\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{2004-Fez conference on Differential Equations and Mechanics \newline
\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 23--32.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{23}

\begin{document}

\title[\hfilneg EJDE/Conf/11 \hfil Heat equation on unbounded domains]
{On the well-posedness of the heat equation on unbounded domains}

\author[W. Arendt, S. Lalaoui R.\hfil EJDE/Conf/11 \hfilneg]
{Wolfgang Arendt, Soumia Lalaoui Rhali}  % in alphabetical order

\address{Wolfgang Arendt \hfill\break
Universit\"{a}t Ulm \\ Angewandte Analysis\\ D-89069 Ulm, Germany}
\email{arendt@mathematik.uni-ulm.de}

\address{Soumia Lalaoui Rhali\hfill\break
Facult\'{e} Polydisciplinaire de Taza\\ 
Universit\'{e} Sidi  Mohamed Ben Abdellah\\
 B.P: 1223 Taza\\ Morocco}
\email {slalaoui@ucam.ac.ma}


\date{}
\thanks{Published October 15, 2004.}
\thanks{The second author gratefully acknowledges his support by the DAAD}
\subjclass[2000]{35K05, 35K20, 47D06} 
\keywords{Unbounded domains; heat equation; Dirichlet problem;
\hfill\break\indent
resolvent positive operators}


\begin{abstract}
 This work concerns the well-posedness of the heat equation
 in an unbounded open domain, under basic regularity
 assumptions on this domain.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

Let $\Omega $ be an open set of $\mathbb{R}^{n}$ with boundary
$\Gamma=\partial\Omega $ and consider the problem
% $P_{\tau }(u_{0},\varphi)$,
\begin{equation} \label{Ptau}
\begin{gathered}
u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\
u(t)\big|_{\Gamma }=\varphi (t), \quad t\in [ 0,\tau ] \\
u(0)=u_{0},
\end{gathered}
\end{equation}
where $u_{0}\in C(\overline{\Omega })$,
$\varphi \in C([0,\tau ];C(\Gamma))$, $\tau >0$.

The aim of this work is to study the well-posedness of \eqref{Ptau}
when $\Omega $ is unbounded. The case where $\Omega $ is
bounded has been studied in \cite[Chapter 6]{ar}, and sufficient conditions
on the initial data $ u_{0}$ and the boundary condition $\varphi $ are
given to show that the problem \eqref{Ptau} is well-posed in
$C(([0,\tau ];C(\overline{\Omega }))$  whenever $\Omega $ is regular (See
definition \ref{1.1}).

We point out here that the regularity assumption is equivalent when $\Omega $
is bounded to that the Dirichlet problem, \eqref{Dp},
\begin{equation} \label{Dp}
\begin{gathered}
u\in C(\overline{\Omega }) \\
\Delta u=0\quad \text{in }\mathcal{D}(\Omega )'
\\ u\big|_{\Gamma}=\phi,
\end{gathered}
\end{equation}
has for all $\phi \in C(\Gamma )$ a classical solution $u$, that means that
$u$ is a solution of \eqref{Dp} and $u\in C^{2}(\Omega )$. (See \cite{dl},
\cite{gt} for instance). The situation is more complicated when $\Omega $ is
unbounded since one must take into account the condition at infinity that
the solution of \eqref{Dp} satisfies (see Theorem \ref{1.2}), and the choice
of the space $X\subset C(\overline{\Omega })$ in which the solution $u(t)$
of \eqref{Ptau} belongs will be imposed by this condition at
infinity and then by the choice of the unbounded regular open set $\Omega$.

In this work, the unbounded set is taken in the case $n=1$ as an interval
of $\mathbb{R}$ and as the exterior of a ball of $\mathbb{R}^{n}$ for $n\geq
2$. When $n=2$, we deal with the heat equation with homogeneous boundary
conditions and for $n\geq 3$, the heat equation with inhomogeneous boundary
conditions is studied for an exterior domain.

The organization of this work is as follows: In Section 2, we
recall some preliminaries results lying between the regularity
property for unbounded sets and the well-posedness of the
Dirichlet problem \eqref{Dp}. We also recall some existence
results for Cauchy problems with resolvent positive operators. We
present in Section 3 our main result, the method of proof consists
on reformulating \eqref{Ptau} as a Cauchy problem with the Poisson
operator. Section 4 is devoted to the study of the well-posedness
of this Cauchy problem, we first show that the Poisson operator
has a positive resolvent in $X\times C(\Gamma )$. Using results of
Section 2, we then show the well-posedness of \eqref{Ptau}.

