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\markboth{\hfil Solutions to nonlinear elliptic equations \hfil EJDE--2002/05}
{EJDE--2002/05\hfil Yuandi Wang \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 05, pp. 1--16. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Solutions to nonlinear elliptic equations with a nonlocal boundary condition
 %
\thanks{ {\em Mathematics Subject Classifications:} 35Q53, 42B35, 37K10, 35K55, 35K57.
\hfil\break\indent
{\em Key words:} nonlocal boundary condition, eigenvalue,
 comparison principle, semigroup.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted June 15, 2001. Published January 8, 2002.} }
\date{}
%
\author{Yuandi Wang}
\maketitle

\begin{abstract}
 We study an elliptic equation and its evolution problem
 on a bounded domain with nonlocal boundary conditions.
 Eigenvalue problems, existence, and dynamic behavior
 of solutions for linear and semilinear equations are investigated.
 We use the comparison principle and a semigroup approach.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}

In this paper we consider the following nonlinear equation with nonlocal
boundary conditions
\begin{equation} \label{En}
\begin{gathered}
L u\equiv  -\sum_{i,j=1}^n \frac {\partial}{\partial x_i}( a_{ij}(x)
\frac {\partial u}{\partial x_j})=f(x,u),  \text{ in } \Omega \\
u|_{\partial\Omega}=  \int_{\Omega }K(x,y)u(y)\,dy
\end{gathered}
\end{equation}
and its corresponding evolution problem. Firstly, we consider the
eigenproblem for the special case
$ u|_{\partial\Omega}= k\int_{\Omega }u(y)\,dy $ with $k$ a constant.
As we know from the literature \cite{d,p95,p97,y}, the comparison
principle may not apply, unless $K(x,y)\ge 0$ and $ \int_\Omega K(x,y)\,dy<1$.
However, using special techniques one can obtain the behavior of solutions
when $K(x,y)$ alternates  signs \cite{d,p95,p97}.
But we wondered how the boundary kernel $K(x,y)$ influences results such as
those on the eigenvalues and on the decay of solutions for evolution
equations. Because these questions are not easy, we expect to have only a
partial answer by considering a simple case.  We will find that there are no negative
eigenvalues unless $k>1/|\Omega|$. Also we will obtain some estimates on
the eigenvalues. In section \ref{s3}, we prove the existence of solutions
for linear problem. In  section \ref{s4}, the method of quasilinearization
is used to prove that monotonic iterative sequences converge quadratically
to the solution of the nonlinear problem.
Lastly, we discuss the long time behavior of solution in
Sobolev-Slobodeckii spaces.

Throughout this paper  we assume that $\Omega\subset\mathbb{R}^n$ is a bounded
domain with $C^{2+\mu}$-boundary $\partial\Omega$, $a_{ij}\in C^{1+\mu}$
($i,j=1,2,\cdots,n$) with $\mu\in(0,1)$
and that there exists a positive number $\alpha$ such that
\begin{equation}
 \sum_{i,j=1}^n a_{ij}(x)\xi_i\xi_j\ge\alpha\sum_{i=1}^n\xi_i^2, \quad\forall\,
  (x,\xi_1,\cdots,\xi_n)\in\overline{\Omega}\times\mathbb{R}^n.\label {el}
\end{equation}

\section{Eigenvalue Problems\label{s2}}

Let us consider a special eigenvalue problem for (\ref{En}) with
$K(x,y)= k$ a constant.
\begin{equation}
\begin{gathered}
L \varphi(x) \equiv  -\sum_{i,j=1}^n \frac {\partial}{\partial x_i}( a_{ij}(x)
\frac {\partial \varphi(x)}{\partial x_j})=\lambda\varphi(x) , \quad
 \text{in } \Omega \\
\varphi|_{\partial\Omega}= k \int_{\Omega }\varphi(y)\,dy.
\end{gathered} \label{eigen}
\end{equation}
We expect to obtain some information about the relation between the
eigenvalue $\lambda$ and the constant $k$.
First integrate over $\Omega$ on the first equation of (\ref{eigen}):
\begin{equation}
-\int_{\partial\Omega} \sum_{i,j=1}^n a_{ij}\frac {\partial \varphi}{\partial x_j}\cos({ \nu}, x_i)\,dS=\lambda\int_{\Omega}\varphi(x)\,dx. \label{1}
\end{equation}
Then multiplying by $\varphi(x)$ and integrate again
\begin{eqnarray}
\int_{\Omega} \varphi L \varphi\,dx&=& - \int_{\partial\Omega}
\sum_{i,j=1}^n a_{ij}\frac {\partial \varphi}{\partial x_j}
\cos({ \nu}, x_i)\,dS\cdot \gamma(\varphi)
+ \int_{\Omega}\sum_{i,j=1}^n a_{ij}\frac{\partial \varphi}{\partial x_i}
\frac{\partial \varphi}{\partial x_j}\, dx\nonumber \\
&=& \lambda\int_{\Omega}\varphi^2(x)\,dx,  \label{2}
\end{eqnarray}
where $\gamma$ is the trace operator $\gamma(\varphi)=\varphi|_{\partial\Omega}$.
 Combining the above equations with the boundary condition in (\ref{eigen}),
 we have
\begin{equation}
\lambda\Big\{ \int_{\Omega}\varphi^2\,dx -k (\int_{\Omega} \varphi\,dx)^2
\Big\}=\int_{\Omega}\sum_{i,j=1}^n a_{ij}\frac{\partial \varphi}{\partial x_i}
\frac{\partial \varphi}{\partial x_j}\, dx\ge \alpha \int_{\Omega} |\nabla
\varphi|^2\,dx.\label{3}
\end{equation}
It follows directly from Jensen's  inequality and (\ref{3}) that if there
exists an eigenvalue $\lambda<0$, then $k>1/|\Omega|$.
Moreover, for $f_1$ and $f_2\in C(\overline{\Omega})$,
Cauchy's inequality
\begin{equation}
\big( \int_\Omega f_1(x)f_1(x)\,dx\big)^2\le \int_\Omega f_1^2(x)\,dx
\int_\Omega f_2^2(x)\,dx \label{cauchy}
\end{equation}
becomes equality if and only if $f_1(x)=lf_2(x)$, in $ \overline{\Omega}$.
Therefore, if $\lambda_0=0$ is an eigenvalue, then its corresponding
eigenfunction is $\varphi_0=1$. This implies that $k=1/|\Omega|$.
On the other hand, if $k=1/|\Omega|$ then $0$ is an eigenvalue of (\ref{eigen}).
Hence, all eigenvalues of (\ref{eigen}) are positive when $k<1/|\Omega|$.
Thus, we have
\begin{proposition} For the linear eigenproblem (\ref{eigen}) the following
holds:
\begin{description}
\item{\text{i)}} $0$ is an eigenvalue (with eigenfunction 1) if and only if
$k=1/|\Omega|$
\item\text{ii)} If there exists one eigenvalue $\lambda<0$,
 then  $k>1/|\Omega|$
\item\text{iii)} If $k<1/|\Omega|$ then all eigenvalues of (\ref{eigen})
are positive.
\end{description}\label{prop}
\end{proposition}

\begin{proposition} The linear eigenproblem (\ref{eigen}) has at most one
negative eigenvalue.
\end{proposition}

\noindent{\it Proof.} First, we claim  that the eigenfunction $\varphi(x)$
corresponding  to one negative eigenvalue $\lambda$ does not alternate its
sign on $\overline\Omega$.

