\|_k\le \sum\|g_j\|_{k+a}$
for some $a\ge 0$. We get
\begin{align*}
\|Rg\|_k & \le
C\sum_i\sum_{j=1}^p\sum_{l=1}^j\sum_{m=0}^{k+a}\|\Phi_{jl}^{i}\|_m
\|\alpha_ig_l\|_{k+j+a-l-m}\\
& \le C' \sum_i\sum_{j=1}^p\sum_{l=1}^j\sum_{m=0}^{k+a}[B_l,\ldots ,
B_j]_{m+j-l}\|g_l\|_{k+j+a-l-m}\\
& \le C''\sum_{l=1}^p[B_l,\ldots , B_p;g_l]_{p+a-1,k}
\end{align*}
which gives the result where $b=p+a-1$.
\end{proof}

Theorem \ref{F4} requires a parameter depending version of Theorem
\ref{E4}. Let $T_1>0,p\ge 1,m\ge 2$ and assume that
$B_j=B_j(t)=B_j(t,x,\partial),j=1,\ldots ,p$ have
$C^\infty$-coefficients $b_{j,\beta}\in C^\infty([0,T_1],
C^\infty(\overline{\Omega}))$. We equip the space
$C_0^\infty([0,T],E)$ for $E=H^\infty(\Omega)$ or
$E=H^\infty(\partial\Omega)^p$ with the norms given by
(\ref{C11eq}) (involving $m$). Since the proof of Theorem \ref{E4}
is constructive we obtain the following result.

\begin{theorem}
\label{E5} Let $T_1>0,m\ge 2$. Assume that $\{B_j(t)\}_{j=1}^p$ is
normal for each $t\in [0,T_1]$. Then there exists for any $0<T\le
T_1$ a linear map
\begin{equation}
\label{E5beq}
R:C_0^\infty([0,T],H^\infty(\partial\Omega)^p)\to
C_0^\infty([0,T],H^\infty({\Omega}))
\end{equation}
such that $B_j(t)Rg(t)=g_j(t)$ for any $t\in [0,T],g=(g_j)_j$.
There is $b\ge 0$ such that {\rm (\ref{E4eq})} holds for any $k$
and $0<T\le T_1$ where the norms in {\rm (\ref{E4eq})} are given
by {\rm (\ref{C11eq})} and where $C$ in {\rm (\ref{E4eq})} depends
only on $k,p,n,m,\Omega,T_1$ and on
\begin{equation}
\label{E5ceq}
\sup\{|B_j^P(t,x,\nu(x))|^{\pm 1}
:x\in\partial\Omega, 1\le j\le p,t\in [0,T_1]\}.
\end{equation}
\end{theorem}

\begin{proof}
On $C^\infty([0,T],C^\infty(\overline{\Omega}))$
we consider the norms defined by
\begin{equation}
\label{E5c1eq}
\|u\|_k^\infty=\sup\{\|u^{(i)}(t)\|_{k-mi}^\infty
:t\in[0,T],0\le i\le k/m \}.
\end{equation}
These norms satisfy $\|uv\|_k^\infty\le C_k\sum
\|u\|_{k-i}^\infty\|v\|_i^\infty $. The rules (\ref{E1eq}),
(\ref{E2eq}) can easily be established for the norms
(\ref{E5c1eq}) as well where the definition (\ref{E1heq}) uses the
norms (\ref{E5c1eq}) on the right hand side in (\ref{E1heq}).
Therefore, Lemma \ref{E3} holds with the same proof also for
$t$-depending differential operators where the estimate
(\ref{E3eq}) is formulated using the norms defined by
(\ref{E5c1eq}). The proof of Theorem \ref{E4} gives the result
since $R$ maps $C_0^\infty$ into $C_0^\infty$.
\end{proof}

In the situation of Theorem \ref{F4} we have the linearized boundary operators
\begin{equation}
\label{E5deq}
\mathcal{B}_u(t,u)=\{\sum_{|\alpha|\le m_j} (B_j)_{\partial^\alpha u}
(t,\cdot, \{\partial^\beta u(\cdot)\}_{|\beta|\le m_j})\partial^\alpha
\}_{j=1}^{m/2}.
\end{equation}
Let $\phi\in H^\infty(\Omega)$ be an initial value such that
$\mathcal{B}_u(0,\phi)$ is a normal system.
By continuity, we can choose $T_1>0$ and
an open neighbourhood $U$ of $\phi$ in $H^\infty(\Omega)$ such that
$\mathcal{B}_u(t,u)$ is normal for $t\in[0,T_1],u\in U$ such that
\begin{equation}
\label{E5eeq}
|\sum_{|\alpha|=m_j}(B_j)_{\partial^\alpha u}(t,x,\{\partial^\beta
u(x)\})\nu(x)^\alpha |\ge \mu >0
\end{equation}
uniformly for all $t\in [0,T_1],x\in \overline{\Omega},u\in
U,j=1,\ldots,m/2$. We get
\begin{equation}
\label{E5feq}
\|\mathcal{B}_u(t,u)\|_k^\infty\le C [u]_{p,k}^\infty,
\quad t\in [0,T_1],\;u\in U
\end{equation}
where $p=\max\{m_j\}$ and the norm on the left hand side in
(\ref{E5feq}) is defined by (\ref{E1heq}), (\ref{E5c1eq}). We
choose $a$ as in \ref{E4} and put $b=\max\{a,p\}+p+[n/2]+1$.
\begin{corollary}
\label{E6}
Let $\phi\in H^\infty(\Omega)$ and let $\mathcal{B}_u(0,\phi)$
be normal. Then there exist $T_1>0$ and a neighbourhood $W$
of $\phi$ in $C^\infty([0,T],H^\infty(\Omega))$ and constants $c_k>0$
and $b$ (as above) such that for any $0<T\le T_1$ and $u\in W$
there exists a mapping $R_u$ satisfying {\rm (\ref{F11aeq}), (\ref{F11beq})},
as required by {\rm Theorem \ref{F4}}.
\end{corollary}

The proof follows from Theorems \ref{E4} and \ref{E5}
using (\ref{E5feq}).\smallskip

Corollary \ref{E6} shows that the assumption of Theorem \ref{F4}
on the existence of right inverses $R_u$
is satisfied for normal boundary conditions.

We prove that the trace operators $T_k^p$ in (\ref{E1ceq}) admit
right inverses which are continuous simultaneously for different
values of $k$. Generalizing techniques of \cite{LioMa} we obtain
additional continuity estimates for lower order derivatives which
will be important in the sequel. Let $X$ and $Y$ be separable
Hilbert spaces such that there is a continuous injection
$X\hookrightarrow Y$ with a dense range. As in \cite{LioMa}, we
shall make use of a representation
\begin{equation}
\label{E6aeq}
\|v\|_{[X,Y]_\theta}^2 = \int_{\lambda_0}^\infty \lambda^{2(1-\theta)}
\|v(\lambda)\|^2_{h(\lambda)} \,d\lambda, \quad 0\le \theta\le 1
\end{equation}
based on a spectral decomposition of $X\hookrightarrow Y$ where
$(h(\lambda), \|\quad \|_{h(\lambda)})$ is a scale of Hilbert
spaces and $\lambda_0>0$ (cf. \cite{LioMa}, Ch. 1, 2.3). The
interpolation space $[X,Y]_\theta=h(1-\theta)$ coincides with
$X,Y$ if $\theta=0,1$. We shall identify elements of
$[X,Y]_\theta$ with functions $v=v(\lambda)$ satisfying
(\ref{E6aeq}). For $s>0$ put
\begin{equation}
\label{E6beq}
W(\mathbb{R},s,X,Y)=\{u:\mathbb{R}\to X| \;\hat{u}\in L^2(\mathbb{R},X),\tau^s \hat{u}\in
L^2(\mathbb{R},Y)\}
\end{equation}
where $\hat{u}=\mathcal{F}_{t\to\tau}(u)$ denotes the Fourier
transform of $u$ (cf. \cite{LioMa}, Ch.1, 4.1).
The space $W(\mathbb{R},s,X,Y)$ is a Hilbert space equipped with the norm
\begin{equation}
\label{E6ceq}
\|u\|_{W(\mathbb{R},s,X,Y)}^2=
\|\hat{u}\|_{L^2(\mathbb{R},X)}^2+\|\tau^s\hat{u}\|_{L^2(\mathbb{R},Y)}^2.
\end{equation}
The following Lemma improves \cite{LioMa}, Ch. 1, Theorem 4.2.

\begin{lemma}
\label{E7} Let $q$ be an integer and $0\le q<s-\frac{1}{2}$. Then
the map
\begin{equation}
\label{E6deq}
W(\mathbb{R},s,X,Y)\to \prod_{j=0}^q [X,Y]_{(j+1/2)/s}, \quad
u\mapsto \{u^{(j)}(0)\}_{j=0}^{q}
\end{equation}
is continuous, linear, surjective and admits a continuous linear
right inverse $R$ where $R:\prod_j [X,Y]_{(j+1/2)/s}\to
W(\mathbb{R},s,X,Y)$. There is $C>0$ only depending on $s$ such that
\begin{equation}
\label{E7aeq}
\|Rg\|_{W(\mathbb{R}, s(1-\mu), [X,Y]_\mu, Y)}
\le C\sum_{j=0}^q \|g_j\|_{[X,Y]_{\mu+(j+1/2)/s}}
\end{equation}
for any $0\le \mu < 1-(q+1/2)/s $ and $g=(g_j)\in\prod_j
[X,Y]_{(j+1/2)/s}$. \end{lemma}

\begin{proof}
The case $\mu=0$ is proved in \cite{LioMa}, Ch.\ 1, Theorem 4.2.
It remains to show (\ref{E7aeq}) for $0\le \mu < 1-(q+1/2)/s$ and
$R$ constructed in \cite{LioMa}. As in \cite{LioMa} (cf. Ch.\ 1,
Theorem 3.2) it is enough to consider the map $u\mapsto
u^{(j)}(0)$ for a fixed $j$. We fix $j$ and write $g=g_j,
g=g(\lambda)$ as above. We choose $\phi\in C^\infty(\mathbb{R} )$ with
compact support such that $\phi^{(j)}(0)=1$. Then the function
\begin{equation}
\label{E7beq}
w(\lambda,t)=\lambda^{-j/s}g(\lambda)\phi(\lambda^{1/s}t)
,\quad \hat{w}(\lambda,\tau)=\lambda^{-(j+1)/s}g(\lambda)\hat{\phi}
(\lambda^{-1/s}\tau)
\end{equation}
satisfies $w\in W(\mathbb{R},s,x,y)$ and $w^{(j)}(0)= g$. We get the
estimates (cf. \cite{LioMa})
\begin{align*} \label{E7ceq}
& \|\hat{w}\|^2_{L^2(\mathbb{R},[X,Y]_\mu)}+ \|\tau^{s(1-\mu)}
\hat{w}\|^2_{L^2(\mathbb{R},Y)} \\
& \le \int_{-\infty}^\infty\int_{\lambda_0}^\infty
|\lambda|^{2(1-\mu-(j+\frac{1}{2})/s)}
|g(\lambda)|^2|\hat{\phi}(\tau)|^2 (1+|\tau|^{2s(1-\mu)})d\lambda d\tau\\
& \le C\|g\|_{[X,Y]_{\mu+(j+1/2)/s}}.
\end{align*}
This proves the assertion.
\end{proof}