\section{Preliminaries}

\subsection*{The Dirichlet Problem}

Let $\Omega $ be an open set of $\mathbb{R}^{n}$ with boundary
$\Gamma =\partial \Omega$.

\begin{definition}[\cite{dl}] \label{1.1} \rm
(a) Let $z\in \Gamma $. we say that $z$ is a regular
boundary point of $\Omega $ if there exists $r>0$, and
$w\in C(\overline{\Omega \cap B(z,r)})$ such that
\begin{gather*}
\Delta w\leq 0, \quad \text{in }\mathcal{D}(\Omega \cap B(z,r))' \\
w(x)>0, \quad x\in (\Omega \cap B(z,r))\backslash \{z\} \\
w(z)=0\,.
\end{gather*}
Then the function $w$ is called a barrier.
\\
(b) We say that $\Omega $ is regular if all boundary  points are regular.
\end{definition}

This regularity property is related to the Dirichlet problem \eqref{Dp} as
follows.

\begin{theorem}[\cite{dl}] \label{1.2}
Let $\Omega $ be an unbounded set, not dense in $\mathbb{R}^{n}(n\geq 2)$
with boundary $\Gamma $. Then the following two assertions are equivalent:
\\
(i) For every continuous $\phi $ with compact support in $\Gamma $, there
exists a classical solution of \eqref{Dp} satisfying the following null
condition at infinity
\begin{itemize}
\item[(NC)] There exists $h$ harmonic on $\Omega$ such that
$h\in C(\overline{\Omega })$,
with $h(x)>0$ for $| x|$ large so that
$\lim_{|x| \to +\infty }\frac{u(x)}{h(x)}=0.$
\end{itemize}
(ii) All boundary points of $\Omega $ are regular.
\end{theorem}

\begin{example}[\cite{dl}]\label{1.3} \rm
Let $n\in \mathbb{N}^{\ast }$.\newline (a) Case of an interval of
$\mathbb{R}$. Let $\Omega _{1}=]1,\infty[ $, then $\Omega _{1}$ is
regular and for all $\phi \in \mathbb{R}$ and all $c\in
\mathbb{R}$, there exists a unique classical solution of
\eqref{Dp} satisfying the condition at infinity:
\begin{equation*}
\lim_{x\rightarrow +\infty }\frac{u(x)}{x}=c.
\end{equation*}
(b) Case of the exterior of a ball of $\mathbb{R}^{n}$, $n\geq 2$.
Let $\Omega _{n}=\mathbb{R}^{n}\setminus \overline{B(0,1)}$, then
$\Omega _{n}$ is regular and given $u$ a bounded classical solution of \eqref{Dp}, then $%
c=\lim_{\left| x\right| \rightarrow \infty }u(x)$ exists and
\begin{equation*}
u(x)=(1-\frac{1}{{|x|}^{n-2}})c+\frac{1}{{\sigma }_{n}}\int_{\partial B}%
\frac{{|x|}^{2}-1}{{|t-x|}^{n}}\phi (t)d\gamma (t)
\end{equation*}
is a classical solution of \eqref{Dp}, with $\sigma _{n}$ being
the total surface area of the unit sphere in $\mathbb{R}^{n}$.
Conversely, the
function $u$ given by the last formula is a classical solution of \eqref{Dp}%
. Moreover, one has:\newline If $n=2$, then for all $\phi \in
C(\partial B)$, there exists a unique classical solution of
\eqref{Dp} satisfying the condition at infinity:
\begin{equation*}
u\text{ is bounded on }\Omega .
\end{equation*}
This solution will have a limit at infinity which is imposed by the giving
$\phi$:
\begin{equation}
\lim_{| x| \to \infty }u(x)=\frac{1}{2\pi }\int_{\partial
B}\phi (t)d\gamma (t).  \label{0}
\end{equation}
If $n\geq 3$, then for all $\phi \in C(\partial B)$ and all
$c\in \mathbb{R}$, there exists a unique classical solution of \eqref{Dp}
satisfying the condition at infinity:
\begin{equation*}
\lim_{| x| \to \infty }u(x)=c.
\end{equation*}
\end{example}

Note that a solution $u$ of \eqref{Dp} satisfying the null
condition at infinity $(\mathbf{NC}) $ does not
necessarily satisfy:
\begin{equation*}
\lim_{| x| \to \infty }u(x)=0.
\end{equation*}
This remains true for the exterior of a compact set of $\mathbb{R}^{n}$,
$n\geq 3$.

\begin{proposition}[\cite{dl}]\label{1.4}
Let $K$ be a compact set of $\mathbb{R}^{n}$,
$n\geq 3$ with boundary $\Gamma$. If $\Omega
=\mathbb{R}^{n}\setminus K$ is regular, then for all $\phi \in
C(\Gamma)$, there exists a unique classical solution of \eqref{Dp}
satisfying the condition at infinity:
\begin{equation*}
\lim_{| x| \to \infty }u(x)=0.
\end{equation*}
\end{proposition}

\subsection*{Cauchy Problem}

Let $X$ be a Banach space and consider the inhomogeneous Cauchy
Problem: %(ACP_{f})
\begin{equation} \label{ACPf}
\begin{gathered}
u'(t)=Au(t)+f(t),\quad t\in [ 0,\tau ] \\
u(0)=u_{0},
\end{gathered}
\end{equation}
where $u_{0}\in X$ and $f\in C([0,\tau ];X)$.