Actually, the positive maximum $\varphi(x_M)$ can not be attained at
$x_M\in\Omega$, otherwise
\begin{equation}
 0\le L\varphi(x_M)= -\sum_{i,j=1}^n \frac {\partial}{\partial x_i}( a_{ij}\frac {\partial \varphi(x_M)}{\partial x_j})=\lambda\varphi(x_M)<0, \label{m}
\end{equation}
this is impossible. So, $\varphi(x_M)>0$ can be attained only on the boundary
$\partial\Omega$. Also  $\varphi(x)$ can not have a negative minimum,
$\varphi(x_m)<0$, in $\Omega$: It is easy to get  contradiction as the one
above. Hence, if $\varphi(x)$ is an eigenfunction with positive maximum
on $\partial\Omega$ for a negative eigenvalue $\lambda$,
then $\varphi(x)\ge 0$ for all $x\in \overline\Omega$.

Similarly, if $\varphi(x)$ is an eigenfunction with negative minimum on
$\partial\Omega$ for a negative eigenvalue $\lambda$, then
$\varphi(x)\le 0$ for all $x\in \overline\Omega$.

For $k>1/|\Omega|$, we suppose that there exist two eigenvalues
$\lambda_1<\lambda_2<0$ and that $\varphi_1(x)$ and $\varphi_2(x)$ are
the corresponding eigenfunctions, with $\varphi_1(x)\ge 0$,
$\varphi_2(x)\ge 0$, satisfying
$\varphi_1|_{\partial\Omega}=\varphi_2|_{\partial\Omega}$.
Then the positive maxima  for  $\varphi_1(x)$ and $\varphi_2(x)$ can be attained
only on $\partial\Omega$. We claim that  $\varphi_1(x)\le \varphi_2(x)$ on
$\overline{\Omega}$. If it is not true, there is $x^*\in\Omega$ such that
$\varphi_1(x^*)>\varphi_2(x^*)$, with  $x^*$ a positive maximum point
for $\varphi_1-\varphi_2$, then
$$0\le L (\varphi_1-\varphi_2)|_{x^*}=\lambda_1\varphi_1(x^*)
-\lambda_2\varphi_2(x^*).
$$
From $\lambda_1<\lambda_2<0$, it follows that
$\varphi_1(x^*)\le |\frac {\lambda_2}{\lambda_1}| \varphi_2(x^*)<\varphi_1(x^*)$,
which is a contradiction.

The inequality  $\lambda_1<\lambda_2$ implies $\varphi_1(x)\le \varphi_2(x)$,
but
$$0=\varphi_1|_{\partial\Omega}-\varphi_2|_{\partial\Omega}
=k \int_\Omega(\varphi_1(y)-\varphi_2(y))\,dy\le 0\,.
$$
There exists only one possibility: $\varphi_1(x)=\varphi_2(x)$ on
$\overline\Omega$.  Therefore, $\lambda_1=\lambda_2$. \hfill $\Box $\smallskip

Naturally, the next step is to estimate the minimal eigenvalue for (\ref{eigen}).
As mentioned, if $k=1/|\Omega|$ then the minimal eigenvalue $\lambda=0$.
Now we consider the issue for $k< 1/|\Omega|$.

\begin{proposition}  Let $d$ be the diameter of  $\Omega$. Then
\begin{description}
\item{\text{i)}}  $ \lambda\ge \frac {2\alpha}{nd^2}
\big[ 1+\frac {|\Omega|}{1-k|\Omega|}(k-\frac 1{d^n})\big]$ for $k\le 0$
\item\text{ii)}  $ \lambda\ge  \frac {2\alpha}{nd^2}
\big[ 1-\frac {|\Omega|}{1-k|\Omega|}(k+\frac 1{d^n})\big]$ for $0<k<1/|\Omega|$.
\end{description}
\end{proposition}

\noindent{\it Proof.}  Let $\varphi(x)$ be an eigenfunction for the minimal
eigenvalue $\lambda$. Let $D$ be the cube in $\mathbb{R}^n$ with edges of
length $d$ containing $\Omega$. Extend $\varphi$ into $D$ with
$\gamma(\varphi)= k\int_\Omega \varphi(y)\,dy$, denote the extension by
\begin{equation}
\tilde\varphi(x)=\left\{ \begin{array}{ll}
\varphi(x), & \text{in } \Omega \\
k \int_\Omega\varphi\,dx, & \text{in } D-\Omega.
\end{array}
\right.
\end{equation}
Define  $\Phi=\int_\Omega\varphi\,dx$, obviously,
\[ \int_D\tilde\varphi^2\, dx=\int_\Omega\varphi^2\,dx+k^2 \Phi^2|D-\Omega|.\]
Applying Poincar\'e's inequality in the cube $D$, we have
\[ \int_\Omega\varphi^2\,dx+k^2 \Phi^2|D-\Omega|
 \le \frac 1{d^n}( k\Phi |D-\Omega|+\Phi)^2+\frac {nd^2}2\int_\Omega|\nabla\varphi|^2\,dx. \]
From the elliptic hypothesis (\ref{el}) and (\ref{3}),
\[ \lambda\frac {nd^2}{2\alpha}\big[\int_\Omega\varphi^2\,dx-k\Phi^2\big]
\ge \int_\Omega\varphi^2\,dx+k^2\Phi^2|D-\Omega|
- \frac {\Phi^2( k|D-\Omega|+1)^2}{d^n}.\]
Let $\int_\Omega\varphi^2\,dx=1$, take note of
$\Phi^2=(\int_\Omega\varphi\,dx)^2<|\Omega|$, Since
$\varphi(x)$ is not constant for $k\not= 1/|\Omega|$
(see Proposition \ref{prop}), then $\Phi^2\in[0,|\Omega|)$.
By the assumption $k<1/|\Omega|$,
\begin{eqnarray}
 \lambda  &\ge& \frac {2\alpha}{nd^2(1-k\Phi^2)}
\big[1+\Phi^2(k^2 |D-\Omega|- \frac {( k|D-\Omega|+1)^2} {d^n})\big]\nonumber \\
 &=& \frac {2\alpha}{nd^2}+\frac {2\alpha \Phi^2}{nd^2(1-k\Phi^2)}( k+k^2|D
 -\Omega|-\frac{k^2|D-\Omega|^2+2k|D-\Omega|+1}{d^n})\nonumber \\
 &\ge& \frac {2\alpha}{nd^2}+ \frac {2\alpha \Phi^2}{nd^2(1-k\Phi^2)}
 ( k-\frac {2k|D-\Omega|+1}{d^n} ),  \label{k}
 \end{eqnarray}
in the last inequality above, the relation $|D-\Omega|\le |D|=d^n$ is used. It is not difficult to get that the nonnegative function $h(t)=\frac t{1-kt}$ reach its maximum $\frac {|\Omega|}{1-k|\Omega|}$ at $t=|\Omega|$ (for $t\in[0,|\Omega|]$). \par
Hence, if $0\le k<1/|\Omega|$, then
\begin{equation}
\lambda\ge \frac {2\alpha}{nd^2}-\frac {2\alpha}{nd^2}\frac {|\Omega|}{1-k|\Omega|}( k+\frac{1}{d^n}).\label{ki}
\end{equation}
If $k<0$, then
\begin{equation}
\lambda\ge \frac {2\alpha}{nd^2}+\frac {2\alpha}{nd^2}\frac {|\Omega|}{1-k|\Omega|}( k-\frac{1}{d^n}).\label{kii}
\end{equation}
The assertion are proved.  \hfill $\Box $ \smallskip