The following Lemma follows from the proof of \cite{LioMa}, Ch.\ 1,
Theorems 8.3, 9.4 using Lemma \ref{E7} instead of \cite{LioMa},
Ch.\ 1, Theorem 4.2.

\begin{lemma}
\label{E8} Let $k\ge p\ge 1$. The trace operator $T_k^p$ in {\rm
(\ref{E1ceq})} admits a continuous linear right inverse which is
simultaneously continuous
\begin{equation}
\label{E8aeq}
Z_k^p:\prod_{j=0}^{p-1} H^{l-j-1/2}(\partial \Omega)\to H^l(\Omega), \quad
p-\frac{1}{2}< l\le k.
\end{equation}
\end{lemma}

\begin{proof}
By usual methods (cf. \cite{LioMa}, Ch.\ 1, Thms.\ 8.3, 9.4) we are
brought back to a half space $\Omega=\{x\in\mathbb{R}^n:x_n>0\},
\partial\Omega=\mathbb{R}^{n-1}_{x'},x=(x',x_n)$. We put $X_l=H^l(\mathbb{R}^{n-1}_{x'}), Y=H^0(\mathbb{R}^{n-1}_{x'})$. For an integer $l\ge 0$ we get
\[
H^l(\Omega)=W(\mathbb{R}_+,l,X_l,Y)=
\{u\in L^2(\mathbb{R}_+,X_l): \frac{\partial^l u}{\partial x_n^l}\in
L^2(\mathbb{R}_+,Y)\}
\]
(cf. \cite{LioMa}, Ch.\ 1, Theorem 7.4). For noninteger values of $l$
we consider analogously restrictions of functions belonging to the space
defined in (\ref{E6beq}) (cf.\ \cite{LioMa}, Ch.\ 1, Theorem 9.4).
By Lemma \ref{E7} the map
\begin{equation}
\label{E8ceq}
W(\mathbb{R},s,X_k,Y)\to \prod_{j=0}^{p-1} [X_k,Y]_{(j+1/2)/k}, \quad
u\mapsto \{u^{(j)}(0)\}_{j=0}^{p-1}
\end{equation}
admits a right inverse $R$ which is simultaneously continuous as a map
\begin{equation}
\label{E9ceq}
R:\prod_{j=0}^{p-1}[X_k,Y]_{\mu+(j+1/2)/k}\to W(\mathbb{R},(1-\mu)k,[X_k,Y]_\mu, Y)
\end{equation}
for $0\le \mu< 1-(p-1/2)/k$. Since
$[H^l(\mathbb{R}^{i}),H^0(\mathbb{R}^{i})]_\theta =H^{(1-\theta)l}(\mathbb{R}^{i})$ we get
\[
R:\prod_{j=0}^{p-1}H^{(1-\mu)k-j-\frac{1}{2}}(\mathbb{R}^{n-1}_{x'})\to W(\mathbb{R},(1-\mu)k,
H^{(1-\mu)k}(\mathbb{R}^{n-1}_{x'}),H^0(\mathbb{R}^{n-1}_{x'}))
\]
By restriction $\mathbb{R}\to \mathbb{R}_+$ and putting $l=(1-\mu)k$
we get a right inverse $Z_k^p$ for $T_k^p$
satisfying (\ref{E8aeq}) for all
$l$ with $p-\frac{1}{2}<l\le k$. The lemma is proved.
\end{proof}

The following lemma follows from the proof of \cite{Wlo1}, Lemma 3.2.

\begin{lemma}
\label{E10}
For any real number $l\ge 0$
there is a constant $C_l>0$ such that
for any $\phi\in C^{[l]+1}(\overline{\Omega})$ and $u\in H^l(\Omega)$
we have $\phi u\in H^l(\Omega)$ and
\[
\|\phi u\|_l\le C_l
\sum_{i=0}^{[l]}\Big(\|\phi\|^\infty_{i}\|u\|_{l-i}
+\|\phi\|^\infty_{i+1}\|u\|_{[l]-i}\Big)\le 2C_l
\sum_{i=0}^{[l]}\|\phi\|_{i+1}^\infty\|u\|_{l-i}.
\]
\end{lemma}

The assertion of Lemma \ref{E10} holds analogously for $u\in
H^l(\partial\Omega)$. In the next section we shall apply results
on maximal regularity for parabolic problems. These results
require H\"older estimates in the time variable. Let $X$ be a
Banach space, let $0\le \delta <1$ and $T>0$, let $i\ge 0$ be an
integer. By $C^{i+\delta}([0,T],X)$ we denote the set of functions
$u$ in $C^{i}([0,T],X)$ having a H\"older continuous derivative
$u^{(i)}$ with exponent $\delta$, equipped with the norm
\begin{equation}
\label{E10beq}
|u|_{i+\delta,X} = \sum_{j=0}^{i}\|u^{(j)}\|_{C([0,T],X)}
+\sum_{j=0}^{i}\sup\limits_{s,t\in [0,T],s\not= t}
\frac{\|u^{(j)}(t)-u^{(j)}(s)\|_X}{|t-s|^\delta}.
\end{equation}
Writing $H^r=H^r(\Omega)$ or $H^r=H^r(\partial\Omega)$,
respectively, we put in particular
\begin{equation}
|u|_{i+\delta ,r}=|u|_{i+\delta, H^r}, \quad
|u|_{i+\delta ,k}^\infty=|u|_{i+\delta, C^k(\overline{\Omega})}.
\end{equation}

\begin{lemma}
\label{E11}
For $f\in C^\infty([0,T],C^\infty(\overline{\Omega})),
g\in C^\infty([0,T], H^\infty)$ we have
\begin{equation}
\label{E11aeq}
|fg|_{i+\delta ,r} \le C
\sum_{j=0}^{i}\sum_{l=0}^{j}\sum_{q=0}^{[r]}
|f|_{l+\delta, q+\epsilon}^\infty|g|_{j-l+\delta,r-q}
\end{equation}
with some constant $C>0$ where $\epsilon =0$ if $r$ is an integer
and $\epsilon =1$ otherwise.
\end{lemma}

\begin{proof}
The case $\delta =0$ follows from Lemma \ref{E10}.
For $0<\delta <1$ we write
\[
\label{E11ceq}
\frac{f^{l}(t)g^{j-l}(t)-f^{l}(s)g^{j-l}(s)}{|t-s|^\delta}
= \frac{f^{l}(t)-f^{l}(s)}{|t-s|^\delta}g^{j-l}(t)+
f^{l}(s)\frac{g^{j-l}(t)-g^{j-l}(s)}{|t-s|^\delta}
\]
An application of Lemma \ref{E10} gives the result.
\end{proof}

For a differential operator $P=\sum_{|\alpha|\le m}
a_\alpha(t,x)\partial_x^\alpha$ we put
\begin{equation}
\label{E10deq}
|P|_{i+\delta ,r}=\sum_{|\alpha|\le m} |a_\alpha|_{i+\delta ,r}^\infty.
\end{equation}
We note that (\ref{E1eq}) holds for the norms $|\quad|_{\delta ,i}$
(replacing $\|\quad\|_i$ in (\ref{E1eq})) as well.
From Lemma \ref{E11} we get for $u\in C^\infty([0,T],H^\infty)$
with $\epsilon$ as in \ref{E11} that
\begin{equation}
\label{E10eeq}
|Pu|_{i+\delta ,r}\le C\sum_{j=0}^{i}\sum_{l=0}^{j}\sum_{q=0}^{[r]}
|P|_{l+\delta, q+\epsilon}|u|_{j-l+\delta,r-q+m}.
\end{equation}
Let $\{B_j\}_{j=1}^p$ be a normal system with smooth
$(t,x)$-dependent coefficients as in Theorem \ref{E5} where
$B_j=B_j(t,x,\partial)$ has order $m_j$.

\begin{lemma}
\label{E12} Let $1\le j\le p,r> m_j+\frac{1}{2}$, let $s\ge 0$ be
an integer. Then
\begin{equation}
\label{E12aeq}
|(\partial_t^s B_j)u|_{\delta,r-m_j-1/2}\le C
\sum_{q=0}^{[r-m_j-1/2]} |B_j|_{s+\delta ,q+1}|u|_{\delta , r-q}
\end{equation}
for $u\in C^\infty([0,T],H^r(\Omega))$ with a constant $C>0$ only
depending on $p,s,r$. \end{lemma}

\begin{proof}
We may assume that $m_j=j-1$. As in the proof of Theorem \ref{E4}
(cf. formula (\ref{E4ceq})) we can choose local representations
$B_j=\sum_{l=1}^j\Lambda_{jl}D_l$ where
$D_l=\partial^{l-1}/\partial\nu^{l-1}$. Then, locally,
$\partial_t^sB_j=\sum_{l=1}^j(\partial_t^s\Lambda_{jl})D_l$ and
$|\Lambda_{jl}|_{s+\delta,q}\le C|B_j|_{s+\delta ,q}$ for any $q$.
Applying Lemma \ref{E11} and observing that $\Lambda_{jl}$ has
order $j-l$ we get locally
\begin{equation}
\label{E12beq}
|(\partial_t^sB_j)u|_{\delta,r-j+1/2}
\le C \sum_{l=1}^j\sum_{q=0}^{[r-j+1/2]}|\Lambda_{jl}|_{s+\delta,q+1}
|D_lu|_{\delta ,r-l+1/2-q}.
\end{equation}
This gives the result since $D_l:H^k(\Omega)\to H^{k-l+1/2}
(\partial \Omega)$ for $k> l-1/2$.
\end{proof}

Let $\{B_j\}_{j=1}^p$ be as above and $M=\max\{m_j:j=1,\ldots,
p\}$. Let $m\ge M+1$. The proofs of Theorem \ref{E4} and Lemma
\ref{E8} give a linear right inverse $R$ for $\{B_j\}$ satisfying
$B_j(t)Rg(t)=g_j(t)$ for every $t,j$ and all $g=\{g_j\}$ such that
$R$ is simultaneously for $M+\frac{1}{2}< k\le m$ defined as a map
\begin{equation}
\label{E12ceq}
R:C_0^\infty([0,T],\prod_{j=1}^{p}H^{k-m_j-\frac{1}{2}}(\partial\Omega))\to
C_0^\infty([0,T],H^k(\Omega)), B_jRg=g_j~
\end{equation}
For Dirichlet systems $\{B_j\}_{j=1}^p$ the map $R$ is locally
given by (\ref{E5eq}) using $Z_m^{p}$ from (\ref{E8aeq}) instead
of $Z^p$. This gives $R$ for normal systems $\{B_j\}_{j=1}^p$ as
well.