\begin{definition} \label{1.5} \rm
A mild solution of $(ACP_{f})$ is a function $u\in
C([0,\tau ];X)$ such that $\int_{0}^{t}u(s)ds\in D(A)$ and
for all $t\in [ 0,\tau ]$,
\begin{equation*}
u(t)=u_{0}+A\int_{0}^{t}u(s)ds+\int_{0}^{t}f(s)ds\,.
\end{equation*}
\end{definition}

We recall now some results on resolvent positive operators and
Cauchy problems, we refer to \cite[Chapter 3]{ar}, for more
details.

\begin{theorem}[\cite{ar}] \label{1.6}
Let $A$ be a resolvent positive operator on a
Banach lattice $X$, that means, there exists $w\in \mathbb{R}$
such that $(w,\infty )\subset \rho (A)$ and $R(\lambda ,A)\geq 0$
for all $\lambda >w$.
\\
(i) Let $u_{0}\in D(A)$,$\;f_{0}\in X$ such that
$Au_{0}+f_{0}\in \overline{D(A)}$.
Let $f(t)=f_{0}+\int_{0}^{t}f'(s)ds$ where
$f'\in L^{1}((0,\tau );X)$. Then $(ACP_{f})$ has a unique mild
solution.
\\
(ii) Let $f\in C([0,\tau ];X_{+})$, $u_{0}\in X_{+}$ and let $u$ be a mild
solution of $(ACP_{f})$. Then $u(t)\geq 0$ for all $t\in [ 0,\tau ]$.
\end{theorem}

Define now the Gaussian semigroup $(G(t)) _{t\geq 0}$
on the space $C_{0}(\mathbb{R}^{n})$ of all continuous functions
vanishing at infinity by:
\begin{equation*}
G(t)f(x)=(4\pi t) ^{-n/2}\int_{\mathbb{R}^{n}}f(x-y)e^{-|y|^{2}/(4t)}dy,
\quad t>0,x\in \mathbb{R}^{n},f\in C_{0}(\mathbb{R}^{n}).
\end{equation*}

\begin{theorem}[\cite{ar}] \label{1.7}
The family $(G(t)) _{t\geq 0}$ defines a
bounded holomorphic $C_{0}-$semigroup of angle $\frac{\pi }{2}$ on $C_{0}(
\mathbb{R}^{n})$. Its generator is the Laplacian $\Delta _{G}$ on $C_{0}(
\mathbb{R}^{n})$ with maximal domain; i.e.,
\begin{gather*}
D(\Delta _{G})=\{ f\in C_{0}(\mathbb{R}^{n}),\;\Delta f\in C_{0}(
\mathbb{R}^{n})\} , \\
\Delta _{G}f=\Delta f,
\end{gather*}
here one identifies $C_{0}(\mathbb{R}^{n})$ with a subspace of
$\mathcal{D}(\mathbb{R}^{n})'$.
\end{theorem}

\begin{proposition}[\cite{ar}]\label{1.8}
Let $A$ be the generator of a bounded $C_{0}$-group
$(U(t)) _{t\in \mathbb{R}}$ on $X$. Then $A^{2}$ generates a
bounded holomorphic $C_{0}$-semigroup $(T(t)) _{t\geq 0}$ of
angle $\frac{\pi }{2}$ on $X$. Moreover, for $t>0$,
\begin{equation*}
T(t)=(4\pi t)^{-1/2}\int_{\mathbb{R}}e^{-| y| ^{2}/(4t)} U(y)\,dy.
\end{equation*}
\end{proposition}

\section{Main result}

We consider the problem \eqref{Ptau} with $\Omega $
presenting the cases in Example \ref{1.3} and Proposition \ref{1.4}. Since
in the case $n=2$, the condition at $\inf $inity (\ref{0}) is imposed by the
boundary function, we restrict our study of \eqref{Ptau} for
$n=2$ to the case where $\varphi =0.$

\begin{theorem} \label{1.9}
Let $n\in \mathbb{N}$. \textbf{Case $n=1$:} Let $\Omega _{1}=]
1,+\infty [ $ with boundary $\Gamma _{1}=\{ 1\} $ and denote by
$\big(C_{\infty }(\overline{\Omega }_{1});\| .\| _{C_{\infty }
(\overline{\Omega }_{1})}\big) $ the Banach space
\begin{equation*}
C_{\infty }(\overline{\Omega }_{1}):=\big\{ u\in C([ 1,+\infty [
),\lim\limits_{x\rightarrow +\infty }\frac{u(x)}{x}\text{
exists}\big\}
\end{equation*}
with the norm $\| u\| _{C_{\infty }(\overline{\Omega }
_{1})}=\max_{x\in [ 1,\infty [ }| u(x)/x|$.

