\chapter{SECOND ORDER EVOLUTION EQUATIONS}
In this chapter, we study
evolution equation,
of parabolic type
\bes u_t + L u =f \label{3.1}\ees
and of hyperbolic type
\bes u_{tt} + L u = f\label{3.2}\ees
where $L$ is a second order elliptic operator, in the
divergence form
\bes Lu=-(a^{ij}u_{x_i}+d^j u)_{x_j}+b^iu_{x_i}+cu \label{3.3}\ees
or in the non--divergence form
\bes Lu=-a^{ij}u_{x_ix_j}+b^iu_{x_i}+cu.\label{3.4}\ees
We notice that if we write $\bu=(u_1,u_2):=(u,u_t)$, then the hyperbolic equation
(\ref{3.2}) can be written as a first order (in time) evolution
\bes \bu_t = \cL \bu \label{3.5} \ees
where $\cL\bu = (u_2, L u_1)$.
Under approriate setting, (\ref{3.5}) can
be regarded as an ODE.
%\section{Curves in Banach Spaces}
In studying evolution equations,
a space time function $u(x,t)$ can be simply viewed
as a funtion from $t$ to $\bu(t):=u(
\cdot,t)$ in certain Sobolev space $\bX$, say $H^1(\Omega)$.
That is, one can regard an evolution equation as an ``ODE",
and the solution $\bu$ as a curve (trajectory) in certain Banach space.
\section{Parabolic Equations in Divergence Form} We shall study the
following initial value problem in the cylinder $\Omega_T=\Omega\times
(0,T]$, where $\Omega$ is a bounded domain in $\R^n$:
\begin{equation}
\left\{\begin{aligned}
u_t+Lu&=f&\text{\ \ \ \ in $\Omega_T$}\\
u&=0&\text{\ \ \ \ in $\dd \Omega\times [0,T]$}\\
u&=g&\text{\ \ \ \ in $\Omega\times \{t=0\}$}
\end{aligned}\right.
\label{p3.1}
\end{equation}
Here $f$ and $g$ are given functions, and $L=L(x,t)$ is a uniformly
elliptic differential operator in the divergence form
\bes Lu=-(a^{ij}u_{x_i}+d^j u)_{x_j}+b^iu_{x_i}+cu \label{3.6} \ees
Parallel to that of ellpitic case, for (a.e.) $t\in[0,T]$, we
define a bilinear form $B[t; \cdot,\cdot]:H^1_0(\Omega)\times H^1_0(\Omega)\rightarrow \R$ by
$$B[t;u,v]=\int_\Omega
\Big\{a^{i,j}(x,t)u_{x_i}v_{x_j}+b^i(x,t)u_{x_i}v+c(x,t)uv\Big\}\,dx$$
Likewise, we may regard a solution $u=u(x,t)$ as a mapping $t\mapsto\bu(t):=
u(\cdot,t)$ from $[0,T]$ into the function space $\bX=H^1_0(\Omega)$.
Thus a solution
can be regarded as a trajectory in $\bX$.
Indeed, when $L$ is in the divergence form,
it is very convenient to work on the space $u\in L^2(0,T;H^1_0(\Omega))$.
Similarly, we regard $\bff(t)=f(\cdot,t)$ as a function in $L^2(0,T;H^{-1}(\Omega))$.
\subsection{Definiton of Weak solutions}
\begin{definition} Assume that $g\in L^2(\Omega)$ and $\bff\in L^2(0,T;H^{-1}(\Omega))$.
We say that a function
$${\bu}\in L^2(0,T;H^1_0(\Omega))\text{\ \ with\ \ } {\bu}'\in
L^2(0,T;H^{-1}(\Omega)),$$
is a weak solution of the initial boundary value problem (\ref{p3.1}) if
$\bu(0)=g$ in $L^2(\Omega)$ and
$$\langle{\bf u}',v\rangle+B[t;{\bf u},v]=({\bf f},v)$$
for each $v\in H^1_0(\Omega)$ and a.e. $0\le t\le T$.