We define the expressions $[\quad]_{k;\delta}$
by the norms $|\quad|_{\delta ,i}$
(which are for differential operators given by (\ref{E10deq})).
Analogously to (\ref{E2eq}) we then have
\begin{equation}
\label{E12deq}
|1/f|_{\delta ,i}^\infty\le C[f]_{i;\delta} , \quad
|P/f|_{\delta ,i}\le C[f;P]_{i;\delta}
\end{equation}
where $C$ depends on $i,m,n$ and on a bound for
$|1/f|_{0,0}^\infty+|f|_{\delta ,0}^\infty$; we here have observed
that $|1/f|_{\delta ,0}^\infty\le (|1/f|_{0,0}^\infty)^2
|f|_{\delta ,0}^\infty$.

\begin{lemma}
\label{E13}
For $M+\frac{1}{2}<k\le m$ the map $R$ in {\rm (\ref{E12ceq})} satisfies
\begin{equation}
\label{E13aeq}
|Rg|_{\delta ,k} \le C_1 \sum_{j=1}^p\sum_{q=0}^{[k-m_j-1/2]}
[B_1,\ldots , B_p]_{M+1+q;\delta}|g_j|_{\delta ,k-m_j-q-1/2}
\end{equation}
where $C_1>0$ depends on $k,m, \sum|B_j|_{\delta ,0}$
and on the constant in {\rm (\ref{E4aeq})} and
\begin{equation}
\label{E13beq}
|Rg|_{1+\delta ,k} \le C_2 \sum_{j=1}^p|g_j|_{1+\delta ,k-m_j-1/2}
\end{equation}
where $C_2>0$ depends on the same data as $C_1$ and on
$\sum|B_j|_{1+\delta, [k+1/2]}$. \end{lemma}

\begin{proof}
We may assume that $m_j=j-1,M=p-1$. Using (\ref{E5eq}) with
$Z_m^p$ in place of $Z^p$ and omitting the index $i$ we get from
\ref{E8} and (\ref{E10eeq}) that
\begin{equation}
\label{E13ceq}
|Rg|_{\delta ,k}
\le C \sum_{1\le l\le j\le p}\sum_{q=0}^{[k-j+\frac{1}{2}]}
|\Phi_{jl}|_{\delta ,q+1}|g_l|_{\delta,k-l-q+\frac{1}{2}}.
\end{equation}
Using (\ref{E12deq}) instead
of (\ref{E2eq}) in the proof of \ref{E3} (cf. (\ref{E3deq})) we get
\begin{equation}
\label{E13deq}
|\Phi_{jl}|_{\delta ,i}\le C[B_l,\ldots ,B_j]_{i+j-l;\delta}.
\end{equation}
The inequalities (\ref{E13ceq}), (\ref{E13deq})
yield (\ref{E13aeq}). Analogously, we get the estimate
\begin{equation}
\label{E13eeq}
|\partial_t \Phi_{jl}|_{\delta ,i}\le C\sum_{s=l}^j
[B_l,\ldots ,B_j; \partial_t B_s]_{i+j-l;\delta}.
\end{equation}
Together with (\ref{E13ceq}) this proves the assertion.
\end{proof}

\section{The linear parabolic problem}
\label{K}

Let $T_1>0$. We consider for $0<T\le T_1$ the linear evolution
equation
\begin{equation}
\label{K1eq}
\begin{gathered}
\partial_tz(t)  =  A(t)z(t)+f(t) , \quad t\in [0,T]  \\
z(0)  =  0\,. \end{gathered}
\end{equation}
We assume that $A(t),t\in [0,T_1]$, is a closed linear operator in
a Banach space $X$ with a (not necessarily dense) domain $\mathcal{D}
(A(t))$ (which may depend on $t$). We assume that there is a
Banach space $Z\hookrightarrow X$ continuously imbedded into $X$
such that $\mathcal{D} (A(t))\subset Z$ for all $t$. In applications we put
$X=L^2(\Omega),Z=H^m(\Omega)$ where $\mathcal{D} (A(t))$ is given by
boundary conditions. We shall suppose the following conditions
(${\rm P}_0$), $\ldots$, (${\rm P_3}$) (cf.\ \cite{Tana1},
\cite{Ama1}, \cite{Tana}).
\begin{itemize}
\item[$({\rm P}_0)$]  There is a constant $M_0>0$ such that
\begin{equation}
\label{K2eq}
\|z\|_Z\le M_0 (\|A(t)z\|_X+\|z\|_X), \quad z\in \mathcal{D} (A(t)),
\quad t\in [0,T_1].
\end{equation}
\item[(${\rm P}_1$)] There is $\theta_0\in (\pi/2,\pi)$ so that
$\rho(A(t))\supset\Sigma:=\{\lambda : |{\rm arg}\, \lambda | <\theta_0\}
\cup\{0\}$ and there is $M_1>0$ such that
$R(\lambda , A(t))=(\lambda I-A(t))^{-1}$ satisfies
\begin{equation}
\label{K3eq}
\|R(\lambda, A(t))\|_{L(X)}  \le  M_1/|\lambda|
,\quad   \lambda\in\Sigma\backslash \{0\}, \quad t\in [0,T_1] .
\end{equation}
\item[(${\rm P}_2$)]  For each $\lambda \in \Sigma$ the operator
valued function $t\mapsto R(\lambda, A(t))$ belongs to the space
$C^1([0,T_1],L(X))$. There is a constant $M_2>0$ such that
\begin{equation}
\label{K4eq}
\|(d/dt)R(\lambda,A(t))\|_{L(X)}  \le  M_2/|\lambda|
, \quad \lambda\in\Sigma\backslash \{0\}  ,\quad t\in [0,T_1] .
\end{equation}
\item[(${\rm P}_3$)] There is a constant $M_3>0$ such that
\begin{equation}
\label{K5eq}
\|(d/dt)A(t)^{-1}-(d/dt)A(\tau)^{-1}\|_{L(X)}
\le M_3|t-\tau|,\quad t,\tau \in [0,T_1].
\end{equation}
\end{itemize}
We take advantage of the following result on maximal regularity
from \cite{Tana1}.

\begin{theorem}
\label{K1} Assume {\rm (${\rm P}_1$), (${\rm P}_2$), (${\rm
P_3}$).} Let $0<\delta <1$. Then there is $C_0>0$ depending only
on $M_1,M_2,M_3,\theta_0,T_1$ such that for any $f\in
C^\delta([0,T],X)$ with $f(0)=0$ and $0<T\le T_1$ any solution
$z\in C^1([0,T],X)$ of {\rm (\ref{K1eq})} with $z(t)\in \mathcal{D} (A(t))$
for all $t$ satisfies $z\in C^{1+\delta}([0,T],X)$ and
\begin{equation}
\label{K7eq}
|z|_{1+\delta,X}^{[0,T]}\le C_0 |f|_{\delta ,X}^{[0,T]}.
\end{equation}
Assuming also {\rm (${\rm P}_0$)} there is $C_1>0$
depending only on $C_0,M_0$ such that
\begin{equation}
\label{K8eq}
|z|_{0,Z}^{[0,T]}\le C_1 |f|_{\delta ,X}^{[0,T]}.
\end{equation}
\end{theorem}

\begin{proof}
We have $z\in C^{1+\delta}([0,T],X)$ by \cite{Tana1}, Theorem 6.4
since $z(0)=f(0)=0$ and since $z$ is a strict solution in the sense
of \cite{Tana1}. The estimate (\ref{K7eq}) follows from
the proof of \cite{Tana1}, Theorem 6.4.
We apply (${\rm P}_0$) and (\ref{K7eq}) and use equation (\ref{K1eq}) to get
\begin{equation}
\label{K9eq}
\|z(t)\|_Z \le M_0(\|A(t)z(t)\|_X+\|z(t)\|_X)
\le M_0(1+2C_0)|f|_{\delta,X}.
\end{equation}
This gives the result.
\end{proof}

Let $\Omega\subset \mathbb{R}^n$ be bounded with $C^\infty$-boundary,
let $m\ge 2$ be even and let
\begin{equation}
\label{K11eq}
A=A(t)= A(t,x,\partial_x)=\sum_{|\alpha|\le m} a_\alpha(t,x)\partial_x^\alpha
\end{equation}
be a differential operator with
coefficients $a_\alpha\in C^\infty([0,T_1],C^\infty(\overline{\Omega}))$
and let
\begin{equation}
\label{K12eq}
B_j=B_j(t)=B_j(t,x,\partial_x)=\sum_{|\beta|\le m_j} b_{j,\beta}(t,x)
\partial_x^\beta, \quad j=1,\ldots,m/2
\end{equation}
be boundary operators with coefficients $b_{j,\beta}\in
C^\infty([0,T_1],C^\infty(\overline{\Omega}))$ where
\begin{equation}
\label{K12aeq}
0\le m_1 <\ldots < m_{m/2}<m.
\end{equation}
We suppose that $\{B_j(t)\}_{j=1}^{m/2}$ is normal for $t\in
[0,T_1]$. Then the constant in (\ref{E5ceq}) is bounded. We put
$X=L^2(\Omega),Z=H^m(\Omega),Z_j=H^{m-m_j-1/2}(\partial\Omega)$.
Then $A(t):Z\to X$ and $B_j(t):Z\to Z_j$ are continuous. We put
\begin{equation}
\label{K13aeq}
\mathcal{D} (A(t))=\{
z\in H^m(\Omega): B_j(t)z(t)=0
\mbox{\rm ~on}~\partial\Omega, \quad j=1,\ldots ,m/2\}.
\end{equation}
We consider for $0<T\le T_1$ the boundary-value problem
\begin{equation}
\label{K13eq}
\begin{gathered}
\partial_t z(t)  =  A(t)z(t)+f(t) \quad \mbox{in } \Omega, \; t\in [0,T]\\
B_j(t)z(t)  =  0 \quad \mbox{on } \partial\Omega,\; t\in [0,T],
1\le j\le \frac{m}{2}\\
z(0)=  0\,.
\end{gathered}
\end{equation}
Solutions of (\ref{K13eq}) correspond to solutions of (\ref{K1eq})
where $z(t)\in\mathcal{D} (A(t)),t\in [0,T]$. In addition, we assume the
following conditions.
\begin{itemize}
\item[(${\rm P}_4$)]
There is $M_4>0$ such that for all $z\in Z$ and $t\in [0,T_1]$ we have
\begin{equation}
\label{K13beq}
\|z\|_Z\le M_4(\|A(t)z\|_X+\|z\|_X+\sum_{j=1}^{m/2}\|B_j(t)z\|_{Z_j}).
\end{equation}
\item[(${\rm P}_5$)]
There exist $0<\delta \le 1,M_5>0$ such that $A \in C^\delta
([0,T_1], L(Z,X))$ and $B_j\in C^\delta ([0,T_1], L(Z,Z_j))$ satisfy
\begin{equation}
\label{K13ceq}
|A|_{\delta ,L(Z,X)} +|B_j|_{\delta , L(Z,Z_j)}\le M_5, \quad j=1,\ldots ,m/2.
\end{equation}
\end{itemize}
We note that $A$ and $B_j$ as above enjoy this condition ${\rm
(P}_5{\rm )}$. In the following condition we use the notation
$[A,B]_{m,k}=[A,B_1,\ldots , B_{m/2}]_{m,k}$. 