Then for all $u_{0}\in $ $C_{\infty }(\overline{\Omega }_{1})$ and all
$\varphi \in C([ 0,\tau ] )$ such that $u_{0}(1)=\varphi (0),$
there exists a unique mild solution
$u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{1}))$ of the problem
\begin{equation} \label{Ptau1}
\begin{gathered}
u_{t}(t,x)=u''(t,x), \quad t\in [ 0,\tau ],\; x\in ] 1,+\infty [ \\
u(t,1)=\varphi (t), \quad t\in [ 0,\tau ] \\
u(0,x)=u_{0}(x).
\end{gathered}
\end{equation}
%
\textbf{Case $n=2$:}  Let $\Omega _{2}=\mathbb{R}^{2}\setminus B(0,1)$
with boundary $\Gamma _{2}=\partial B$ and set
$$
C_{\infty }(\overline{\Omega}_{2}):=\{ u\in C(\overline{\Omega }_{2}),
\text{ }u\big|_{ \Gamma _{2}}=0
\text{ and }\lim_{| x| \to +\infty }u(x)=0\}
$$
with the supremum norm$ \| u\| _{C_{\infty }(\overline{\Omega }
_{2})}=\max_{x\in \Omega _{2}}| u(x)|$.
Then for all $u_{0}\in $ $C_{\infty }(\overline{\Omega }_{2})$, there exists
a unique mild solution $u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{2})) $
 of the problem
\begin{equation} \label{Ptau2}
\begin{gathered}
u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\
u\big|_{ \Gamma _{2}}=0,   \\
u(0)=u_{0}.
\end{gathered}
\end{equation}
\textbf{Case $n\geq 3$:} Let $\Omega _{n}=\mathbb{R}^{n}\setminus
B(0,1) $ or more generally $\Omega _{n}=\mathbb{R}^{n}\setminus K$  with
$K$ being a compact set of $\mathbb{R}^{n}$ with boundary $\Gamma _{n}$,
and set
$$
C_{\infty }(\overline{\Omega }_{n}):=\{ u\in C(\overline{\Omega }_{n}),
\;\lim_{| x| \to +\infty }u(x)=0\}
$$
with the supremum norm. If $\Omega _{n}$ is regular, then for all
$u_{0}\in C_{\infty }(\overline{\Omega }_{n})$ and all
$\varphi \in C([ 0,\tau] ;C(\Gamma _{n}))$ such that
$u_0\big|_{\Gamma_{n}}=\varphi (0)$,
there exists a unique mild solution
$u\in C([0,\tau];C_{\infty }(\overline{\Omega }_{n}))$ of the problem
\begin{equation} \label{Ptaun}
\begin{gathered}
u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\
u(t)\big|_{ \Gamma _{n}}=\varphi (t), \quad t\in [ 0,\tau ] \\
u(0)=u_{0}.
\end{gathered}
\end{equation}
\end{theorem}

Let $\Omega _{n}$, $n\geq 1$ be defined as in Theorem \ref{1.9} and define
the operator $\Delta _{\max }^{n}$ on $C_{\infty }(\overline{\Omega }_{n})$
as follows
\begin{gather*}
D(\Delta _{\max }^{n})=\{ u\in C_{\infty }(\overline{\Omega }
_{n}),\Delta u\in C_{\infty }(\overline{\Omega }_{n})\} \\
\Delta _{\max }^{n}u=\Delta u \quad \text{in }\mathcal{D}(\Omega_{n})'.
\end{gather*}
We mean by mild solution of \eqref{Ptaun} a function
$u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))$ such that
$\int_{0}^{t}u(s)ds\in D(\Delta _{\max }^{n})$ and
for all $t\in [ 0,\tau]$,
\begin{gather*}
u(t)=u_{0}+\Delta \int_{0}^{t}u(s)ds\quad\text{in }\mathcal{D}(\Omega_{n})' \\
u(t)\big|_{\Gamma _{n}}=\varphi (t).
\end{gather*}

To prove Theorem \ref{1.9}, we will reformulate the problem
\eqref{Ptaun} as an inhomogeneous Cauchy problem with resolvent
positive operator.