Because the domain is smooth, $|\Omega|/d^n<1$. Certainly, (\ref{kii})
deduces $\lambda>0$ for $k<0$. On the other hand, the estimate (\ref{k})
is more accurate than ({\ref{ki}), one can obtain easily from  (\ref{k})
that  $\lambda>0$ provided with  $k<1/(2|\Omega|)$.
Now we see a special example in one-dimension:
\begin{equation}
-\phi^{\prime\prime}=\rho\phi, \quad x\in (-\pi,\pi);\quad
\phi(-\pi)=\phi(\pi)=k\int^\pi_{-\pi}\phi(x)\,dx. \label{e1}
\end{equation}
For this problem, the relationship between $k$ and $\rho$ is
as follows:
If $\rho<0$, then $k=\frac12 \sqrt{-\rho}\coth (\pi\sqrt{-\rho})$;
if $\rho=0$, then $k=1/(2\pi)$; and if $\rho>0$ and not the square
of an integer, then $k=\frac12 \sqrt{\rho}\cot (\pi\sqrt{\rho})$. See
Figure 1, where the eigenvalues correspond to the values $k$ for which
the graph crosses the horizontal axis.

\begin{figure}[ht]
 \setlength{\unitlength}{8mm}
\begin{picture} (10,4)(-5,-2)
\put(0,-1){\vector(0,1){3}}
\put(-3,0){\vector(1,0){11}}
\qbezier(-3,.9)(-1,.7)(0,.159)
\qbezier(0,0.159)(0.125,0.09)(0.25,0)
\qbezier(0.25,0)(.7,-.3)(0.8,-2)
\qbezier(1.16,2)(1.4,.3)(2.25,0)
\qbezier(2.25,0)(3.2,-.3)(3.48,-2)
\qbezier(4.64,2)(5.25,.3)(6.25,0)
\qbezier(6.25,0)(7.25,-.3)(7.88,-2)
\put(-2,-.1){\line(0,1){.2}} \put(-2.25,-.4){$\scriptstyle -2$}
\put(-1,-.1){\line(0,1){.2}} \put(-1.25,-.4){$\scriptstyle -1$}
\put(0,-.1){\line(0,1){.2}} \put(-.058,-.4){$\scriptstyle 0$}
\put(1,-.1){\line(0,1){.2}} \put(.92,-.4){$\scriptstyle 1$}
\put(2,-.1){\line(0,1){.2}} \put(1.92,-.4){$\scriptstyle 2$}
\put(3,-.1){\line(0,1){.2}} \put(2.92,-.4){$\scriptstyle 3$}
\put(4,-.1){\line(0,1){.2}} \put(3.92,-.4){$\scriptstyle 4$}
\put(5,-.1){\line(0,1){.2}} \put(4.92,-.4){$\scriptstyle 5$}
\put(6,-.1){\line(0,1){.2}} \put(5.92,-.4){$\scriptstyle 6$}
\put(7,-.1){\line(0,1){.2}} \put(6.92,-.4){$\scriptstyle 7$}
 \put(7.92,-.3){$\scriptstyle \rho $}
 \put(-.4,1.8){$k$}
\end{picture}
\caption{$k$ as a function of $\rho$ for problem (\ref{e1}) }
\end{figure}



\section{Linear Problems\label{s3}}

We investigate the linear problem before using the monotonic iteration method
for nonlinear equations. Throughout this sections we assume that $k<1/|\Omega|$.
To get the existence of solutions for the linear problem
\begin{equation} \label{P1}
\begin{gathered}
(L+c) u\equiv  -\sum_{i,j=1}^n \frac {\partial}{\partial x_i}
\big( a_{ij}(x)\frac {\partial u}{\partial x_j}\big)+c(x)u=F(x),
\quad \text{in } \Omega \\
u|_{\partial\Omega}= k \int_{\Omega }u(y)\,dy,
\end{gathered}
\end{equation}
we discuss the Dirichlet problem
\begin{equation} \begin{gathered}
 (L+c)U+\frac {k\,c(x)}{1-k|\Omega|}\int_\Omega U(x)\,dx=F(x), \quad
  \text{in }\Omega \\
U|_{\partial\Omega}=0 . \end{gathered}\label{diri}
\end{equation}

\begin{lemma}
For  $F(x),c(x)\in C^\mu(\overline{\Omega})$ and $c(x)\ge 0$, the
linear problem (\ref{diri}) admits a unique solution $u\in C^{2+\mu}$.
\end{lemma}

\noindent{\it Proof.} From the theory on elliptic equations \cite{gt},
we know that (\ref{diri}) has a unique solution when $k=0$,
{\it i.e.} the operator $L+c$ has a compact inverse operator $(L+c)^{-1}$.
According to Riesz-Schauder theory \cite{yosida}, if 0 is not an eigenvalues
for the eigenproblem
\begin{equation} \begin{gathered}
 (L+c)\varphi+\frac {k\,c(x)}{1-k|\Omega|}\int_\Omega \varphi(x)\,dx
 =\lambda\varphi , \quad \text{in } \Omega \\
\varphi|_{\partial\Omega}=0 ,
\end{gathered} \label{e}
\end{equation}
then (\ref{diri}) has a unique solution. Now we show that 0 is
an eigenvalue of (\ref{e}). Otherwise, the problem
\begin{equation}
\begin{gathered}
 (L+c)\varphi+\frac {k\,c(x)}{1-k|\Omega|}\int_\Omega \varphi(x)\,dx=0 ,
 \quad \text{in }\Omega \\
\varphi|_{\partial\Omega}=0
\end{gathered} \label{0}
\end{equation}
has a solution $\varphi(x)\not\equiv 0$
($l\varphi$ is also a solution for all $l\in\mathbb{R}$).
From the maximum principle, $\int_\Omega \varphi(x)\,dx\not=0$.
Denote $\varphi_0= {\varphi}/\int_\Omega \varphi(x)\,dx$.
Then the Dirichlet problem
\[ (L+c)\varphi_0(x)=-\frac{k\,c(x)}{1-k|\Omega|},\quad
\varphi_0|_{\partial\Omega}=0 \]
has a unique solution $\varphi_0(x)$ for any $c(x)\ge0$ and $k$.
The maximal principle for nonhomogeneous equations  \cite[chpater 3]{gt}
shows that there is a constant $C$, independent of the nonhomogeneous term
$-\frac {k\,c(x)}{1-k|\Omega|}$, such that
\begin{equation*}
\sup_\Omega \varphi_0(x)\le \sup_{\partial\Omega}\varphi_0(x)
+\frac {C}{\alpha}\sup_\Omega\big\| \frac{-k\,c(x)}{1-k|\Omega|}
\big\| = \big\| \frac{Ck}{\alpha(1-k|\Omega|)}\big\|
\sup_\Omega c(x) \stackrel{k\to 0}{\longrightarrow}0.
\label{nonh}
\end{equation*}
But $\int_\Omega \varphi_0(x)\, dx=1$ for all $k< 1/|\Omega|$ and $c\ge 0$,
this contradicts the above inequality.
Therefore, there is no eigenfunction $\varphi\not\equiv 0$ and
$0$ is not the eigenvalue of (\ref{e}).
It follows that (\ref{diri}) has a unique solution  $U\in C^{2+\mu}$.
\hfill$\Box$ \smallskip

In the above proof, we observe that the mapping
$\tilde k(\varphi)\equiv \frac {k\,c(x)}{1-k|\Omega|} \int_\Omega\varphi(x)\,dx$
with the domain and the range $C^{\mu}(\overline\Omega)$, 
is  linear  and bounded.

The proof consists of finding an $H^1_0(\Omega)$-solution, then to
strengthening the regularity by estimates and Sobolev inequalities.