\noindent ${\rm
(P}_6{\rm )}$ For $k\ge m$ there is $M_k$ so that for all $z\in
H^k(\Omega),t\in [0,T_1]$ we have
\begin{align*}
\|z\|_{k}\le& M_k \Big\{\sum_{i=m}^k[A,B]_{m,k-i}\big(\|A(t)z\|_{i-m}
+\sum_{j=1}^{m/2} \|B_j(t)z\|_{i-m_j-\frac{1}{2}}\big)\\
&+[A,B]_{m,k-m}\|z\|_{0}\Big\}.
\end{align*}

\begin{lemma}
\label{K2} Assume {\rm (${\rm P}_0$), $\ldots $, (${\rm P_5}$)}
Let $f,z,w_0\in C_0^\infty([0,T],H^\infty(\Omega))$ where $0<T\le
T_1$. Assume that $z$ is a solution of problem {\rm (\ref{K1eq})}
and that
\begin{equation}
\label{K14eq}
B_j(t)w_0(t)=B_j(t)z(t)\quad\mbox{\rm on}\quad \partial\Omega, \quad
j=1,\ldots ,m/2, \quad t\in [0,T].
\end{equation}
There is a constant $C_2>0$ depending only
on $M_0,\ldots,M_5,\theta_0,T_1$ such that
\begin{equation}
\label{K15eq}
|z|_{1+\delta ,0}+|z|_{\delta ,m}\le C_2(|f|_{\delta ,0} +
|w_0|_{1+ \delta ,0}+|w_0|_{\delta ,m}).
\end{equation}
\end{lemma}

\begin{proof}
We note that $v(t)=z(t)-w_0(t)\in \mathcal{D} (A(t))$ for all $t$ and
\begin{equation}
\label{K16eq}
v_t(t)=A(t)v(t)+f(t)+A(t)w_0(t)-(w_0)_t(t).
\end{equation}
 
 From Lemma \ref{K1} we obtain the estimate
\begin{equation}
\label{K17eq}
|v|_{1+\delta ,0}+|v|_{0 ,m}\le C
(|f|_{\delta ,0}+|A(t)w_0(t)|_{\delta ,0}+|w_0|_{1+\delta ,0}).
\end{equation}
Using (${\rm P_5}$) we get $|A(t)w_0(t)|_{\delta ,0}\le CM_5|w_0|_{\delta ,m}$
and thus
\begin{equation}
\label{K17aeq}
|z|_{1+\delta ,0}+|z|_{0 ,m}\le C(|f|_{\delta ,0} +
|w_0|_{1+ \delta ,0}+|w_0|_{\delta ,m}).
\end{equation}
To estimate $|z|_{\delta ,m}$ we apply (${\rm P_4}$) and obtain
\[
\frac{\|z(t)-z(s)\|_Z}{|t-s|^\delta} \le
M_4(\frac{\|A(t)(z(t)-z(s))\|_X}{|t-s|^\delta}+|z|_{\delta,0}
+\sum_{j=1}^{m/2}\frac{\|B_j(t)(z(t)-z(s))\|_{Z_j}}{|t-s|^\delta}).
\]
We further get
\begin{equation}
\label{K17ceq}
\frac{\|A(t)(z(t)-z(s))\|_X}{|t-s|^\delta}\le
|z|_{1+\delta ,0}+|f|_{\delta ,0} +M_5|z|_{0,m}
\end{equation}
which gives the desired estimate by means of (\ref{K17aeq}).
Finally we obtain
\begin{equation}
\label{K17eeq}
\frac{\|B_j(t)(z(t)-z(s))\|_{Z_j}}{|t-s|^\delta}\le M_5
(|w_0|_{\delta ,m}+|w_0|_{0,m}+|z|_{0,m}).
\end{equation}
We proved estimate (\ref{K15eq}) and thus the result.
\end{proof}

In the sequel, the symbol $H^\infty$ is used simultaneously to
denote $H^\infty(\Omega)$ or $H^\infty(\partial\Omega)$. For $f\in
C^\infty([0,T],H^\infty)$ and an integer $k$ we put 
\begin{equation}
\label{K18eq}
\|f\|_{k;\delta}=\sup\{|f|_{i+\delta,k-mi}:
i=0,1,\ldots, [k/m]\} .
\end{equation}
For $f\in C^\infty([0,T],C^\infty(\overline{\Omega}))$ we define
$\|f\|_{k;\delta}^\infty$ analogously using
$|f|_{i+\delta,k-mi}^\infty$ in (\ref{K18eq}). For a differential
operator $P$ then $\|P\|_{k;\delta}$ is given by (\ref{K18eq})
replacing $f$ by $P$ while $[P]_{k;\delta}$ and $[P]_{m,k;\delta}$
are defined using the norms $\|P\|_{i;\delta}$. We write
\[
[B]_{m,k;\delta}=[B_1,\ldots ,B_{m/2}]_{m,k;\delta},
[A,B]_{m,k;\delta}=[A,B_1,\ldots ,B_{m/2}]_{m,k;\delta}.
\]

\begin{lemma}
\label{K3} Assume conditions {\rm (${\rm P}_0$), $\ldots$, (${\rm
P_6}$).} Let $k\ge 1,0<T\le T_1$ and $f,z$, $w_{i}\in
C_0^\infty([0,T],H^\infty(\Omega)),0\le i\le k-1. $ Let $z$ be a
solution of {\rm (\ref{K13eq})} and
\begin{equation}
\label{K19eq}
B_r(t)w_{i}(t)=B_r(t)\partial^{i}_tz(t)\quad\mbox{\rm on}\quad \partial\Omega
\end{equation}
for $r=1,\ldots ,m/2, i=0,\ldots ,k-1,t\in [0,T]$. Then there is
$C>0$ depending only on $k,M_0,\ldots
,M_5,M_{(k+1)m},\theta_0,T_1,
\|A\|_{2m;\delta},\sum_r\|B_r\|_{2m;\delta}$ such that
\begin{eqnarray}
%\split
\label{K20eq}
\sum_{i=0}^k |z|_{i+\delta ,(k-i)m}\le C\sum_{i=0}^{k-1}
[A,B]_{2m,m(k-1-i);\delta}\Big\{\|f\|_{mi;\delta}\\
+|w_{i}|_{1+\delta ,0}+\sum_{j=0}^{i}|w_{j}|_{\delta ,m(i+1-j)}\Big\}.
\end{eqnarray}
\end{lemma}