\section{Inhomogeneous Cauchy Problem}

Define for $n\geq 1$ the Poisson operators $A_{n}$ with domain
$D(A_{n})=D(\Delta _{\max }^{n})\times \{ 0\} $ by
\begin{gather*}
A_{1}(u,0)=(\Delta u,-u(1)), \\
A_{2}(u,0)=(\Delta u,0), \\
A_{n}(u,0)=(\Delta u,-u\big|_{\Gamma _{n}}),\quad n\geq 3,
\end{gather*}
and consider the Cauchy problem
\begin{equation} \label{ACPn}
\begin{gathered}
U'(t)=A_{n}U(t)+\Phi _{n}(t),\quad t\in [0,\tau ] \\
U(0)=U_{0},
\end{gathered}
\end{equation}
where $U_{0}=(u_{0},0)$, $u_{0}\in C_{\infty
}(\overline{\Omega}_{n})$ is the initial data , $\Phi _{2}=(0,0)$
and for $n\neq 2$, $\Phi _{n}(t)=(0,\varphi (t)), \varphi \in
C([0,\tau ];C(\Gamma _{n}))$ is the boundary condition.

\begin{proposition} \label{1.10}
Let  $n\geq 1$ and
$U\in C([ 0,\tau ] ;C_{\infty }(\overline{\Omega }_{n})\times C(\Gamma _{n}))$.
 Then $U$ is a mild solution of \eqref{ACPn} if
and only if $U(t)=(u(t),0)$ where $u\in C([0,\tau ];C_{\infty
}(\overline{ \Omega }_{n}))$ is the mild solution of \eqref{Ptau}.
\end{proposition}

The proof is immediate from the definition of $A_{n}$ and the fact that
$\overline{D(A_{n})}=C_{\infty }(\overline{\Omega }_{n})\times \{0\}$.

To show the well-posedness of \eqref{ACPn}, we first prove that $A_{n}$ is a
resolvent positive operator.

\begin{theorem} \label{1.11}
Let $\lambda >0$, if $n\neq 2$ then for all $(f,\phi) \in C_{\infty }(\overline{\Omega }_{n})\times C(\Gamma _{n})$ there
exists a unique function $u\in D(\Delta _{\max }^{n})$ such that
\begin{equation}
\begin{gathered}
(\lambda -\Delta )u=f\quad\text{in }\mathcal{D}(\Omega _{n})'\\
u\big|_{\Gamma _{n}}=\phi .
\end{gathered}  \label{a1}
\end{equation}
Moreover, if $f\leq 0$, $\phi \leq 0$, then $u\leq 0$.
If $n=2$, then for all $f\in C_{\infty }(\overline{\Omega }_{2})$, there
exists a unique function $u\in D(\Delta _{\max }^{2})$ such that
\begin{equation}
\begin{gathered}
(\lambda -\Delta )u=f\;\;\quad\text{in }\mathcal{D}(\Omega _{2})'\\
u\big|_{ \Gamma _{2}}=0.
\end{gathered} \label{a2}
\end{equation}
Moreover, if $f\leq 0$, then $u\leq 0$.
\end{theorem}

\begin{proof}
(1) \textbf{Existence.} (a) Case $n=1:$ \newline Set $C_{\infty
}(\mathbb{R})=\{ f\in C(\mathbb{R}),\;\lim_{x \to -\infty
}f(x)=\text{0 and }\lim_{x\to +\infty } \frac{f(x)}{x}\text{
exists}\} $ and define on $C_{\infty }(\mathbb{R}) $ the
translation group
\begin{equation*}
T(t)f(x)=f(x-t),\quad t\in \mathbb{R},\;x\in \mathbb{R.}
\end{equation*}
Then $(T(t))_{t\in \mathbb{R}}$ is a $C_{0}-$group with generator $A_{T}$
defined by
\begin{gather*}
D(A_{T})=\{ f\in C_{\infty }(\mathbb{R}),f'\in C_{\infty }(
\mathbb{R})\} \\
A_{T}f=f'.
\end{gather*}
It follows from Proposition \ref{1.8} that $A_{T}^{2}$ generates a $C_{0}-
$semigroup $(G(t))_{t\geq 0}$ which is the Gaussian semigroup:
\begin{equation*}
G(t)f(x)=(4\pi t)^{-1/2}\int_{\mathbb{R}}e^{-| y|^{2}/(4t)}f(x-y)dy.
\end{equation*}
Moreover, $G(t)C_{\infty }(\overline{\Omega }_{1})\subset C_{\infty }(
\overline{\Omega }_{1})$ for all $t\geq 0$. Let $\lambda
>0$ and $(f,\phi ) \in C_{\infty }(\overline{\Omega
}_{1})\times \mathbb{R}$, and take
\begin{gather*}
v_{0}(x)=\int_{0}^{+\infty }e^{-\lambda t}G(t)f(x)dt, \\
v(x)=(\phi -v_{0}(1))e^{-\sqrt{\lambda }(x-1)},
\end{gather*}
Then $u=v+v_{0}$ is a solution of (\ref{a1}).
\\
(b) Case $n\geq 2$: Let $f\in C_{\infty }(\overline{\Omega }_{n})$. Then $f$
can be extended to $C_{0}(\mathbb{R}^{n})$. Since the Gaussian semigroup
generates an holomorphic $C_{0}-$semigroup on $C_{0}(\mathbb{R}^{n})$, we
get that
\begin{equation*}
v_{0}(x)=\int_{0}^{+\infty }e^{-\lambda t}G(t)f(x)dt,
\end{equation*}
is a solution of
\begin{equation}
(\lambda -\Delta )v=f\text{   for all }\lambda >0.  \label{a3}
\end{equation}
Moreover, if  $f\in C_{\infty }(\overline{\Omega }_{n})$, then
$v_{0}\in C_{\infty }(\overline{\Omega }_{n})$.
\\
If $n=2$, then $v_{0}$ is a solution of $(\lambda -A_{2})(u_{0},0)=(f,0).$
\\
If $n\geq 3$, it remains to show that there exists a solution of
\begin{equation}
\begin{gathered}
(\lambda -\Delta )v=0, \quad\text{in }\mathcal{D}(\Omega_{n})'\\
v\big|_{ \Gamma _{n}}=\phi -v_0\big|_{ \Gamma _{n}}=:\psi .
\end{gathered} \label{a4}
\end{equation}
Let $\Omega _{nk}=\Omega _{n}\cap B(0,R_{k})\;$where $(R_{k})
_{k\geq 1}$ is an increasing sequence of positif reals such that
$R_{k}\to \infty $ as $k\to \infty \;$and consider the following
problem on $C(\overline{\Omega }_{nk})$.
\begin{equation}
\begin{gathered}
(\lambda -\Delta )v_{k}=0 \quad\text{in }\mathcal{D}(\Omega_{nk})' \\
v_k\big|_{ \Gamma _{k}}=0\quad \text{on }\Gamma_{k}=\partial B(0,R_{k}) \\
v_k\big|_{ \Gamma _{n}}=\psi .
\end{gathered}  \label{a5}
\end{equation}
Since $\Omega _{n}$ is regular, $\Omega _{nk}$ is regular and it
follows from \cite{lsw}, \cite{st} that (\ref{a5}) has a solution$
v_{k}\in C(\overline{ \Omega }_{nk})$. Our aim now is to show that
the sequence $(v_{k}) _{k\geq 1}$
 converges to the solution of (\ref{a4}),
for that, we use the following maximum principle due to \cite{ar}.