Take $u=U+\frac k{1-k|\Omega|}\int_\Omega U\,dx$ with $U$ being the solution
(\ref{diri}), then
\[ (L+c)u=(L+c)\big(U+\frac k{1-k|\Omega|}\int_\Omega U\,dx\big)=F(x), \]
and
\begin{eqnarray*} u|_{\partial\Omega}&=&\frac k{1-k|\Omega|}
\int_\Omega U\,dx
=\frac {k(1-k|\Omega|)+k^2|\Omega|}{1-k|\Omega|}\int_\Omega U\,dx \\
&=& k\int_\Omega U dx+\frac {k^2|\Omega|}{1-k|\Omega|}\int_\Omega U\,dx\\
&=&k\int_\Omega\big(U+\frac k{1-k|\Omega|}\int_\Omega U\,dx\big)dx
=k\int_\Omega u dx.
\end{eqnarray*}

\begin{theorem}
For $c>0$ and $F(x)\in C^\mu$, the linear nonlocal boundary problem (\ref{P1})
admits a unique solution $u\in C^{2+\mu}$.
\end{theorem}

\noindent{\it Proof}. We prove only the uniqueness. If there are two solutions,
 then the problem
\[ L u+cu= 0, \quad \text{for }\ x\in\Omega;\quad
u|_{\partial\Omega}= k\int_\Omega u(x)dx \]
has nonzero solution. This is not possible for $c(x)>0$ and $u(x)$ being
constant on $\partial\Omega$. \hfill$\Box$\smallskip

From the above discussion, one can see that when $k\to 0$, the solution of
(\ref{P1}) approaches $U_0$, the solution of
\begin{equation} \begin{gathered}
(L+c)u\equiv  -\sum_{i,j=1}^n \frac {\partial}{\partial x_i}( a_{ij}(x)
\frac {\partial u}{\partial x_j})+c(x)u=F(x), \quad \text{in }\Omega \\
u|_{\partial\Omega}=0.
\end{gathered} \label{o}
\end{equation}
More generally, if $k=K(x,y)$ is smooth enough on
$\overline\Omega\times\overline\Omega$, then the solution of the corresponding
linear problem with boundary condition
$u|_{\partial\Omega}=\int_\Omega K(x,y)u(y)dx$ approaches the solution of
(\ref{o}) when
$\epsilon\equiv  \max_{\overline\Omega\times\overline\Omega}|K(x,y)| \to 0$.
We assume that $K\in C^{1+\mu}(\overline\Omega)\times C(\overline\Omega)$
satisfies
\begin{equation}
K(x,y)\ge 0,\quad \int_{\Omega} K(x,y)\,dy <1, \quad \text{for }
 x\in\partial\Omega,y\in\overline\Omega. \label{K}
\end{equation}
Now we give a comparison and an existence result.

\begin{lemma} \label{max}
Let $K(x,y)$ satisfy (\ref{K}), and $u\in C^{2}(\Omega)\cap C(\overline\Omega)$ satisfy
\begin{equation*}
L u+cu\le 0, \quad \text{for }\ x\in\Omega;\quad
u|_{\partial\Omega}\le \int_\Omega K(x,y)u(y)dx ,
\end{equation*}
with $c(x)\ge 0$. Then $u(x)\le 0$ for all $x\in\overline\Omega$.
\end{lemma}

\begin{lemma} Let $K(x,y)$ satisfy (\ref{K}), $C\in C^\mu(\overline{\Omega})$,
 and $c(x)\ge 0$, then the linear problem
\begin{equation*}
L u+cu=F(x), \quad \text{for } x\in\Omega;\quad
u|_{\partial\Omega}\le \int_\Omega K(x,y)u(y)dx \label{LK}
\end{equation*}
has a unique solution $u\in C^{2+\mu}(\overline\Omega)$ for all
$F\in C^\mu(\overline\Omega)$.
\end{lemma}

\noindent{\it Proof}. The assertion in Lemma \ref{max} can be proved
using a method similar to the one in  \cite[Lemma 3.1]{p95}.
The existence is deduced from \cite[Theorem 3.3]{p95}. \hfill$\Box$\smallskip

\section{Nonlinear Problems\label{s4}}

We use the method of upper and lower solutions to discuss the existence of
solutions for nonlinear problem (\ref{En}). In this section, we assume
that $K(x,y)$ satisfies (\ref{K}).

A pair of a lower solution $\underline{u}(x)$ and an upper solution
$\overline{u}(x)$ in $C^2(\Omega)\cap C(\overline{\Omega})$ of (\ref{En})
is defined as
\begin{gather}
L \underline{u}\le f(x,\underline{u}), \quad
\underline{u}|_{\partial\Omega}\le \int_{\Omega}K(x,y)\underline{u}(y)\,dy; \\
L \overline{u}\ge f(x,\overline{u}), \quad
\overline{u}|_{\partial\Omega}\ge \int_{\Omega}K(x,y)\overline{u}(y)\,dy.
\end{gather}
We construct two iteration sequences $\{\underline{u}_n\}$ and
$\{\overline{u}_n\}$ starting with $\underline{u}=\underline{u}_0$
and  $\overline{u}=\overline{u}_0$ as follows
\begin{gather}
L \underline{u}_{n}+c\underline{u}_{n}= c\underline{u}_{n-1}+f(x,
\underline{u}_{n-1}), \quad \underline{u}_{n}|_{\partial\Omega}
= \int_{\Omega}K(x,y)\underline{u}_n(y)\,dy; \label{it1}\\
L \overline{u}_{n}+c\overline{u}_{n}= c\overline{u}_{n-1}
+f(x,\overline{u}_{n-1}), \quad \overline{u}_{n}|_{\partial\Omega}
= \int_{\Omega}K(x,y)\overline{u}_n(y)\,dy. \label{it2}
\end{gather}
Though the construction of iteration sequences are not the same as that
in \cite{p95}, the convergence can be proved by an analogous argument.

\begin{theorem}
If there exists one ordered pair of a lower and an upper solution
$\underline{u}$ and $\overline{u}$, $\underline{u}\le\overline{u}$,
and there is a constant $c>0$ such that
\begin{equation*}
f(x,u)-f(x,v)\ge - c(u-v),\quad \text{for}\ u\geq v,\quad
 \text{and}\quad u,v\in[\underline{u},\overline{u}],
\end{equation*}
where $u\in [\underline{u},\overline{u}]$ means
$\underline{u}(x)\le u(x)\le\overline{u}(x)$, for all $x\in\overline{\Omega}$.
Then the problem (\ref{En}) has  solutions $\underline{u}_s$ and
$\overline{u}_s$ satisfying
$\underline{u}(x)\le \underline{u}_s\le \overline{u}_s\le \overline{u}(x)$.
\end{theorem}