\begin{proof}
The case $k=1$ follows from Lemma \ref{K2}. We fix $k\ge 1$. We
assume that (\ref{K20eq}) is proved for $k$ and that (\ref{K19eq})
holds for $i=0,\ldots ,k$. We have to show (\ref{K20eq}) for
$k+1$. Differentiating (\ref{K1eq}) we obtain
\begin{equation}
\label{K21eq}
(\partial_t^kz)_t(t)=A(t)\partial_t^kz(t)+\sum_{i=0}^{k-1}
\binom{k}{i}(\partial_t^{k-i}A(t))\partial_t^{i}z(t)+\partial_t^kf(t).
\end{equation}
Applying Lemma \ref{K2} to $\partial_t^kz$ we get
\[  %\label{K22eq}
|\partial_t^kz|_{1+\delta ,0}+|\partial_t^kz|_{\delta ,m}
\le C \Big\{\sum_{i=0}^{k-1}|A|_{k-i+\delta ,0}|z|_{i+\delta ,m}
+\|f\|_{mk;\delta}
+|w_{k}|_{1+\delta ,0}+|w_{k}|_{\delta ,m}\Big\}.
\]
The hypothesis of induction gives for $0\le i\le k-1$ the estimates
\[  %\label{K23eq}
|A|_{k-i+\delta ,0}|z|_{i+\delta ,m}
\le C\sum_{l=0}^{i}[A,B]_{2m,m(k-l);\delta}\Big\{\|f\|_{m;l}
|w_{l}|_{1+\delta ,0}
+\sum_{j=0}^{l}|w_{j}|_{\delta ,m(l+1-j)}\Big\}
\]
and thus the desired estimate for
$|z|_{k+1+\delta,0}+|z|_{k+\delta, m}$. Next we fix $0\le i\le
k-1$ and assume that the estimate in (\ref{K20eq}) is proved in
the case $k+1$ for all terms $|z|_{l+\delta,(k+1-l)m}$ with
$i+1\le l\le k+1$; we have to show the estimate in (\ref{K20eq})
in the case $k+1$ for the term $|z|_{i+\delta, (k+1-i)m}$. We fix
$0\le j\le i$. Using (${\rm P}_6$) we first get
  %\label{K25eq}
\begin{align}
&\|\partial^j_tz(t)\|_{(k+1-i)m} \nonumber\\
&\le M\{\sum_{l=m}^{(k+1-i)m}[A,B]_{m,(k+1-i)m-l}(\|A(t)
\partial_t^jz(t)\|_{l-m}  \label{K25eq} \\
&\quad+\sum_{r=1}^{m/2}\|B_r(t)w_j(t)\|_{l-m_r-\frac{1}{2}})
+[A,B]_{m,(k-i)m}\|\partial^j_tz(t)\|_0\} \label{K25aeq}
\end{align}
The last term
enjoys the desired estimate by induction. Using (\ref{K21eq}) we get
\[ %\label{K26eq}
\|A(t)\partial_t^jz(t)\|_{l-m}\le |z|_{i+1,l-m}+|f|_{i,l-m}+
C\sum_{q=0}^{j-1} \|(\partial_t^{j-q}A(t))\partial_t^qz(t)\|_{l-m}.
\]
To estimate the term $[A,B]_{m,(k+1-i)m-l} |f|_{i,l-m}$ we
consider two cases. If $l\ge (k-i)m$ then this term is $\le
C\|f\|_{mk;\delta}$ since $[A,B]_{m,m}\le C$; in the case $l\le
(k-i)m$ we can use the estimate $[A,B]_{m,(k+1-i)m-l}\le
C[A,B]_{2m, (k-i)m-l}$ to estimate this term appropriately.
Analogously, we can apply in the case $l\ge (k-i)m$ the hypothesis
of the induction (case $k+1,i+1$) to the term
$|z|_{i+1,l-m}\le|z|_{i+1,(k-i)m}$. In the case $l\le (k-i)m$ and
$am< l\le (a+1)m$, thus $i+a+1\le k+1$, we obtain
\[
[A,B]_{2m, (k-i)m-l} |z|_{i+1,l-m}\le
[A,B]_{2m, (k-i-a)m} |z|_{i+1,(i+1+a-i-1)m}
\]
and induction gives the desired estimate. We further get
\begin{equation}
\label{K28eq}
\|(\partial_t^{j-q}A(t))\partial_t^qz(t)\|_{l-m}\le C
\sum_{r=0}^{l-m}|A|_{j-q,l-m-r}|z|_{q,r+m}.
\end{equation}
For $am< r\le (a+1)m$ we have $|z|_{q,r+m}\le |z|_{q,(a+2)m}$
where $q+a+2\le k$. If $(k+1-i)m-l\le m$ and $(i-q-a-1)m+l\le m$
then $l=(k-i)m,k=q+a+2$ and $|z|_{q,(a+2)m}=|z|_{q,(k-q)m}$ can be
estimated by induction where
$[A,B]_{m,(k+1-i)m-l}[A,B]_{(i-q-a-1)m+l}\le C$; otherwise we
observe
\begin{equation}
\label{K28aeq}
[A,B]_{m,(k+1-i)m-l}[A,B]_{(i-q-a-1)m+l}\le C[A,B]_{2m,(k-q-a-1)m}
\end{equation}
and can apply induction to $|z|_{q,(a+2)m}$. This yields the
necessary estimate for the term on the right hand side in
(\ref{K25eq}). By Lemma \ref{E12} we have
\begin{equation}
\label{K29eq}
\|B_r(t)w_j(t)\|_{l-m_r-\frac{1}{2}}\le C \sum_{q=0}^{l-m_r-1}
|B_r|_{0, q+1}|w_j|_{0,l-q}
\end{equation}
which gives the desired estimate for the first term in (\ref{K25aeq})
since for $am < l-q\le (a+1)m$ we have $i+a \le k$ and
$|B_r|_{0,q+1}\le [B_r]_{m,q}$ and thus
\[
[A,B]_{m,(k+1-i)m-l}|B_r|_{0,q+1}|w_j|_{0,l-q}\le
C[A,B]_{2m,(k-j-a)m}|w_j|_{0,(a+1)m}.
\]
It remains to prove H\"older estimates for $\delta>0$ for the term
$|z|_{i+\delta,(k+1-i)m}$. For that we replace in (\ref{K25eq}),
(\ref{K25aeq}) the term $\partial_t^jz(t)$ by the term
$(\partial_t^jz(t)-\partial_t^jz(s))/|t-s|^\delta$. The last term
$[A,B]_{m,(k-i)m}|z|_{j+\delta ,0}$ in this inequality satisfied
the desired estimate by induction. For the first term we write
\begin{equation}
\label{K31eq}
A(t)\frac{\partial_t^jz(t)-\partial_t^jz(s)}{|t-s|^\delta}
=\frac{A(t)\partial_t^jz(t)-A(s)\partial_t^jz(s)}{|t-s|^\delta}
+\frac{(A(s)-A(t))}{|t-s|^\delta}\partial_t^jz(s).
\end{equation}
The first term $|A(\cdot)\partial_t^jz(\cdot)|_{\delta , l-m}$
resulting from (\ref{K31eq}) is estimated like \\
$\|A(t)\partial_t^jz(t)\|_{l-m}$ using H\"older norms in the above
estimates and observing the estimates $|A|_{j-q+\delta,l-m-r}\le
[A]_{m,m(j-q)+l-m-r}$. For the other term we have
\begin{equation}
\label{K32eq}
\big\|\frac{(A(s)-A(t))}{|t-s|^\delta}\partial_t^jz(s)\big\|_{l-m}
\le C\sum_{r=0}^{l-m}[A]_{m,l-m-r}|z|_{i,r+m}
\end{equation}
since $j\le i, |A|_{\delta , l-m-r}\le [A]_{m,l-m-r}$. The proved
case (for $|z|_{i,(k+1-i)m}$) gives the required estimate for the
terms appearing on the right hand side in (\ref{K32eq}). Finally
we use for the term
$B_r(t)(\partial_t^jz(t)-\partial_t^jz(s))/|t-s|^\delta$ a
decomposition as in (\ref{K31eq}). Lemma \ref{E12} gives for
$|B_r(\cdot)\partial_t^jz(\cdot)|_{\delta, l-m_r-\frac{1}{2}}=
|B_r(\cdot) w_j(\cdot)|_{\delta, l-m_r-\frac{1}{2}}$ an estimate
as in (\ref{K29eq}) involving $\delta$ on both sides; using
$|B|_{\delta ,q+1}\le [B]_{m,q+1}$ we get the necessary estimate
for this term as above. On the other hand, we have
\begin{equation}
\label{K32aeq}
\Big\|\frac{B_r(s)-B_r(t)}{|t-s|^\delta}\partial_t^jz(t)
\Big\|_{l-m_r-\frac{1}{2}}\le C\sum_{q=0}^{l-m_r-1}
|B_r|_{\delta,q+1}|z|_{i,l-q}.
\end{equation}
Since $|B_r|_{\delta , q+1}\le [B]_{m,q+1}$ we must estimate
$[A,B]_{m,(k+1-i)m-l+q+1}|z|_{i,l-q}$ which is $\le
[A,B]_{m,(k+1-i-a)m}|z|_{i,(a+1)m}$ if $am < l-q \le (a+1)m$.
Since $i+a+1\le k+1$ we can apply the above proved estimate for
$|z|_{i,(a+1)m}$. This gives the result.
\end{proof}

It remains to choose and estimate the terms $w_i$ in
(\ref{K19eq}). We put $w_0=0$. For $i\ge 1$ we use the linear
right inverse $R$ for $\{B_r\}_{r=1}^{m/2}$ from Lemma \ref{E13}
and (\ref{E12ceq}). Since $\partial_t^{i}(B_j(t)z(t))=0$ on
$\partial\Omega$ we may define
\begin{equation}
\label{K33eq}
w_i=R\Big\{
\Big(-\sum_{r=0}^{i-1} \binom{i}{r}(\partial_t^{i-r}B_j)\partial_t^r z
\Big)_{j=1}^{m/2}\Big\}, \quad i\ge 1.
\end{equation}

\begin{theorem}\label{K4}
In the situation of {\rm Lemma \ref{K3}} there is $C>0$
depending on the same data as $C$ in {\rm Lemma \ref{K3}}
and on the constant in {\rm (\ref{E4aeq})} such that
\begin{equation}
\label{K34eq}
|w_i|_{1+\delta,0}+\sum_{j=0}^{i}|w_j|_{\delta ,m(i+1-j)} \le
C \sum_{l=0}^i [A,B]_{2m,(i-l)m;\delta}\|f\|_{ml;\delta}
\end{equation}
for $i=0,\ldots ,k-1$. In addition, we have the inequality
\begin{equation} \label{K35eq}
\sum_{i=0}^k|z|_{i+\delta ,(k-i)m}\le
C \sum_{i=0}^{k-1} [A,B]_{2m,(k-1-i)m;\delta}\|f\|_{mi;\delta}.
\end{equation}
\end{theorem}