\begin{theorem}[Maximum Principle for distributional solutions] \label{1.12}
Let $\Omega _{0}$ be a bounded open set of $\mathbb{R}^{n}$ with boundary
$\Gamma $. Let $M\geq 0$, $\lambda \geq 0$, $u\in C(\overline{\Omega }_{0})$
such that\newline
$(i)$ $\lambda u-\Delta u\leq 0$, in $\mathcal{D}(\Omega _{0})'
$\newline
$(ii)$ $u\big|_{ \Gamma }\leq M$,\newline Then $u\leq M$ on
$\overline{\Omega }_{0}$.
\end{theorem}

Without loss of generality, we can assume that $\psi
\geq0$.\newline \textbf{Claim 1}: $(v_{k}) _{k\geq 1}$ is an
increasing bounded sequence. Indeed, by applying the Maximum
principle in $\Omega _{nk}$ to $v_{k}$ and $v_{k}-v_{k+1}$
respectively, we obtain$: $\begin{equation*} 0\leq v_{k}\leq \|
\psi \| ,
\end{equation*}
and
\begin{gather*}
(\lambda -\Delta )(v_{k}-v_{k+1})=0,\;\;\text{in
}\mathcal{D}(\Omega _{nk})' \\ (v_{k}-v_{k+1})\big|_{ \Gamma
_{k}}=-v_{k+1}\leq 0, \\ (v_{k}-v_{k+1})\big|_{ \Gamma_{n}}=0.
\end{gather*}
Hence $v_{k}\leq v_{k+1}\text{ in }\Omega _{nk}$.