\noindent{\it Proof}. According to the definition of iteration sequences $\{ \underline{u}_n\}$ and $\{\overline{u}_n\}$ in (\ref{it1}) and (\ref{it2}), we get
$$
L (\underline{u}_{1}-\underline{u}_0)+c(\underline{u}_{1}-\underline{u}_0)
\geq 0 , \quad (\underline{u}_{1}-\underline{u}_0)|_{\partial\Omega}\geq
\int_{\Omega}K(x,y)(\underline{u}_{1}-\underline{u}_0)(y)\,dy.
$$
 From Lemma \ref{max}, it follows that $\underline{u}_{1}\ge\underline{u}_0$.
Similarly
\begin{gather*}
L (\underline{u}_{2}-\underline{u}_1)+c(\underline{u}_{2}-\underline{u}_1)
= c(\underline{u}_{1}-\underline{u}_0)+f(x,\underline{u}_{1})
-f(x,\underline{u}_0)\geq 0 ,  \\
(\underline{u}_{2}-\underline{u}_1)|_{\partial\Omega} \geq  \int_{\Omega}K(x,y)(\underline{u}_{2}-\underline{u}_1)(y)\,dy.\nonumber
\end{gather*}
As in the discussion above, one can prove that the sequence
$\{\underline{u}_n\}$ is monotone nondecreasing, the sequence
$\{\overline{u}_n\}$ is monotone non-increasing. and
\begin{gather*}
L (\overline{u}_1-\underline{u}_{1})+c(\overline{u}_1-\underline{u}_{1})
= c(\overline{u}_1-\underline{u}_{0})+f(x,\overline{u}_{0})-f(x,
\underline{u}_0)\geq 0 ,  \\
(\overline{u}_{1}-\underline{u}_1)|_{\partial\Omega}
 \geq \int_{\Omega}K(x,y)(\overline{u}_{1}-\underline{u}_1)(y)\,dy.
\end{gather*}
Then, $\overline{u}_1\geq\underline{u}_1$, generally,
\[ \underline{u}=\underline{u}_0\le \underline{u}_1\le\cdots \le
\underline{u}_n\le \overline{u}_n \le \cdots\le \overline{u}_1=\overline{u}_0,
\]
it follows that $\{\underline{u}_n\}$ and  $\{\overline{u}_n\}$ converge,
respectively, to some limits $\underline{u}_s$ and $\overline{u}_s$,
and satisfy the relation  $\underline{u}_s\le \overline{u}_s$.
A regularity argument shows that  $\underline{u}_s$ and $\overline{u}_s$
are solutions of (\ref{En}) \cite{pbook}, the details are omitted here.
\hfill$\Box$\smallskip

In fact, $\underline{u}_s$ and $\overline{u}_s$ are the minimal and the maximal
solution in $[\underline{u},\overline{u}]$, it is easy to obtain that
$\underline{u}_s\le u\le\overline{u}_s$ if (\ref{En}) has another solution
$u\in [\underline{u},\overline{u}]$. Certainly, $\underline{u}_s$ and
$\overline{u}_s$ may be equal, for example, when $f(x,u)$ is monotone
non-increasing on $u$, $\underline{u}_s=\overline{u}_s$ \cite{p95}.

Furthermore, assume that
\begin{description}
\item{H1}: $ f(x,u)=F(x,u)+G(x,u) $ and that $F_u$, $G_u$, $F_{uu}$, $G_{uu}$
exist, are continuous, and $F_{uu}\ge 0$, $G_{uu}\le 0$ on
$\overline\Omega\times \text{R}$.
\end{description}
Employing the quasilinearization idea in \cite{lv}, we have
\begin{theorem}
Under assumption H1, if there exist one pair of ordered lower and upper
solutions $\underline{u}$ and $\overline{u}$ for the problem
(\ref{En}), and there is a positive constant $c$ such that
\begin{equation*}
 F_u(x,\overline{u})+G_u(x,\underline{u})\le -c<0. \label{hf}
\end{equation*}
Then there exist monotone sequences $\{ \underline{u}_n\}$,
$\{\overline{u}_n\}\in C^{2+\mu}(\overline\Omega)$ such that
$\underline{u}_n\to u \gets\overline{u}_n$, $u$ is the unique solution of
(\ref{En}) satisfying $\underline{u}\le u\le \overline{u}$, and the
convergence is quadratic.
\end{theorem}

\noindent{\it Proof}. The hypotheses $ F_{uu}\ge 0$ and
$G_{uu}\le 0$, yield inequalities
\begin{equation}
\begin{gathered}
F(x,u)-F(x,v)\ge F_u(x,v)(u-v),\\
G(x,u)-G(x,v)\ge G_u(x,u)(u-v),
\end{gathered}
 \quad \text{for } u\ge v. \label{convex}
\end{equation}
We construct new iterative sequences $\{\underline{u}_n\}$ and
$\{ \overline{u}_n\}$, starting with $\underline{u}_0=\underline{u}$
and $\overline{u}_0=\overline{u}$, by linear equations
\begin{equation}
\begin{gathered}
L\underline{u}_{n}
=F(x,\underline {u}_{n-1})+G(x,\underline {u}_{n-1})
 +(F_u(x,\underline {u}_{n-1})+G_u(x,\overline {u}_{n-1}))
 (\underline{u}_n-\underline {u}_{n-1}), \\
L\overline{u}_{n}
= F(x,\overline {u}_{n-1})+G(x,\overline {u}_{n-1})
 +(F_u(x,\underline {u}_{n-1})+G_u(x,\overline {u}_{n-1}))(\overline{u}_n
 -\overline {u}_{n-1}); \\
\underline{u}_{n}|_{\partial\Omega}
= \int_{\Omega}K(x,y)\underline{u}_{n}\,dx,\quad
  \overline{u}_{n}|_{\partial\Omega}=\int_{\Omega}K(x,y)\overline{u}_{n}\,dx.
\end{gathered} \label{i1}
\end{equation}
It is obvious that
\begin{equation}
 F_u(x,\overline{u}_n)+G_u(x,\underline{u}_n)\le -c <0 \ \ \text{for \ } \underline{u}\le \underline{u}_{n-1}, \,\overline{u}_{n-1}\le\overline{u};
\label{-c}
\end{equation}
$(n=1,2,\cdots,)$. As we know, for $\eta\in C^{2}(\overline\Omega)$ with
$\underline{u}\le\eta\le \overline{u}$, the function
$h(x)=F(x,\eta)+G(x,\eta)-F_u(x,\eta)\eta-G_u(x,\eta)\eta $
belongs to $C^{\mu}(\overline\Omega)$ \cite{lv}.
Hence the linear problems (\ref{i1}) have unique solutions
$\{\underline{u}_n\}$ and $\{ \overline{u}_n\}$ in $C^{2+\mu}(\overline\Omega)$.
Also,
\begin{gather*}
 L(\underline{u}_{1}-\underline{u}_0)\ge (F_u(x,\underline{u}_0)+G_u(x,
\underline{u}_0))(\underline{u}_1-\underline{u}_{0}), \\
(\underline{u}_{1}-\underline{u}_0)|_{\partial\Omega}\ge k\int_\Omega
(\underline{u}_1-\underline{u}_0)\,dx .
\end{gather*}
Taking notice of (\ref{-c}), Lemma \ref{max} yields
$\underline{u}_{0}\le\underline{u}_1$.