\begin{proof}
The case $k=1$ follows from Lemma \ref{K2}.
If (\ref{K34eq}) is already proved for $i=0,\ldots ,k-1$
then (\ref{K35eq}) follows from Lemma \ref{K3} since
\begin{equation}
\label{K36eq}
[A,B]_{2m,(k-1-i)m;\delta} [A,B]_{2m, (i-l)m;\delta}\le
[A,B]_{2m,(k-1-l)m;\delta}.
\end{equation}
Let $k\ge 1$ and assume that (\ref{K35eq}) is proved for $k$; we
show that this implies (\ref{K34eq}) for $i=k$. We choose $R$ in
(\ref{E12ceq}) depending on $k$ so that (\ref{E13aeq}),
(\ref{E13beq}) hold for $M+\frac{1}{2}<K\le km$ (replacing $k$ by
$K$ in (\ref{E13aeq}), (\ref{E13beq})) where $M=\max\{m_j\}$; note
that $R$ depends on $m$ in Lemma \ref{E13}.
 From (\ref{E13aeq}) we get
\begin{align*}
\sum_{i=1}^k|w_i|_{\delta ,m(k+1-i)}
&\le C \sum_{i=1}^k
\sum_{r=0}^{i-1}\sum_{j=1}^{m/2}\sum_{q=0}^{(k+1-i)m-m_j-1}
\Big\{[B]_{m,q;\delta} \\
&\quad \times|(\partial_t^{i-r}B_j)\partial_t^rz|_{\delta,(k+1-i)m-m_j-q
-\frac{1}{2}}\Big\}.
\end{align*}
Using Lemma \ref{E12} we obtain
\begin{align*}
&|\partial_t^{i-r}B_j\partial_t^rz|_{\delta,(k+1-i)m-m_j-q-\frac{1}{2}}\\
&\le C
\sum_{l=0}^{(k+1-i)m-m_j-1} |B_j|_{i-r+\delta , l+1}
|z|_{r+\delta ,(k+1-i)m-l-q}
\end{align*}
For $am\le l+q<(a+1)m$ we have $k+r+1-i-a\le k$ and (\ref{K35eq}) shows
\[
|z|_{r+\delta,(k+1-i-a)m}\le C\sum_{s=0}^{(k+r-i-a)m}
[A,B]_{2m, (k+r-i-a-s)m;\delta} \|f\|_{ms;\delta}.
\]
We have $|B_j|_{i-r+\delta, l+1}\le C[B]_{2m,m(i-1-r)+l;\delta}$ and thus
\[
[B]_{m,q;\delta}|B_j|_{i-r+\delta,l+1}[A,B]_{2m,(k+r-i-a-s)m}\le
C[A,B]_{2m,(k-s)m;\delta}.
\]
We proved (\ref{K34eq}) for the second term in (\ref{K34eq}) for
$i=k$. For the other term we fix a real number $k_0$ with
$M+\frac{1}{2}<k_0< m$. Applying (\ref{E13beq}) to $k_0$ we get
\[
\label{K40eq}
|w_k|_{1+\delta ,0} \le C \sum_{j=1}^{m/2}\sum_{r=0}^{k-1}
|(\partial_t^{k-r}B_j)\partial_t^rz|_{1+\delta ,k_0-m_j-1/2}
\]
where $|B_j|_{1+\delta ,[k-\frac{1}{2}]}\le
|B_j|_{1+\delta ,m}\le \|B_j\|_{2m;\delta}\le C$. We have to
estimate
\begin{equation}
\label{K41eq}
|(\partial_t^{k+1-r}B_j)\partial_t^rz|_{\delta ,k_0-m_j-1/2}+
|(\partial_t^{k-r}B_j)\partial_t^{r+1}z|_{\delta ,k_0-m_j-1/2}.
\end{equation}
The above yields the required estimate for the first term
in (\ref{K41eq}) since
\begin{equation}\label{K42eq}
|B_j|_{k+1-r+\delta , m}\le
[B_j]_{2m,(k-r-1)m+q+1;\delta}\le C[B_j]_{2m,(k-r)m;\delta}
\end{equation}
with $C$ depending on $\|B_j\|_{2m;\delta}$. For the second term
in (\ref{K41eq}) we get
\begin{equation}
|(\partial_t^{k-r}B_j)\partial_t^{r+1}z|_{\delta ,k_0-m_j-1/2}
\le C |B_{j}|_{k-r+\delta,m}|z|_{r+1+\delta, k_0}.
\end{equation}
Induction does not apply. Since $k_0<m$ we get for $\epsilon >0$
by interpolation
\begin{equation}
\label{K43eq}
|z|_{r+1+\delta, k_0}\le \epsilon |z|_{r+1+\delta,m}+ C(\epsilon)
|z|_{r+1+\delta ,0}
\end{equation}
where the constant $C(\epsilon)$ depends on $\epsilon$. Since
$r\le k-1$ we get from (\ref{K35eq})
\begin{equation}
\label{K44eq}
|B_j|_{k-r+\delta ,m}|z|_{r+1+\delta, 0}
\le C\sum_{i=0}^{k-1} [A,B]_{2m, (k-i)m;\delta}\|f\|_{mi;\delta}
\end{equation}
observing $|B_j|_{k-r+\delta ,m}\le \|B_j\|_{(k+1-r)m;\delta}\le
[B]_{m,(k-r)m;\delta}$. Note that (\ref{K35eq}) does not apply to
the other term.
%The other term in (\ref{K43eq}) cannot be estimated by means of
%(\ref{K35eq}).
We thus apply Lemma \ref{K3} and get
\begin{align*} %\label{K45eq}
&\epsilon|B_j|_{k-r+\delta ,m} |z|_{r+1+\delta ,m}\\
&\le \epsilon C \sum_{i=0}^{k}
[A]_{m,(k-i)m;\delta}\big\{\|f\|_{mi;\delta}
+|w_i|_{1+\delta,0}+\sum_{j=0}^{i}|w_j|_{\delta,(i+1-j)m}\big\}
\end{align*}
since $|B_j|_{k-r+\delta ,m}\le [B]_{2m,(k-r-1;\delta}$. We here
can estimate all terms appropriately except $|w_k|_{1+\delta ,0}$.
However, the proved cases give
\begin{equation}
\label{K46eq}
|w_k|_{1+\delta ,0}\le
C\sum_{i=0}^k [A,B]_{2m,(k-i)m;\delta}\|f\|_{mi;\delta}
+\epsilon C |w_k|_{1+\delta ,0}.
\end{equation}
Choosing $\epsilon >0$ small enough we get (\ref{K34eq})
for $i=k$ and thus the result.
\end{proof}

\section{Elliptic a priori estimates}
\label{D}

We formulate sufficient conditions of elliptic type for (${\rm
P}_0$), $\ldots$, (${\rm P}_6$). The classical elliptic a priori
estimates due to Agmon, Douglis, Nirenberg \cite{AgDoNi} are well
known. We accomplish these estimates including the dependence of
the constants from the coefficients, as required by the
Nash-Moser technique. Uniform dependence as stated in
\cite{AgDoNi}, Theorem 15.2 is not sufficient for (${\rm P_6}$).

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with
$C^\infty$-boundary and $n\ge 2$. Let
\begin{equation} \label{D1eq}
L=L(x,\partial)=\sum_{|\alpha|\le m}\,a_\alpha(x) \partial^\alpha
\end{equation}
and let $B_j=B_j(x,\partial),j=1,\ldots ,m/2$ be given by (\ref{E1aeq})
where $a_\alpha, b_{j,\beta}\in C^\infty(\overline{\Omega})$ are
$\mathbb{C}$-valued. Let $m\ge 2$ be and assume (\ref{K12aeq}).
Write $L=L^P+L^R, B_j=B_j^R+B_j^{R}$ where
$L^P, B_j^P$ denote the principal parts.

\begin{definitiion} \label{D0} \rm
The pair $(L,B_j)$ is called {\em elliptic} if the
following holds:
\begin{itemize}
\item[(i)] {\em Ellipticity:} $L$ is uniformly elliptic
on $\overline{\Omega}$, i.e., there is $\mu>0$ so that
\begin{equation}
|L^P(x,\xi)|\ge \mu|\xi|^{m},\quad x\in\overline{\Omega},\xi\in\mathbb{R}^n.
\end{equation}
\item[(ii)] {\em Root Condition:}
For every $x\in\partial\Omega$ and $\xi\not=0$
tangential to $\partial\Omega$ at $x$
the polynomial $\tau\mapsto L^P(x,\xi+\tau\nu)$ has exactly $m/2$ roots
with positive imaginary part denoted by
$\tau_1^+(x,\xi),\ldots , \tau_{m/2}^+(x,\xi)$
($\nu=\nu(x)=$ inner normal vector).

\item[(iii)] {\em Complementing Condition:} For every
$x\in\partial\Omega$ and $\xi\not=0$ as in (ii) the polynomials
$\{B_j^P(x,\xi+\tau\nu)\}_{j=1}^{m/2}$ in $\tau$ are linearly
independent modulo $\prod_{j=1}^{m/2}
(\tau-\tau_j^+(x,\xi))$.
\end{itemize}
\end{definitiion}


For $n\ge 3$ all elliptic operators satisfy the root condition.
We consider in the half space $H_+=
\{x=(x_1,\ldots ,x_n)\in\mathbb{R}^n:x_n> 0\}$ the problem
\begin{equation}
\label{D3eq}
\begin{gathered}
Lu  =   F  , \quad  x_n>0\\
B_ju  =  \Phi_j  , \quad  x_n=0,\; j=1,\ldots m/2\,.
\end{gathered}
\end{equation}

We first assume that the elliptic pair $(L,B_j)$ has constant
coefficients. As in \cite{AgDoNi}, (1.9) we define a determinant
constant $\Delta =\min\{|{\rm det}(b_{jk}(\xi))|:|\xi|=1\}>0$.
Here $\sum_{k=1}^{m/2} b_{jk}(\xi)\tau^{k-1}= B_j^P(\xi,\tau)
\mbox{\rm~ mod~} \prod_{j=1}^{m/2} (\tau-\tau_j^+(\xi))$ and thus
$\Delta >0$ by means of the complementing condition. If $L,B_j$
have variable coefficients then $\Delta$ means a lower bound for
the determinant constants of the frozen operators. If $L,B_j$
depend continuously on additional parameters then $\Delta$ depends
continuously on these parameters as well. As in \cite{AgDoNi}
(2.12) we introduce the characteristic constant
\begin{equation}
\label{D4eq}
E=\mu^{-1}+\Delta^{-1}+\|A\|_m+\sum_j\|B_j\|_m+n+m+\sum_j m_j.
\end{equation}

\begin{lemma}[cf. \cite{AgDoNi}, Thm. 14.1] \label{D1}
 Let the elliptic pair
$(L,B_j)=(L^P,B_j^P)$ have constant coefficients. Let $u\in
H^k(H_+),k\ge m$, satisfy $u(x)=0$ for $|x|\ge 1$. Then
\begin{equation}
\label{D5eq}
\|u\|_k\le C\Big(\|Lu\|_{k-m}+\sum_{j=1}^{m/2}\|B_j u\|_{k-m_j-1/2}\Big)
\end{equation}
where $C$ depends only on $k$ and on the characteristic constant
$E$.
\end{lemma}

\begin{lemma}[cf. \cite{AgDoNi}, Theorem 15.1] \label{D2}
Let $(L,B_j)$
be an elliptic pair. Let $k\ge m$. Then there exist $C>0,r>0$
depending only on $k,E$ such that
\begin{equation} \label{D6eq}
\begin{aligned}
\|u\|_k & \le C\Big\{\sum_{i=m}^k[L,B]_{m,k-i}
\big(\|Lu\|_{i-m}+\sum_{j=1}^{m/2}\|B_j u\|_{i-m_j-1/2}\big)\\
&\quad +[L,B]_{m,k-m}\|u\|_0\Big\}.
\end{aligned}
\end{equation}
for all $u\in H^{k}(H^+)$ satisfying $u(x)=0$ for $|x|\ge r$.
\end{lemma}