\noindent\textbf{Claim 2:} Let $v=\lim_{k\to \infty }v_{k}$, then
$v\in C_{\infty }(\overline{\Omega }_{n})$.
Indeed, denote by $w_{k}\;$the solution of the problem
\begin{gather*}
\Delta w_{k}=0, \quad\text{in }\mathcal{D}(\Omega_{nk})' \\
w_k\big|_{ \Gamma _{k}}=0, \\
w_k\big|_{\Gamma _{n}}=\psi .
\end{gather*}
Then $w_{k}\geq 0$. Define the Poisson operator $B_{k}$ on
$C(\overline{\Omega }_{nk})\times C(\Gamma _{n}\cup \Gamma _{k})$
by
\begin{gather*}
D(B_{k})=\{ w\in C(\overline{\Omega }_{nk}),\Delta w\in
C(\overline{
\Omega }_{nk})\} \times \{0\}, \\
B_{k}(w,0)=(\Delta w,-(w\big|_{ \Gamma _{n}},w\big|_{ \Gamma _{k}})).
\end{gather*}
Since $\Omega _{nk}$ is regular, we deduce from \cite[Chapter
6]{ar}, that $B_{k}$ is a resolvent positive operator and then
\begin{equation}
(w_{k},0)  =  R(\lambda ,B_{k})(\lambda w_{k},(\psi ,0))
 \geq   R(\lambda ,B_{k})(0,(\psi ,0))
 =  (v_{k},0). \label{a6}
\end{equation}
On the other hand, it follows from Proposition \ref{1.4} that for
all $\Phi \in C(\Gamma _{n})$, the Dirichlet problem
\eqref{Dp}(with $\phi=\Phi$) has a unique solution $w$ satisfying
the condition at infinity $\lim_{| x| \to \infty }w(x)=0$.
Moreover, if $\Phi \leq 0$, then $w\leq 0.$ Indeed, let
$\varepsilon >0$, since $w\in C^{1}(\Omega _{n})$, there exists
$\Omega _{0}\subset \subset \Omega _{n}$ such that $\mathop{\rm
supp}(w-\varepsilon ) ^{+}\subset \Omega _{0}$, Thus
$(w-\varepsilon) ^{+}\in H_{0}^{1}(\Omega _{0})$ and $\int_{\{
w>\varepsilon \} }| \nabla w| ^{2}=0$. Hence,
\begin{equation*}
w\leq \varepsilon .
\end{equation*}
Denote by $w_{0}$ the solution of \eqref{Dp} (with $\phi=\psi$)
vanishing to zero at infinity$, $ then
\begin{gather*}
\Delta (w_{k}-w_{0})=0,\;\;\quad\text{in }\mathcal{D}(\Omega _{nk})'\\
(w_{k}-w_{0})\big|_{ \Gamma _{k}}=-w_{0\mid \Gamma _{k}}\leq 0,
\\ (w_{k}-w_{0})\big|_{ \Gamma _{n}}=0.
\end{gather*}
Theorem \ref{1.12} and (\ref{a6}) imply that
$0\leq v_{k}\leq w_{k}\leq w_{0}$.
Hence
\begin{equation*}
\lim_{| x| \to \infty }v(x)=\lim_{| x|\to \infty }w_{0}(x)=0.
\end{equation*}
Finally, $u=v_{0}+v$ is a solution of (\ref{a1}).
\\
(2)\textbf{Positivity and Uniqueness.}
Let $(f,\phi ) \in C_{\infty }(\overline{\Omega }_{n})\times
C(\Gamma _{n})$ such that $f\leq 0$, $\phi \leq 0$ and $u$ a solution of
(\ref{a1}).\\
Case $n=1$: Since in that case
$u\in C^{2}(\Omega _{1})\cap C(\overline{\Omega }_{1})$, we apply the
Phragm\`{e}n-Lindel\"{o}f principle to deduce
that $u\leq 0$ whenever $f\leq 0$, $\phi \leq 0$.
(See \cite[Chapter 2]{pw}). By applying this maximum principle to $u$
and $-u$ respectively when $f=0$, we get uniqueness.
\newline
Case $n\geq 2$: Since $u\in D(\Delta _{\max }^{n}),$ we get
$ u\in C^{1}(\Omega _{n})$. Let $\Omega _{0}\subset \subset \Omega _{n}$
such that  $\mathop{\rm supp}(u-\varepsilon ) ^{+}\subset \Omega _{0}$,
$\varepsilon >0$. then $(u-\varepsilon ) ^{+}\in H_{0}^{1}(\Omega _{0})$ and
\begin{align*}
\int f(u-\varepsilon ) ^{+}
& =  \lambda \int u(u-\varepsilon ) ^{+}+\int \nabla u\nabla (u-\varepsilon )^{+} \\
& =  \lambda \int (u-\varepsilon ) (u-\varepsilon )
^{+}+\varepsilon \lambda \int (u-\varepsilon ) ^{+}
+\int_{\{ u>\varepsilon \} }| \nabla u| ^{2}  \\
& \leq   0.
\end{align*}
Hence $ u\leq \varepsilon $.
\end{proof}

We are now in position to show the well-posedness of the Cauchy problem
\eqref{ACPn}.
 If $n\neq 2,$ let $\varphi \in W^{1,1}((0,\tau );C(\Gamma _{n}))$ and
$U_{0}=(u_{0},0)\in D(A_{n})=D(\Delta _{\max }^{n})\times \{0\}$, then
\begin{equation*}
A_{n}U_{0}+\Phi _{n}(0)=(\Delta u_{0},-u_{0\mid \Gamma _{n}}+\varphi (0)).
\end{equation*}
Hence $A_{n}U_{0}+\Phi _{n}(0)\in \overline{D(A_{n})}=C_{\infty }(\overline{
\Omega }_{n})\times \{0\}$ if and only if
\begin{equation}
u_0\big|_{\Gamma _{n}}=\varphi (0). \label{a7}
\end{equation}
Assumption (\ref{a7}) becomes trivial in the case $n=2$ since we
have assumed $\varphi =0$. $\newline $On the other hand, it
follows from Theorem \ref{1.11} that $A_{n}$ is a resolvent
positive operator. Hence, by applying Theorems \ref{1.6} we obtain the
following result.