A similar argument gives $\overline{u}_1\le\overline{u}_0$. We show next
that $\underline{u}_1\le\overline{u}_0$ on $\overline\Omega$.
Using the inequalities in (\ref{convex}), we get
\begin{eqnarray*}
\lefteqn{L (\overline{u}_0-\underline{u}_1)}\\
&\ge & F(x,\overline{u}_0)+G(x,\overline{u}_0)- F(x,\underline{u}_0)-G(x,
\underline{u}_0)\\
&& -(F_u(x,\underline{u}_0)+G_u(x,\underline{u}_0))(\underline{u}_1
-\underline{u}_{0}) \\
&\ge & (F_u(x,\underline{u}_0)+ G_u(x,\overline{u}_0))(\overline{u}_0
-\underline{u}_0)-(F_u(x,\underline{u}_0)+G_u(x,\underline{u}_0))
(\underline{u}_1-\underline{u}_{0}) \\
&\ge& (F_u(x,\overline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_0
-\underline{u}_{1})+(G_u(x,\overline{u}_0)-G_u(x,\underline{u}_0)
(\underline{u}_1-\underline{u}_0) \\
&\ge &  (F_u(x,\overline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_0
-\underline{u}_{1}).
\end{eqnarray*}
The condition $G_{uu} \le 0$ is used for the last inequality.
Lemma \ref{max} implies $\underline{u}_1\le\overline{u}_0$.
Similarly one can get that $\underline{u}_0\le\overline{u}_1$.
Also, since that $F_u(x,u)$ and $G_u(x,u)$ are nondecreasing and
non-increasing in $u$ respectively, from (\ref{convex}) we arrive at
\begin{eqnarray*}
 \lefteqn{ L(\underline{u}_{2}-\underline{u}_1) }\\
&=&   F(x,\underline{u}_1)+G(x,\underline{u}_1)- F(x,\underline{u}_0)
+G(x,\underline{u}_0) \\
&& +(F_u(x,\underline{u}_1)
+G_u(x,\overline{u}_1))(\underline{u}_2-\underline{u}_{1})-
(F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))(\underline{u}_1
-\underline{u}_{0})  \\
&\ge& (F_u(x,\underline{u}_1)+G_u(x,\underline{u}_1))(\underline{u}_1
-\underline{u}_{0})-(F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))
(\underline{u}_1-\underline{u}_{0})  \\
&& +(F_u(x,\underline{u}_1)+G_u(x,\overline{u}_1))(\underline{u}_2
-\underline{u}_{1}) \\
&\ge&  (F_u(x,\underline{u}_1)+G_u(x,\overline{u}_1))(\underline{u}_2
-\underline{u}_{1}).
\end{eqnarray*}
It then follows  by Lemma \ref{max} that $\underline{u}_1\le\underline{u}_2$
on $\overline{\Omega}$. And $\overline{u}_2\le\overline{u}_1$ can be
obtained similarly. In the same way, we get
\begin{eqnarray}
L(\overline{u}_{1}-\underline{u}_1)
&=&   F(x,\overline{u}_0)+G(x,\overline{u}_0)
+(F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_1
-\overline{u}_{0})\nonumber\\
&&- F(x,\underline{u}_0)-G(x,\underline{u}_0) - (F_u(x,\underline{u}_0)
+G_u(x,\overline{u}_0))(\underline{u}_1-\underline{u}_{0}) \nonumber \\
&\ge& (F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_0
-\underline{u}_{0})\nonumber \\
&& +(F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_1
-\overline{u}_{0}-\underline{u}_1+\underline{u}_{0})\nonumber \\
&\ge&  (F_u(x,\underline{u}_0)+G_u(x,\overline{u}_0))(\overline{u}_1
-\underline{u}_1).\nonumber
\end{eqnarray}
Hence, $\underline{u}_1\le\overline{u}_1$. From a similar argument,
we can show $\underline{u}_2\le\overline{u}_2$. By the above process,
step by step, we have
\[ \underline{u}_0\le \underline{u}_1\le\underline{u}_2\le\cdots
\le\underline{u}_n\le\overline{u}_n\le\cdots\le \overline{u}_2
\le\overline{u}_1\le\overline{u}_0. \]
The convergence for $\{\underline{u}_n\}$ and $\{\overline{u}_n\}$, and
regularity for the limits can be proved by a similar process to \cite{lv}
or \cite{pbook}, we omit the details.
The uniqueness of the solution  follows from the assumption  (\ref{hf}).
Hence, we obtain that $\{\underline{u}_n\}$ and $\{\overline{u}_n\}$
converge, nondecreasing and nonincreasing respectively, to the unique solution
$u\in C^{2+\mu}(\overline\Omega)$ between  $\underline{u}$ and $\overline{u}$.

To prove the quadratic convergence of $\{\underline{u}_n\}$ and $\{\overline{u}_n\}$, we define $P_n=u-\underline{u}_n$, and $Q_n=\overline{u}_n-u$, then
\begin{eqnarray*}
LP_{n}&=&F(x,u)+G(x,u)-[F(x,\underline{u}_{n-1})+G(x,\underline{u}_{n-1})\\
&& +(F_u(x,\underline{u}_n)+G_u(x,\overline{u}_n)(\underline{u}_n
-\underline{u}_{n-1})] \\
&\le& [F_u(x,u)-F_u(x,\underline{u}_{n-1})]P_{n-1}+[G_u(x,
\underline{u}_{n-1})-G_u(x,\overline{u}_{n+1})]P_{n-1}  \\
& &+[F_u(x, \underline{u}_{n-1})+G_u(x,\overline{u}_{n-1})]P_{n}  \\
& = &  F_{uu}(x,\xi)P^2_{n-1}+G_{uu}(x,\zeta)(\underline{u}_{n-1}
-\overline{u}_{n-1})P_{n-1} \\
&& +[F_u(x, \underline{u}_{n-1})+G_u(x,\overline{u}_{n-1})]P_{n},
\end{eqnarray*}
where $\underline{u}_{n-1}\le \xi\le u$, $\underline{u}_{n-1}
\le \zeta\le \overline{u}_{n-1}$. Because
\begin{eqnarray*}
\lefteqn{ F_{uu}(x,\xi)P^2_{n-1}+G_{uu}(x,\zeta)(\underline{u}_{n-1}
-\overline{u}_{n-1})P_{n-1}  }\\
&\le& F_{uu}(x,\xi)P^2_{n-1} -G_{uu}(x,\zeta)(P_{n-1}+Q_{n-1})P_{n-1} \\
&\le& \delta_1 (P^2_{n-1}+P_{n-1}Q_{n-1})
\le \frac {3\delta_1}2(P_{n-1}^2+Q^2_{n-1})
\end{eqnarray*}
where $\delta_1=\max\{ |G_{uu}(x,u)|:\, x\in\overline\Omega,\underline{u}\le u\le \overline{u}\}$. Take $\delta= {3\delta_1}/2$, then
$$ LP_n-[F_u(x, \underline{u}_{n-1})+G_u(x,\overline{u}_{n-1})]P_{n}\le
\delta(P_{n-1}^2+Q^2_{n-1}).
$$
Hence $$ LP_n+cP_{n}\le   \delta (P_{n-1}^2+Q^2_{n-1}).
$$
On the other hand, $\phi(x)\equiv  {\delta}  [\max_{\overline\Omega} P^2_{n-1}
+\max_{\overline\Omega} Q^2_{n-1}]/{c}$ satisfies
\begin{gather*}
 L(\phi-P_n)+c(\phi-P_{n})\ge c\phi(x)-\delta (P_{n-1}^2+Q^2_{n-1}) \ge 0  \\
(\phi(x)-P_n(x))|_{\partial\Omega}\ge \int_\Omega K(x,y)(\phi(y)-P_n(y))\,dy.
\end{gather*}
By Lemma \ref{max}, we have $\phi(x)\ge P_n(x)$, that is
\begin{equation}
0\le u-\underline{u}_n=P_n\le  \frac {\delta} {c} [\max_{\overline\Omega}
P^2_{n-1}+\max_{\overline\Omega} Q^2_{n-1}].
\end{equation}
A similar estimate for $Q_n$ can be obtained. Therefore, the assertion is
proved. \hfill$\Box$\smallskip

\section{Parabolic Equations\label{s5}}

In this section we study the large time behavior of solutions for the
evolution equation
\begin{equation} \label{Pn}
\begin{gathered}
u_t+(L+c)\,u\equiv  u_t-\sum_{i,j=1}^n \frac {\partial}{\partial x_i}
( a_{ij}(x)\frac {\partial u}{\partial x_j})+c(x)u=f(x,u), \quad
\text{in } \Omega \times (0,T], \\
u|_{\partial\Omega}=  \int_{\Omega }K(x,y)u(y,t)\,dy.   \\
u(x,0)=u_0(x), \quad\text{on} \overline\Omega.
\end{gathered}
\end{equation}
The authors of \cite{d,f,p95,p97} have obtained some results.
Here, we use semigroup methods to discuss the decay of solutions.
We should also mention the work of Triggiani \cite{tr}, Lasiecka \cite{la, la1},
 and  Amann \cite{amann}.