\begin{proof}
We put $x=(x',x_n)\in\mathbb{R}^n,x'\in\mathbb{R}^{n-1}$, and write (\ref{D3eq})
as
\begin{gather*}
L^P(0,\partial)u(x) = F(x)+(L^P(0,\partial)-L^P(x,\partial))u(x)-
L^R(x,\partial)u(x) \\
B_j^P(0,\partial)u(x',0)
= \Phi_j(x')+(B_j^P(0,\partial)-B_j^P(x',\partial))u(x',0)
 -B_j^R(x',\partial)u(x',0).
\end{gather*}
For $L$ defined by (\ref{D1eq})
we obtain the estimates
\begin{align*}
&\|(L^P(0,\partial)-L^P(x,\partial))u\|_{k-m}+\|L^R(x,\partial)u\|_{k-m}\\
&\quad +C\Big\{r\|u\|_k+
\sum_{|\alpha|=m}\sum_{i=m}^{k-1}\|a_\alpha\|_{k-i}^\infty\|u\|_i
+\sum_{|\alpha|<m}\sum_{i=m-1}^{k-1}\|a_\alpha\|_{k-1-i}^\infty\|u\|_i\Big\}
\end{align*}
and for $B_j=B_j(x',\partial)$ defined by (\ref{E1aeq})
the definition of the norms imply
\begin{align*}
&\|(B_j^P (0,\partial) -B_j^P(x',\partial)-
B_j^R(x',\partial))u(x',0)\|_{k-m_j-1/2}\\
&\le C\Big\{r\|u\|_k+
\sum_{|\beta|=m_j}\sum_{i=m_j}^{k-1}\|b_{j,\beta}\|_{k-i}^\infty\|u\|_i+
\sum_{|\beta|<m_j}\sum_{i=m_j-1}^{k-1}\|b_{j,\beta}\|_{k-1-i}^\infty\|u\|_i
\Big\}.
\end{align*}
Applying Lemma \ref{D1} we get the estimates
\[
\|u\|_k\le C\Big\{r \|u\|_k+\|F\|_{k-m}+\sum_{j=1}^{m/2}
\|\Phi_j\|_{k-m_j-1/2}+\sum_{i=0}^{k-1}[L,B]_{k-i}\|u\|_i\Big\}
\]
and hence, choosing $r$ sufficiently small, the inequalitity
\begin{equation}
\label{D6ceq}
\|u\|_k\le C\Big\{\|F\|_{k-m}+\sum_{j=1}^{m/2}\|\Phi_j\|_{k-m_j-1/2}
+\sum_{i=0}^{k-1}[L,B]_{k-i}\|u\|_i\Big\}.
\end{equation}
This gives the case $k=m$ by interpolation. If the assertion is
proved for $k\ge m$ then we can apply (\ref{D6ceq}) with $k+1$ in
place of $k$. For $m\le i\le k$ the terms $[L,B]_{k+1-i}\|u\|_i$
satisfy the desired estimate by induction observing that
$[L,B]_{k+1-i}[L,B]_{m,i-l}\le [L,B]_{m,k+1-l}$. For $0\le i<m$ we
get
\begin{equation} \label{D6deq}
[L,B]_{k+1-i}\|u\|_i\le C[L,B]_{m,k+1-m}\|u\|_m
\end{equation}
and the proved case $k=m$ gives (\ref{D6eq}) for $k+1$
and thus the result.
\end{proof}

Now let $\Omega\subset\mathbb{R}^n$ be a bounded open set
with $C^\infty$-boundary
(we note that the following Theorem \ref{D3} holds for uniformly regular
sets of class $C^k$ as well, cf.\ \cite{Tana1}, Theorem 4.10).
We consider the boundary value problem
\begin{equation} \label{D7eq}
\begin{gathered}
Lu  =   F  \quad  \mbox{in } \Omega \\
B_ju  =  \Phi_j  \quad  \mbox{on } \partial\Omega,\; j=1, \ldots m/2\,.
\end{gathered}
\end{equation}

\begin{theorem}[cf. \cite{AgDoNi}, Theorem 15.2]\label{D3}
Let $(L,B_j)$ be an elliptic pair. Let $k\ge m,u\in H^k(\Omega)$.
Then {\rm (\ref{D6eq})} holds with $C>0$ depending on $\Omega,k,E$.
\end{theorem}

\begin{proof}
We use the notation of \cite{AgDoNi}, Theorem 15.2. Let $(U_i)$ be
a finite open covering of $\partial\Omega$ and let
$T_i:\overline{U_i}\cap\overline{\Omega}\to\Sigma_{R_i}$ be
bijective $C^\infty$-maps onto the hemisphere
$\Sigma_{R_i}=\{x\in\mathbb{R}^n:x_n\ge 0,|x|\le R_i\}$ such that
$S_i=T_i^{-1}$ is $C^\infty$ and
$\overline{U_i}\cap\partial\Omega$ is mapped onto the part $x_n=0$
of $\Sigma_{R_i}$. We may assume that $R_i <r$ where $r$ is the
constant from Lemma \ref{D2}. We choose a finite
$C^\infty$-partition of unity $(\omega_\sigma)$ in
$\overline{\Omega}$ such that the support of each $\omega_\sigma$
is either contained in $\Omega$ or in one of the sets $U_i$
denoted by $U_{i(\sigma)}$. We want to estimate $u=\sum
\omega_\sigma u$. As in \cite{AgDoNi} we consider here the case
that the support of $\omega_\sigma$ is not contained in $\Omega$;
the other case follows analogously using \cite{AgDoNi}, Theorem
14.1' instead of Lemma \ref{D1}. Let $\omega_\sigma$ be such an
element and $T=T_{i(\sigma)},S=T^{-1}$. We put $v=u\circ S$ and
$\omega=\omega_\sigma\circ S$ where the support of $\omega$ is
contained in $\Sigma_R$ with $R<r$. The transformed operators
$L,B_j$ are denoted by $\tilde{L},\tilde{B_j}$. For
$Lu=L(x,\partial_x)u$ given by (\ref{D1eq}) we obtain from the
chain rule
\begin{equation}
\label{D6feq}
\tilde{L}(y,\partial_y)v =\sum_{1\le |\beta|\le |\alpha|\le m}
a_\alpha( S(y))A_{\beta,\alpha}(S(y))\partial_y^\beta v(y)+a_0( S(y))v(y)
\end{equation}
with smooth $A_{\beta,\alpha}$ depending on $T$. This gives for
$i\ge m$ the estimates
\begin{equation} \label{D6geq}
\|\tilde{L}(\omega v)\|_{i-m}\le C\Big\{\|Lu\|_{i-m}+\sum_{\alpha}
\sum_{l=m-1}^{i-1}\|a_\alpha\|^\infty_{i-l-1}\|u\|_l\Big\}.
\end{equation}
Analogously we obtain the estimates
(cf.\ the proof of \cite{AgDoNi}, Theorem 15.2)
\[
\|\tilde{B_j}(\omega v)\|_{i-m_j-\frac{1}{2}}\le
C\Big\{\|B_ju\|_{i-m_j-\frac{1}{2}}+\sum_{\beta}
\sum_{l={m_j-1}}^{i-1}\|b_{j,\beta}\|^\infty_{i-1-l}\|u\|_l\Big\}.
\]
Since $\|u\|_k\le \sum\|\omega_\sigma u\|_k\le
C\sum \|\omega v\|_k$ we get from Lemma \ref{D2} that
\begin{equation}\label{D6ieq}
\begin{aligned}
\|u\|_k&\le C\Big\{\sum_{i=m}^k [L,B]_{m,k-i}\Big(\|Lu\|_{i-m}+
\sum_{j=1}^{m/2}\|B_j\|_{i-m_j-\frac{1}{2}}\\
&\quad +\sum_{l=0}^{i-1}[L,B]_{i-1-l}\|u\|_l\Big)+[L,B]_{m,k-m}\|u\|_0\Big\}
\end{aligned}
\end{equation}
Inequality (\ref{D6ieq}) gives the case $k=m$. The general case
follows from (\ref{D6ieq}) by induction on $k$ as in the proof of
Lemma \ref{D2}. The theorem is proved.
\end{proof}


Theorem \ref{D3} gives (${\rm P_0},$) (${\rm P_4},$) (${\rm P_6}$) of section \ref{K}.
The resolvent estimates (${\rm P_1}$), (${\rm P_2}$), (${\rm P_3}$)
require stronger ellipticity assumptions due to Agmon \cite{Agm1}.

\begin{definitiion} \label{G1} \rm
(cf. {\rm \cite{Agm1}, \cite{LioMa}, Ch. 4, \cite{Luna}, 3.2, \cite{Tana}, 3.8,
\cite{Tana1}, 5.2}).
The pair $(L,B_j)$ is called a {\em regular elliptic pair}
if the following holds.
\begin{itemize}
\item[(i)] {\em Smoothness:} $L,B_j$ are given by {\rm
(\ref{D1eq}), (\ref{E1aeq})} with $a_\alpha,b_{j,\beta}\in
C^\infty(\overline{\Omega})$.
\item[(ii)] {\em Normality:} The set $\{B_j\}_{j=1}^{m/2}$ is normal and $m_j$
satisfy (\ref{K12aeq}).

\item[(iii)] {\em Strong ellipticity:}
The order $m\ge 2$ of $L$ is even and there exists $\mu>0$
such that for each $\theta\in [-\frac{\pi}{2},\frac{\pi}{2}]$ and any
$x\in \overline{\Omega}, \xi\in\mathbb{R}^n, r\ge 0$ we have
\begin{equation} \label{G2eq}
|L^P(x,\xi)- (-1)^{m/2}r^me^{i\theta}|
\ge \mu(|\xi|^m+r^m).
\end{equation}
\item[(iv)] {\em Root and Complementing Condition:} For each
$\theta\in[-\frac{\pi}{2} ,\frac{\pi}{2}],r\ge 0,
x\in\partial\Omega$ and for any $\xi\in\mathbb{R}^n$ tangential to
$\partial\Omega$ at $x$ with $(\xi,r)\not=0$ the polynomial
$\tau\mapsto L^P(x,\xi+\tau \nu(x))-(-1)^{m/2} r^me^{i\theta}$ has
exactly $m/2$ roots with positive imaginary part
$\{\tau_j^+(x,\xi,r,\theta)\}_{j=1}^{m/2}$, and the polynomials
$\{B_j^P(x,\xi+\tau\nu(x))\}_{j=1}^{m/2}$ are linearly independent
modulo $\prod_{j=1}^{m/2}(\tau-\tau_j^+(x,\xi,r,\theta))$.
\end{itemize}
\end{definitiion}

Taking $r=0$ we get back Definition \ref{D0}. The above
assumptions are made such that for
$\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ the operator
$L_\theta=L-(-1)^{m/2}e^{i\theta}\partial_t^m$ in ($n+1$)
variables is elliptic in $\overline{\Omega}\times\mathbb{R}$ and satisfies
together with $(B_j)$ the root and complementing condition in
\ref{D0}. Let $E_0$ denote the maximum of the characteristic
constants of the frozen operator $L_\theta(x,t,\partial),
t\in[-1,1], x\in \overline{\Omega},\theta\in
[-\frac{\pi}{2},\frac{\pi}{2}]$. Condition (iii) holds iff $L$ is
strongly elliptic, i.e., if
\begin{equation}
\label{G4eq}
-(-1)^{m/2}{\rm Re}\,L^P(x,\xi)>0
,\quad x\in\overline{\Omega},\quad  \xi\in\mathbb{R}^n\backslash\{0\}.
\end{equation}
Any strongly elliptic operator satisfies the root condition
(cf.\ \cite{Tana1}, Theorem 5.4) and together with the
Dirichlet boundary conditions the complementing condition
(cf. \cite{LioMa}, Ch. 4).
For instance, $-(-\Delta)^{m/2}$ is strongly elliptic.