\begin{proposition} \label{1.13}
Let $n\in \mathbb{N}$.\newline \textbf{Case $n=1$:} Let $\Omega
_{1}=] 1,+\infty [.$ Then for all $u_{0}\in D(\Delta _{\max
}^{1})$ and all $\varphi \in W^{1,1}((0,\tau ))$ such that
$u_{0}(1)=\varphi (0)$, there exists a unique mild solution of
\eqref{ACPn} with $n=1$.
\newline
\textbf{Case $n=2$:} Let $\Omega _{2}=\mathbb{R}^{2}\setminus B(0,1)$.
Then for all $u_{0}\in $ $D(\Delta _{\max }^{2})$, there exists a
unique mild solution of \eqref{ACPn} with $n=2$. \newline
\textbf{Case $n\geq 3$:} Let $\Omega _{n}=\mathbb{R}^{n}\setminus K$  with
boundary $\Gamma _{n}$,  $K$ being a compact set of
$\mathbb{R}^{n}$. If $\Omega _{n}$ is regular, then for all
$u_{0}\in $ $D(\Delta _{\max }^{n})$ and all $\varphi \in
W^{1,1}((0,\tau );C(\Gamma _{n}))$ such that
$u_0\big|_{ \Gamma_{n}}=\varphi (0)$, there exists a unique mild solution of
\eqref{ACPn}.
\end{proposition}

The proof of Theorem \ref{1.9} will be complete by combining Theorem \ref
{1.10} and the following result.

\begin{proposition} \label{1.14}
(i) Let $u_{0}\in C_{\infty }(\overline{\Omega }_{1})$ and $\varphi \in
C([0,\tau ])$ such that $u_{0}(1)=\varphi (0)$, then there exists a unique
mild solution of \eqref{ACPn} with $n=1$.\newline
(ii)  Let $u_{0}\in C_{\infty }(\overline{\Omega }_{2})$, then there exists
a unique mild solution of \eqref{ACPn} with $n=2$.\newline
(iii) Assume that $\Omega _{n}=\mathbb{R}^{n}\setminus K$ is regular and
let $u_{0}\in C_{\infty }(\overline{\Omega }_{n})$ and $\varphi \in
C([0,\tau ];C(\Gamma _{n}))$ such that $u_0\big|_{ \Gamma n}=\varphi (0)$,
then there exists a unique mild solution of \eqref{ACPn}.
\end{proposition}

\begin{proof}
Choose $u_{0_{k}}\in D(\Delta _{\max }^{n})$ such that
$\,u_{0_{k}} \to u_{0} $ as $k\to \infty $ in $C_{\infty
}(\overline{ \Omega }_{n})$. Choose $\varphi _{k}\in
W^{1,1}((0,\tau);C(\Gamma _{n}))$ such that $\varphi
_{k}(0)=u_{0_k}\big|_{\Gamma _{n}}$ and $\varphi _{k}\to \varphi
$as $k\to \infty $ in $C([0,\tau ];C(\Gamma_{n} ))$. By applying
Proposition \ref{1.13} and Theorem \ref{1.10}, we deduce that
there exists a unique mild solution $u_{k}\in C_{\infty
}(\overline{\Omega }_{n})$ of $P_{\tau }(u_{0_{k}},\varphi _{k})$.
We can show that
$$
\| u_{k}\| _{C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))}\leq \max \{ \|
\varphi _{k}\| _{C([0,\tau ];C(\Gamma _{n}))},\| u_{0k}\|
_{C_{\infty }(\overline{\Omega }_{n})}\} .
$$ where $$\left\| \varphi
_{k}\right\| _{C([0,\tau ];C(\Gamma _{n}))}=\sup_{0\leq t\leq
\tau}\left\| \varphi _{k}(t)\right\| _{C(\Gamma _{n})}$$
$$\left\| u_{k}\right\| _{E_{\infty }(\overline{\Omega
}_{n})}=\sup_{0\leq t\leq \tau}\left\| u_{k}(t)\right\|
_{E_{\infty }(\overline{\Omega }_{n})}.$$
 Hence $(u_{k})_{k\geq 1}$ is a Cauchy sequence in
 $C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))$.
Let $u=\lim_{k\to \infty }u_{k}$, then
$\int_{0}^{t}u(s)ds=\lim_{k\to \infty }\int_{0}^{t}u_{k}(s)ds
\in D(\Delta _{\max }^{n})$ and
\begin{gather*}
u(t)=u_{0}+\Delta \int_{0}^{t}u(s)ds\quad\text{in }\mathcal{D}(\Omega
_{n})' \\
u(t)\big|_{ \Gamma _{n}}=\lim_{k\to \infty }\varphi
_{k}(t)=\varphi (t).
\end{gather*}
for all $t\in [ 0,\tau ]$.
\end{proof}

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\end{document}