Let $W^s_p(\Omega)$ be the standard Sobolev-Slobodeckii spaces for
$s\in \mathbb{R}^+$, $p>1$, $1/p+1/p'=1$, and
\begin{equation*}
W^{2\beta}_{p,\gamma}\equiv \left\{
\begin{array}{ll}
W^{2\beta}_p, &\text{for } 2\beta\in [0,1/p), \\[3pt]
(W^{-2\beta}_{p'})^{\prime}, &\text{for } 2\beta\in [-2,0]\setminus \{ -2+1/p, -1+1/p\}
 \end{array} \right.
\end{equation*}
where $X^{\prime}$ is the duality space of $X$ with respect to the duality
pairing which is obtained naturally from $\int_\Omega v(x)u(x)\,dx$,
$v\in L_{p'}$, $u\in L_p$. Hence $W^{2\beta}_{p,\gamma}$ is a closed linear
subspace of $W^{2\beta}_{p}$. And the boundary space is defined as
\begin{equation*}
\partial W^{2\beta}_p\equiv W^{2\beta-1/p}_p(\partial\Omega),\quad
  \text{for}\ 2\beta\in [0,1/p).
\end{equation*}
Denote  ${\bf K}(u)=\int_\Omega K(x,y)u(y,t)dy$ and
${\cal F}(u)=f(x, u)$, a $W^{2\beta}_p$-{\it weak solution} on $J$ of
(\ref{Pn}) is defined as one function $u\in C(J, W^{2\beta}_p)$
satisfying the initial data $u(x,0)=u_0$, where $J$ is one perfect
subinterval of $\mathbb{R}^+$ containing $0$, such that
\begin{gather*}
\int^t_0\!\! \int_\Omega \big\{-\dot{\phi} u +\sum_{i,j=1}^n a_{ij}
\frac {\partial u}{\partial x_j} \frac {\partial \phi}{\partial x_i}
+ cu\phi  dx
+\int_{\partial\Omega}\sum_{i,j=1}^n a_{ij}
\frac {\partial \phi}{\partial x_i}{\bf K}(u) \cos(\nu,x_i)dS\,\big\}dt \\
= \int^t_0 \int_\Omega\phi {\cal F}(u) dx\,dt+\int_\Omega \phi(0)u_0 dx
\end{gather*}
for every $t\in J\setminus \{0\}$ and every
$\phi\in C([0,t],W^{2(1-\beta)}_{p',\gamma})\cap C^1([0,t],
W^{-2\beta}_{p',\gamma})$ satisfying $\phi(t)=0$.

The above definition of solution of (\ref{Pn}) is meaningful.
By using Green's formula, if $u\in C(J, W^{2}_{p}) \cap C^1(J, L_p)$
satisfies (\ref{Pn}) (pointwise in $t$) then $u$ is a solution
of (\ref{Pn}) on $J$. Let  ${\cal H}(X)$ be the infinitesimal generator
of a strongly continuous analytic semigroup $\{ e^{-tA};\, t\ge0\}$ on a
Banach space $X$. Let $\sigma(A)$ be the spectrum of $A$.

\begin{lemma}[{\cite[Lemma 4.1]{amann}}]
Put $A_0=(L+c)|W^2_{p,\gamma}\equiv \{u\in W^2_{p};\gamma u=0\}$. Then
\begin{description}

\item{\text{1)}} $A_0\in {\cal L}(W^2_{p,\gamma}, L_p)$ with compact resolvent
belongs to ${\cal H}(L_p)$, where ${\cal L}(X,Y)$ is defined as the bounded linear operators from Banach spaces $X$ to $Y$.

\item\text{2)} There exists a unique $A_{\beta-1}\in {\cal H}((W^{2(1-\beta)}_{p'})')\cap {\cal L}(W^{2\beta}_{p,\gamma},(W^{2(1-\beta)}_{p'})')$ with compact resolvent, so that $A_{\alpha-1}$ is the $(W^{2(1-\alpha)}_{p'})'$-realization of $A_{\beta-1}$, {\it i.e.} $D(A_{\alpha-1})\equiv \{y\in (W^{2\alpha}_{p'})'\cap D(A_{\beta-1});\, A_{\alpha-1}y\in (W^{2\alpha}_{p'})'\}$, for $\alpha\ge \beta$.

\item{\text{3)}} There exist $\sigma\in\mathbb{R}$ and
${\cal R}_\beta \in{\cal L}(\partial W^{2\beta}_p,W^{2\beta}_{p,\gamma})$,
$\beta\in [0,1/p)$, so that ${\cal R}_\alpha ={\cal R}_\beta|\partial
W^{2\alpha}_p$ for $\alpha\geq\beta$ and
\[
 \int_\Omega v (\sigma +A_{\beta-1}){\cal R}_\beta u dx
 =\int_{\partial\Omega}
 \sum_{i,j=1}^n a_{ij}\frac {\partial \phi}{\partial x_i}\gamma u
 \cos(\nu,x_i)dS,
 \]
for $(v,u)\in W^{2(1-\beta)}_{p',\gamma}\times W^{2\beta}_{p,\gamma}$.

\item{\text{4)}} Let $U_\beta$ be a nonempty open subset of
$W^{2\beta}_{p,\gamma}$, ${\cal F}\in C(U_\beta, (W^{2(1-\beta)}_{p'})')$
and $u_0\in {U}_\beta$. Then $u$ is a solution of (\ref{Pn}) on
$J$ if and only if $u$ is a solution on $J$ of the evolution equation
\begin{equation}
\dot u +(A_{\beta-1}-(\sigma +A_{\beta-1}){\cal R}_{\beta}{\bf K})u={\cal F}(u), \ t\in J, \ u(0)=u_0.
\label {Phi}
\end{equation}

\item{\text{5)}} $E_{\alpha-1}$ is imbedded densely into
$(E_{\beta-1}, E_\beta)_\theta$ for $ \alpha-\beta>\theta>0$.

\item{\text{6)}} $\sigma (A_{\beta-1})=\sigma(A_0)$ for $\beta\in [0,1/p]$ and the geometric eigenspaces, $\text{ker} (\lambda+A_{\beta-1})$, and the algebraic eigenspaces, $ \mathop{\cup}_{k\ge 1} \text{ker} (\lambda+A_{\beta-1})^k$, are independent of $\beta$ for $\lambda\in \sigma(A_{\beta-1})=\sigma(A_0)$.
\end{description}
\label{lamann}\end{lemma}

In fact, for the linear elliptic problem
\begin{equation}
A_{\beta-1}u=f_1,\quad \gamma u=\psi, \label{ell}
\end{equation}
there exists one $\sigma\in\mathbb{R}$ and ${\cal R}_\beta$ such that the
problem (\ref{ell}) has solution if and only if the equation
\begin{equation}
A_{\beta-1}u=f_1+(\sigma+A_{\beta-1}){\cal R}_\beta\psi
\end{equation}
has solution \cite{a88}. So, we can treat the linear nonlocal problem
\begin{equation}
A_{\beta-1}u=f_1,\ \ \ \gamma u={\bf K}u, \label{elln}
\end{equation}
as
\begin{equation}
(A_{\beta-1}-(\sigma+A_{\beta-1}){\cal R}_\beta{\bf K})u=f_1.\label{-1}
\end{equation}
{\it i.e.} by a solution $u$ of (\ref{elln}) we mean a
$W^{2\beta}_{p,\gamma}$-solution of (\ref{-1}). The existence of a solution
for (\ref{elln}) is changed into  the existence for a new operator
equation (\ref{-1}). Of course, this is a generalization of the discussion
in section \ref{s3}. Particularly, when $\sigma=0$, the problem (\ref{-1})
becomes
\begin{equation}
(A_{\beta-1}-A_{\beta-1}{\cal R}_\beta{\bf K})u=A_{\beta-1}
(I-{\cal R}_\beta{\bf K})u=f_1,\label{-0}
\end{equation}
in which $I$ is the identity operator. We make an observation on (\ref{-0}),
if $1$ does not belong to the eigenvalue set of ${\cal R}_\beta{\bf K}$ and
$A_{\beta-1}(I-{\cal R}_\beta{\bf K})$ is invertible, then (\ref{-0}) has a
unique solution in $W^{2\beta}_{p,\gamma}$. This is consistent with the
discussion in the section \ref{s2} when ${\bf K}u=k\int_\Omega udx$.