$L$ is a closed operator in $L^2(\Omega)$ with
\begin{equation} \label{G5eq}
\mathcal{D} (L)=\{u\in H^m(\Omega):B_ju=0\mbox{ on }\partial\Omega
,j=1,\ldots ,m/2\}.
\end{equation}

We state the following result of Agmon \cite{Agm1},
Theorem 2.1. (cf. \cite{Luna}, Theorem 3.1.3, \cite{Paz}, 7.3.2,
\cite{Tana}, 3.8, \cite{Tana1}, Theorem 5.5).

\begin{theorem}\label{G2}
Let $(L,B_j)$ be a regular elliptic pair.
Then there exist $C>0,\gamma >0$ depending on $E_0,\Omega$
such that for any $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ we have
\begin{equation}\label{G6aeq}
\rho(L)\supset\Gamma_{\theta,\gamma}=
\{\lambda\in\mathbb{C}:\mathop{\rm arg}\lambda=\theta,|\lambda|\ge \gamma\}.
\end{equation}
For $\lambda\in \Gamma_{\theta,\gamma},
u\in H^{m}(\Omega),g_j\in H^{m-m_j}(\Omega)$
with $B_ju=g_j$ on $\partial\Omega$ we have
\[
\label{G6eq}
\sum_{j=0}^{m}|\lambda|^{\frac{m-j}{m}}\|u\|_j \le C \Big\{
\|\lambda u -Lu\|_0+ \sum_{j=1}^{m/2}
\big(|\lambda|^{\frac{m-m_j}{m}}\|g_j\|_0 +  \|g_j\|_{m-m_j}\big)\Big\}.
\]
In particular, we have for any $\lambda\in \Gamma_{\theta,\gamma}$
and $u\in \mathcal{D} (L)$ the estimate
\begin{equation}
\label{G6beq}
|\lambda|\|u\|_0+\|u\|_m\le C\|\lambda u-Lu\|_0.
\end{equation}
\end{theorem}

\begin{proof}
The estimates in Theorem \ref{G2} are proved in \cite{Tana}, Lemma
3.8.1 by applying (\ref{D6eq}) with $L_\theta$ and $k=m$ only to
functions $u=u(x,t)$ which vanish for $|t|\ge 1$. This and the
proof of \cite{Tana}, Lemma 3.8.1 give the statement on the
constants $C,\gamma$. These estimates imply
(\ref{G6aeq}) (cf.\ \cite{Agm1}, \cite{Tana}).
\end{proof}

\begin{corollary}\label{G3}
Let $(L,B_j)$ be a regular elliptic pair.
Then there exist $\omega>0,\theta_0\in (\pi/2,\pi),M_1>0$
depending on $E_0,\Omega$ such that $A=L-\omega I$ satisfies
$\rho (A)\supset \Sigma_{\theta_0}\cup\{0\}$ where
$\Sigma_{\theta_0}=\{\lambda:-\theta_0 <{\rm arg}\,\lambda < \theta_0\}$ and
\begin{equation}
\|(\lambda I-A )^{-1}\|_{L(H^0)}\le M_1/|\lambda |, \quad
\lambda \in \Sigma_{\theta_0} .
\end{equation}
\end{corollary}

The proof of this corollary follows immediately from Theorem \ref{G2}
choosing $\omega=2\gamma$. \smallskip

We assume that $L,B_j$ depend on $t$. Let $a_\alpha,b_{j,\beta}\in
C^\infty([0,T_1]\times \overline{\Omega})$ where $T_1>0$ is fixed.
Let $(L(t),B_j(t))$ be a regular elliptic pair for each $t\in
[0,T_1]$ where $m,m_j$ do not depend on $t$. Let $E_0(t)$ denote
the corresponding characteristic constant defined as $E_0$ above.
We then have $E_0=\max\{E_0(t):t\in [0,T_1]\}<+\infty$ by
continuity. Hence the constants entering in Theorem \ref{G2} and
Corollary \ref{G3} can be chosen uniformly for $t\in [0,T_1]$. We
put $A(t)=L(t)-\omega I$ as in Corollary \ref{G3}.

\begin{lemma}
\label{G5} Let $(L(t),B_j(t))$ be a regular elliptic pair for each
$t\in [0,T_1]$ and let $A(t)=L(t)-\omega I$. Then for every
$\lambda\in \Sigma_{\theta_0}\cup\{0\}$ the mapping $t\mapsto
(\lambda I-A(t))^{-1}$ belongs to $C^1([0,T_1], L(H^0))$. There
exist $M_2,M_3>0$ depending only on $E_0,\Omega$ and on
$\|\partial_tA\|_0+\sum_j\|\partial_t B_j\|_{m-m_j}$ such that for
all $\lambda \in \Sigma_{\theta_0}$ and $t,\tau\in [0,T_1]$ we
have
\begin{gather}
\label{G8eq}
\|\partial_t(A(t)-\lambda I)^{-1}\|_{L(H^0)}  \le   M_2/|\lambda|\\
\label{G8aeq}
\|\partial_t(A(t)^{-1})-\partial_t(A(\tau)^{-1})\|_{L(H^0)} \le  M_3 |t-\tau|.
\end{gather}
\end{lemma}

The proof of this lemma can be found in  \cite{Tana}, Lemma 5.3.6.
It is based on Theorem \ref{G2}.

\section{The nonlinear parabolic problem}
\label{I}

We consider the nonlinear initial boundary value problem
(\ref{F3eq}). We assume that $\Omega$ is a bounded open subset of
$\mathbb{R}^n$ with boundary $\partial\Omega$ of class $C^\infty$ and that
$\mathcal{F}$ and $\mathcal{B}=(\mathcal{B}_j)_{j=1}^{m/2}$ are smooth
differential operators defined by (\ref{F1beq}), (\ref{F1eeq}) in
$[0,T]\times U$ as in section \ref{F}. We fix an initial value
$\phi\in U\subset H^\infty(\Omega)$ and a boundary value $h\in
C^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$. We suppose the
necessary compatibility conditions (\ref{F4eq}) coupling $\phi$
and $h$. By Theorem \ref{F4} we have to solve the linear problem
(\ref{F12eq}).

We assume that the pair
$(\mathcal{F}_u(0,\phi),\mathcal{B}_u(0,\phi))$ is a regular
elliptic pair in the sense of Definition \ref{G1}.
We can choose $0<T_1\le T$ and an open neighbourhood
$V$ of $\phi$ in $H^\infty(\Omega)$ such that
$(\mathcal{F}_u(t,u(t)),\mathcal{B}_u(t,u(t)))$ is a regular
elliptic pair with a uniform characteristic constant $E_0$
for all $t\in [0,T_1],u\in W$ where
\begin{equation}
W=\{u\in C^\infty([0,T_1],H^\infty(\Omega)): u(t)\in V, t\in [0,T_1]\}.
\end{equation}

\begin{theorem} \label{I2}
Let $\mathcal{F},\mathcal{B}$ be smooth differential operators.
Let the initial value $\phi\in H^\infty(\Omega)$ and
the boundary value $h\in C^\infty([0,T],H^\infty(\partial\Omega)^{m/2})$
satisfy the compatibility conditions {\rm (\ref{F4eq})}.
Assume that $(\mathcal{F}_u(0,\phi),\mathcal{B}_u(0,\phi))$
is a regular elliptic pair.
Then there exist $T_0>0$ and a unique solution
$u\in C^\infty([0,T_0],H^\infty(\Omega))$
of the nonlinear initial value problem {\rm (\ref{F3eq})}.
\end{theorem}

\begin{proof}
We have to verify the assumptions of Theorem \ref{F4}. The
existence of the required mappings $R_u$ is proved in Corollary
\ref{E6}. We choose $T_1,V,W$ as above such that $(\mathcal{F}_u(t,u(t)),\mathcal{B}_u(t,u(t)))$ is a regular elliptic pair for
every $t\in [0,T_1],u\in W$. We fix $0<T\le T_1$ and consider the
linear problem (\ref{F12eq}) where $f_1\in
C_0^\infty([0,T],H^\infty(\Omega))$. Since $f_1^{(j)}(0)=0$ for
all $j$ this is a problem with trivial (vanishing) compatibility
relations. By classical results on linear parabolic boundary value
problems (cf.\ \cite{LioMa}, Ch.\ IV, 6.4) there is a unique
solution $w\in C_0^\infty([0,T],H^\infty(\Omega))$ of problem
(\ref{F12eq}).

We have to show estimates (\ref{F13eq}). We write $L(t)=\mathcal{F}_u(t,u(t)),B(t)=\mathcal{B}_u(t,u(t))$ observing that the following
holds uniformly for $u\in W$. Using \ref{G3}, \ref{G5} we choose
$\omega,\theta_0,M_1,M_2,M_3$ such that $A(t)=L(t)-\omega I$
satisfies conditions ${\rm (P}_1{\rm )}$, ${\rm (P}_2{\rm )}$,
${\rm (P}_3{\rm )}$. By Theorem \ref{D3} conditions ${\rm
(P}_0{\rm )}$, ${\rm (P}_4{\rm )}$, ${\rm (P}_6{\rm )}$ hold for
$A(t)$ with uniform constants $M_0,M_4,M_k$. Choosing $W$
sufficiently small we obtain condition ${\rm (P}_5{\rm )}$ for
$A(t),B(t)$ with a uniform constant $M_5$. Hence Theorem \ref{K4}
applies to the pair $(A(t),B(t))$. We put $f(t)=e^{-\omega
t}f_1(t)$ and $z(t)=e^{-\omega t}w(t)$. Then $z$ is a solution of
problem (\ref{K13eq}). By Theorem \ref{K4} thus $z$ satisfies the
estimate (\ref{K35eq}) with a uniform constant $C$ depending on
$k$. Replacing $(z,f)$ in (\ref{K35eq}) by $(w,f_1)$ we get the
estimate
\[
\|w\|_{km} = \sum_{i=0}^k|w|_{i,k(m-i)}
\le C\sum_{i=0}^{k} [A,B]_{3m, (k-i)m}\|f_1\|_{mi}
\le C[A,B;f_1]_{3m,k}
\]
and thus $\|w\|_k\le C[A,B;f_1]_{4m,k}$ for any $k$, shrinking
$V,W$ if necessary. Since $\|A\|_i+\|B\|_i\le C[u]_{m+b,i}$ for
$b=[n/2]+1$ (cf.\ (\ref{F1eq}), (\ref{F2eq})) this implies
$\|w\|_k\le C[u;f_1]_{5m+b,k}$ for any $k$. We proved (\ref{F13eq})
and thus the result.
\end{proof}

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\end{thebibliography}

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