\end{definition}
Here the requirement $\bu(0)=g$
makes sense since by Theorem \ref{th3.6}, ${\bf u}\in C([0,T];L^2(\Omega)).$
\subsection{Existence of Weak Solutions}
The method we employ for proving existence is known as the Galerkin
method\footnote{Another useful method
is to discritize the time
by either an Euler forward or an Euler backward scheme, and solving
the resulting elliptic
equations. In the current situation, it seems that
the Galerkin method offers some
advantages, although in general both methods seem very robust.}
which approximates the Banach $\bX=H^1_0(\Omega)$
by finite space dimensional subspace $\bX_m$
and project the evolution equation over $L^2(0,T;\bX_m)$
Let
$\{w_k\}_{k=1}^\infty$ be an orthonormal basis for $L^2(\Omega)$
which is also an orthogonal
basis for $H^1_0(\Omega)$. Consider
$${\bf u}_m(t):=\sum_{k=1}^m d^k_m(t)w_k,$$
where the $d^k_m(t)$ are real-valued functions of $t$. We shall choose
coefficients $d^k_m(t)$ in such a way that ${\bf u}_m(t)$ is an
approximate solution of our boundary value problem.
Specifically, for each fixed $m\geq1$,
we seek $d^k_m(t), k=1,\cdots,m$ such that for all $k=1,\cdots,m$,
\bes({\bf u}'_m,w_k)+B[t;{\bf u}_m, w_k]&=&\langle{\bf f},w_k\rangle\label{3.7}
\\
d^k_m(0)&=&(g,w_k)_{L^2}.\label{3.8}\ees
\begin{lemma} For each $m$, there is a unique function ${\bf u}_m$ of
the prescribed form satisfying (\ref{3.7}) and (\ref{3.8}) for all $k=1,\cdots,m$.
\end{lemma}
\begin{proof} We have
$$({\bf u}'_m(t),w_k)={d^k_m}'(t),$$
and
$$B[t;{\bf u}_m,w_k]=\sum_{l=1}^m e^{kl}(t)d^l_m(t)$$
for $e^{kl}(t)=B[t;w_l,w_k]$. Set $f^k(t)=({\bf f}(t),w_k)$. The system
becomes
$${d^k_m}'(t)+\sum_{l=1}^me^{kl}(t)d^l_m(t)=f^k(t)$$
with the appropriate initial conditions. By the existence and uniqueness
theorem from ODEs, there exists an absolutely continuous function
${\bf d}_m(t)=(d^1_m(t),...,d^m_m(t))$ which solves this system.
\end{proof}
We now show that as $m\to\infty$, $\bu_m$ converges to a weak solution of
(\ref{p3.1}).
To do this, we assume that $\cL$ is elliptic in the sense that
there exists positive constants $\alpha,\beta$ and $\mu$ such that
for (a.e.) $t\in[0,T]$
\bess B[t;u,v] &\leq & \alpha \|u\|_{H^1_0(\Omega)} \| v\|_{H^1_0(\Omega)} \quad\forall \,u,v\in
H^1_0(\Omega),
\\ B[t;u,u] &\geq & \beta \|u\|^2_{H^1_0(\Omega)} - \mu \|
u\|_{L^2(\Omega)}^2\quad\forall \, u\in H^1_0(\Omega) . \eess
We make use of the following estimate.
\begin{lemma} [Energy estimate] There exists $C=C(\Omega,T,L)$ such that
for each integer $m\geq1$,
\begin{equation*}
\begin{aligned}
\max_{0\le t\le T}\nn{{\bf u}_m(t)}_{L^2(\Omega)}&+\nn{{\bf
u}_m}_{L^2(0,T;H^1_0(\Omega))}+\nn{{\bf u'}_m}_{L^2(0,T;H^{-1}(\Omega))}\\
&\le C(\nn{\bf f}_{L^2(0,T;H^{-1}(\Omega))}+\nn{g}_{L^2(\Omega)})
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}[Proof sketch]
Multiplying (\ref{3.7}) by $d_m^k$ and sum over $k=1,\cdots,m$ one obtains
$$ (u_m,u_m')+ B[t;u_m,u_m]=\langle \bff,u_m\rangle. $$
First of all, we have
$$({\bf u'}_m,{\bf u}_m)={d\over dt}\biggl({1\over 2}\nn{{\bf
u}_m}^2_{L^2(\Omega)}\biggr).$$
Secondly, by the ellipticity of $\cL$,
$$ B[t;{\bu}_m,{\bu}_m]\leq \beta \|u_m\|_{H^1_0(\Omega)}^2-\gamma\nn{{\bf u}_m}^2_{L^2(\Omega)}$$
Finally,
$$|\langle{\bf f},{\bf u}_m\rangle|
\le \|\bff\|_{H^{-1}(\Omega)} \|u_m\|_{H^1_0(\Omega)} \leq
\tfrac \beta 2 \|\bu_m\|^2_{H^1_0(\Omega)} + \frac 1{2\beta}
\|\bff\|^2_{H^{-1}(\Omega)}.$$
Using these inequalities, we readily obtain an inequality of the form
$$\eta'(t)\le C_1\eta(t)+C_2\xi(t)$$
where
$$\eta(t)=\nn{{\bf u}_m}^2_{L^2(\Omega)},\qquad
\xi(t)=\nn{{\bf f}}^2_{L^2(\Omega)},$$
whence by Gronwall's inequality,
$$\eta(t)\le e^{C_1 t}\biggl(\eta(0)+C_2\int_0^t\xi(s)ds\biggr).$$
Noticing that $\eta(0)=\nn{{\bf u}_m(0)}_{L^2(\Omega)}\le \nn{g}_{L^2(\Omega)}$
implies
$$\max_{0\le t\le T}\nn{{\bf u}_m(t)}_{L^2(\Omega)}\le C(\nn{\bf
f}_{L^2(0,T;L^2(\Omega))}+||g||_{L^2(\Omega)}).$$
Estimates for the other terms follow via elementary calculations.