The asymptotic behavior of a solution to the evolution equation  (\ref{Phi})
can be investigated through the study of properties of $A_{\beta-1}$,
${\bf K}$ and ${\cal F}$. This idea appeared in \cite{h,p,la1}.
Let $u_s\in U_\beta$ satisfy
$$ (A_{\beta-1}-(\sigma+A_{\beta-1}{\cal R}_\beta {\bf K})u_s=
{\cal F}(u_s),
$$
{\it i.e.} $u_s$ is an equilibrium point, and suppose that:
\begin{description}
\item\text{(H2)}  ${\cal F}(u)=f(x, u)$ is locally Lipschitzian in $u$ on
$U_\beta$ and
\begin{equation}
f(x,u)=f(x,u_0)+B(u-u_0)+g(x,u-u_0)
\end{equation}
where $B$ is a bounded linear map from $W^{2\beta}_{p,\gamma}$ to $L_p$
and $\|g(x,v)\|_{L_{p}}=o(\|v\|_{W^{2\beta}_{p,\gamma}})$ as
$\|v\|_{W^{2\beta}_{p,\gamma}}\to 0$, uniformly in $x\in\overline{\Omega}$.
\end{description}

\begin{theorem}
Let ${\cal F}$ be as in \text{(H2)} and $u_s$ be an equilibrium point.
If $A_{\beta-1}-(\sigma+A_{\beta-1}{\cal R}_\beta {\bf K})\in
{\cal H}((W^{2(1-\beta)}_{p'})')$ and the spectrum
$\sigma(A_{0}-(\sigma+A_{0}{\cal R}_0 {\bf K})-B)\subset
\{ \mathop{\rm Re} \lambda>\lambda_0 \}$ for some $\lambda_0>0$,
then there exist $\rho>0$, $M>1$ such that if
$\|u_0-u_s\|_{W^{2\beta}_{p,\gamma}} \le \rho/2M$ then a unique solution of
\text{(\ref{Phi})} exists and satisfies
\begin{equation}
\Vert u(x,t)-u_s(x)\Vert_{W^{2\beta}_{p,\gamma}}\le 2Me^{-\lambda_0t}\|u_0-u_s\|_{W^{2\beta}_{p,\gamma}},\ \ \text{\ for}\ t\geq 0.
\end{equation}
\end{theorem}

\noindent{\it Proof}. Denote
$\bar A_{\beta-1}=A_{\beta-1}-(\sigma+A_{\beta-1}{\cal R}_\beta {\bf K})-B$.
By using semigroup theories and Lemma \ref{lamann}, there exists one semigroup
$\{ e^{-\bar A_{\beta-1}t}\}$ such that
\begin{gather}
u(x,t)=e^{-\bar A_{\beta-1}t} u_0+\int^t_0 e^{-\bar A_{\beta-1}(t-\tau)}
{\cal F}(u(x,\tau))\,d\tau;\label{g} \\
u(x,t)-u_s=e^{-\bar A_{\beta-1}t} (u_0-u_s)+\int^t_0 e^{-\bar A_{\beta-1}
(t-\tau)}g(x,u(x,\tau))\,d\tau \label{g1}
\end{gather}
and there exists $\lambda\in(\lambda_0,\mathop{\rm Re}\sigma(\bar A_{\beta-1}))$,
 $M\ge 1$ such that for $t>0$ and $v\in {W^{2\beta}_{p,\gamma}}$,
$$
\|e^{-\bar A_{\beta-1}t}\|\le Me^{-\lambda t}\|v\|_{W^{2\beta}_{p,\gamma}},
\quad \|e^{-\bar A_{\beta-1}t} v\|_{W^{2\beta}_{p,\gamma}}\le
Mt^{-2\beta}e^{-\lambda t}\|v\|_{L_p},
$$
One can choose $\delta>0$ and $\rho>0$ small so that
\begin{equation}
M\delta\int^{+\infty}_0 \tau^{-2\beta}e^{-(\lambda-\lambda_0)\tau}\,d\tau
<\frac12
\end{equation}
and
\begin{equation}
 \|g(x,v)\|_{L_p}\le \delta\|v\|_{W^{2\beta}_{p,\gamma}}\quad
  \text{for }\|v\|_{W^{2\beta}_{p,\gamma}}\le\rho\,.
\end{equation}
Take $u_0$ with $\|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}\le \rho/2M$, then the
local solution for (\ref{g}) exists and satisfies
$\|u(x,t)-u_s\|_{W^{2\beta}_{p,\gamma}}\le \rho$ for $t\in J$.
On the other hand, from (\ref{g1}),
\begin{eqnarray*}
\lefteqn{\|u-u_s\|_{W^{2\beta}_{p,\gamma}}}\\
&\le&  Me^{-\lambda t} \|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}
 +\int^t_0 \|e^{-\bar A_{\beta-1}(t-\tau)}g(x,u(x,\tau))
 \|_{W^{2\beta}_{p,\gamma}}\,d\tau  \\
&\le& Me^{-\lambda t} \|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}
+\delta M\int^t_0 (t-\tau)^{-2\beta}e^{-\lambda(t-\tau)}\|u-u_s
\|_{W^{2\beta}_{p,\gamma}}d\tau \\
&\le& \rho\left( \frac {1}{2}+\delta M \int_0^t (t-\tau)^{-2\beta}
e^{-\lambda(t-\tau)}\,d\tau\right) <\rho\,.
\end{eqnarray*}
In the last inequality above, (\ref{g}) and (\ref{g1}) are used. Moreover,
\begin{eqnarray*}
\lefteqn{ e^{\lambda_0 t}\|u(x,t)-u_s(x)\|_{W^{2\beta}_{p,\gamma}} } \\
&\le& M \|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}+\delta M\int^t_0
(t-\tau)^{-2\beta}e^{-(\lambda-\lambda_0)(t-\tau)}e^{\lambda_0\tau}\|u-u_s
\|_{W^{2\beta}_{p,\gamma}}d\tau \\
&\le& M \|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}+\frac 12 \sup_{0\le\tau\le t}
\left\{ e^{\lambda_0\tau}\|u(x,\tau)-u_s(x)\|_{W^{2\beta}_{p,\gamma}}\right\}.
\end{eqnarray*}
Hence, $  \|u(x,\tau)-u_s(x)\|_{W^{2\beta}_{p,\gamma}}\le
2M e^{-\lambda_0 t}\|u_0-u_s\|_{W^{2\beta}_{p,\gamma}}$. \hfill$\Box$\smallskip

\subsection*{Acknowledgments}
The author wants to thank the anonymous referee and Prof. Julio G. Dix 
for their suggestions and improvements on the presentation of this article.  

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\noindent\textsc{Yuandi Wang}\\
Department of Mathematics, Shanghai University, \\
Shanghai 200436, China\\
e-mail: ydliwang@online.sh.cn


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