\end{proof}
\begin{theorem} There exists a weak solution of the initial boundary
value problem (\ref{p3.1}).
\end{theorem}
\begin{proof}[Proof sketch] Using the Galerkin approximation, one obtains
a sequence $\{{\bf u}_m\}_{m=1}^\infty$ of approximation solutions. By the energy estimates,
this sequence of approximate solutions is bounded in $L^2(0,T;H^1_0(\Omega))$,
and therefore weakly precompact, so there exists a weakly convergent
subsequence ${\bf u}_{m_l}$ such that
\begin{equation*}
\begin{aligned}
{\bf u}_{m_l}\xrightarrow{w} {\bf u}&\text{\ \ \ \ weakly in
$L^2(0,T;H^1_0(\Omega))$}\\
{{\bf u}'}_{m_l}\xrightarrow{w} {{\bf u}'}&\text{\ \ \ \ weakly in
$L^2(0,T;H^{-1}(\Omega))$}
\end{aligned}
\end{equation*}
Now one uses the density of functions
$$\sum_1^N d^k(t)w_k$$
in the space $L^2(0,T;H^1_0(\Omega))$ to prove that the weak limit is a weak
solution of the problem.
\end{proof}
\subsection{Uniqueness}
\begin{theorem} A weak solution of the problem is unique.
\end{theorem}
\begin{proof}[Proof sketch] Assume that we have a weak solution $\bf u$
with
$${\bf f}=g=0.$$
We have
$${d\over dt}\biggl({1\over 2}\nn{\bf u}^2_{L^2(\Omega)}\biggr) +B[t;{\bf u},{\bf
u}]=\langle{\bf u}',{\bf u}\rangle+B[t;{\bf u},{\bf u}]=0$$
and also
$$B[t;{\bf u},{\bf u}]\ge\beta\nn{\bf u}^2_{H^1_0(\Omega)}-\gamma\nn{\bf
u}^2_{L^2(\Omega)}\ge-\gamma\nn{\bf u}^2_{L^2(\Omega)}$$
whence the result follows from Gronwall's inequality.
\end{proof}
\subsection{Regularity in Hilbert Spaces}
By regarding $-(b^j u)_{x_i}+b^i u_{x_i}+cu$ as source terms, we assume that
$$ L u = -(a^{ij} u_{x_i})_{x_j}. $$
We assume that $L$ is uniformly elliptic and for $m$ under consideration,
$D^{\alpha}_x D^l_t a^{ij} \in C^0(\bar
\Omega_T)$ for all $|\alpha|+2 l\leq 2m+2$.
Also, we assume that $\partial \Omega\in C^{m+2}$.
\begin{theorem} Let $m$ be a non--negative integer. Assume
$$g\in H^{2m+1}(\Omega),\ {d^k{\bf f}\over dt^k}\in L^2(0,T;H^{2m-2k}(\Omega))
\quad\forall \,k=0,\cdots,m.$$
Suppose also we have the compatibility conditions
\begin{equation*}
\left\{\begin{aligned}
&g_0:=g\in H^1_0(\Omega),\ g_1:={\bf f}(0)-Lg_0\in H^1_0(\Omega),\\
&\dots, g_m:={d^{m-1}{\bf f}\over dt^{m-1}}(0)-Lg_{m-1}\in H^1_0(\Omega)
\end{aligned}\right.
\end{equation*}
Then
$${d^k{\bf u}\over dt^k}\in L^2(0,T; H^{2m+2-2k}(\Omega))\text{\ \ \
$k=0,1,...,m-1$}$$
and we have the estimate
\begin{equation*}
\begin{aligned}
\sum_{k=0}^{m+1}\nn{d^k{\bf u}\over dt^k}_{L^2(0,T;H^{2m+2-2k}(\Omega))}
&\le C\biggl(\sum_{k=0}^m \nn{d^k{\bf f}\over
dt^k}_{L^2(0,T;H^{2m-2k}(\Omega))}+||g||_{H^{2m+1}(\Omega)}\biggr)
\end{aligned}
\end{equation*}
\end{theorem}
Using Sobolev embeddings, this theorem allows us to prove smoothness of
solutions (for instance) under the assumption that the data of the problem
are smooth.
\section{Parabolic Equation in Non-divergence form}
Not discussed in class.
\section{Parabolic A Priori Estimates}
We include here the precise statements (without proof) of some of the
regularity theorems for parabolic PDEs, although such theorems had been
only suggested in the lectures.
In each of the estimates given below, there are two versions, one is called local,
which does not need initial boundary values, and the
other called global, which need initial and boundary values. We state the global version
only.
\subsection{The H\"older Estimate}
\begin{theorem} Assume that $\cL$ is uniformly elliptic in divergence or
non-divergence form. Then the solution is H\"older continuous.
\end{theorem}
\subsection{The $L^p$ Estimate}
\begin{theorem} [\bf $L^p$ estimate] Assume that
$$\cL u = - a^{ij} u_{x_ix_j} + b^iu_{x_i} + c u$$
where $\cL$ is uniformly elliptic and
$$a^{ij}\in C^0(\bar{\Omega}_T),\ \ b^i\in L^\infty,\ \ c\in L^\infty,\ \
f\in L^p,\ \ g\in W^{2,1}_p,$$
and that $\dd\Omega\in C^2$.
Also asume that $g\in W^{2,p}\cap H^1_0$. Then we have the estimate
$$\| u\|_{W^{2,1}_p}
:=\sum_{|\alpha|+2 l \leq 2} \|D^\alpha_x D^l_t u\|_{L^p}
\le C\{\nn{f}_{L^p}+\nn{g}_{W^{2,p}}\}.$$
\end{theorem}
\subsection{The Schauder Estimate}
\begin{theorem} [Schauder estimate] Let $\alpha\in(0,1)$. Set $Q=\Omega\times(0,T]$
Assume that $a^{ij},b^i,c\in
C^{\alpha,\alpha/2}(\bar Q)$ and that $f\in C^{\alpha,\alpha/2}(\bar Q)$,
$g\in C^{2+\alpha}(\bar \Omega)\cap H^1_0(\Omega)$, $\partial\Omega\in C^{2+\alpha}$,
then we $u\in C^{2+\alpha,1+\alpha/2}(\bar Q)$ and
$$\nn{u}_{C^{2+\alpha,1+\alpha/2}(\bar Q)}\le
C[\ \Big\{\|f\|_{C^{\alpha,\alpha/2}(\bar Q)}+
\|g\|_{C^{2+\alpha}(\bar\Omega)} \Big\}.$$
\end{theorem}
\section{Hyperbolic Equations}
As before, let $\Omega_T=\Omega\times (0,T]$. We shall be interested in the
initial
boundary-value problem
\begin{equation*}
\left\{\begin{aligned}
u_{tt}+Lu&=f &\text{\ \ in $\Omega_T$}\\
u&=0 &\text{\ \ in $\dd \Omega\times [0,T]$}\\
u=g,\ u_t&=h &\text{\ \ in $\Omega\times \{t=0\}$}
\end{aligned}\right.
\end{equation*}
For simplicity, we shall assume that $L$ is a second order
uniformly elliptic operator. In this case,
we say that the problem is (uniformly) hyperbolic.
\subsection{Definition of Weak Solutions}
For simplicity, we assume that $\cL$ is in divergence form.
As in the case of parabolic equations, we define weak solutions in terms
of curves in Banach spaces. Specificially, let
$${\bf u}:[0,T]\rightarrow H^1_0(\Omega)$$
be defined by
$${\bf u}(t)=u(.,t)\in H^1_0(\Omega),$$
and
$${\bf f}:[0,T]\rightarrow L^2(\Omega)$$
by
$${\bf f}(t)=f(.,t)\in L^2(\Omega).$$
\begin{definition}
A function
$${\bf u}\in L^2(0,T;H^1_0(\Omega))$$
with
$${\bf u}'\in L^2(0,T;L^2(\Omega)),\ \ {\bf u}''\in L^2(0,T;H^{-1}(\Omega))$$
is said to be a weak solution of the hyperbolic problem if
$$({\bf u}'',v)+B[t;{\bf u},v]=\langle{\bf f},v\rangle\quad \forall v\in
H^1_0(\Omega)$$
for (a.e.) $ t\in[0,T]$, and
$${\bf u}(0)={\bf g},\ \ {\bf u}'(0)={\bf h}.$$
\end{definition}
Note that in light of an earlier theorem on curves in Banach spaces, the
latter two conditions make sense, becase ${\bf u}$ and ${\bf u}'$ are {\it
continuous}.
\subsection{Existence Weak Solutions}
Again we can employ the Galerkin approximation. Thus, we attempt to
construct a sequence ${\bf u}_m$ of approximate solutions of the form
$${\bf u}_m(t) = \sum_{k=1}^m d^k_m(t)w_k$$
where $w_k$ is an orthogonal basis for $H^1_0(\Omega)$, and an orthonormal
basis of $L^2(\Omega)$. The coefficients are real-valued functions which are
required to obey the initial conditions
$$d^k_m(0)=(g,w_k)$$
$${d^k_m}'(0)=(h,w_k),$$
and the ${\bf u}_m$ are to obey the differential equation
$$({\bf u}_m'',w_k)+B[{\bf u}_m, w_k;t]=({\bf f}, w_k).$$
By using the fundamental existence and uniqueness theorem from ODEs, this
system of initial-value problems is uniquely solvable for the coefficients
$d^k_m(t)$.
Moreover, as before we obtain the energy estimate which is key to proving
that the sequence ${\bf u}_m$ has a weak limit point which is a weak
solution of the desired problem:
\begin{lemma} (Energy estimate) We have
\bess &&\max_{0\le t\le T}\Big\{\nn{{\bf u}_m(t)}_{H^1_0(\Omega)}+ \nn{{\bf
u'}_m(t)}_{L^2(\Omega)}Big\}
\\ && \qquad +\nn{{\bf u''}_m}_{L^2(0,T;H^1_0(\Omega))}+\nn{{\bf
u'}_m}_{L^2(0,T;H^{-1}(\Omega))} \\ && \qquad\qquad
\le C(\nn{\bf f}_{L^2(0,T;L^2(\Omega))}+\nn{g}_{H^1_0(\Omega)}+\nn{h}_{L^2(\Omega)})
\eess
\end{lemma}
\begin{proof}[Idea of Proof] The idea of the proof is
similar to the parabolic version: obtain
estimates which enable one to apply Gronwall's inequality.
\end{proof}
\vspace{12pt}
Using the energy estimates, we may prove that $u_m$ converges to a weak
solution. Hence, we have the following existence result.
\begin{theorem} A weak solution of the hyperbolic problem exists.
\end{theorem}
The proof is completely analogous to the proof for the parabolic case.
\subsection{Uniqueness}
\begin{theorem} Solutions are unique.
\end{theorem}
Again, the proof relies upon deriving an estimate in the form of
Gronwall's inequality. The details are somewhat tricky, and we leave them
out.
\subsection{Regularity}
\section{Introduction to Semigroup Theory}
%\author{A.A.Dunca \\Department Of Mathematics\\University Of
%Pittsburgh\\Pittsburgh, PA 15260}
%\date{Thursday,April 10, 2001}
In studying evolution equations,
a space time function $u(x,t)$ can be simply viewed
as a funtion from $t$ to $\bu(t):=u(
\cdot,t)$ in certain Sobolev space $\bX$, say $H^1(\Omega)$.
That is, a linear evolution equation can be written as
\bes \bu_t = A \bu \ees
where $A$ is a linear operator. Hence, one can pretty much regard
the evolution equation as an ode and
the solution $\bu$ as a curve (trajectory) in certain Banach space.
Indeed, a lot of the ode ideas can
be used.
In this chapter, we introduce one of
the very powerful tools in studying
evolution equations, the semigroup theory.
\subsection{Semigroups\ and Their Generators}
\begin{definition} Let $\bX$ be a Banach space.
A familly $\{S(t)\}_{t\geq 0}$ of
bounded linear operators from $\bX$ to $\bX$ is called a semigroup if
the following holds:
\begin{enumerate}
\item $S(0)u=u$ for every $u \in \bX$;
\item $S(t+s)u=S(t)S(s)u=S(s)S(t)u$ for every $u\in \bX$ and $t,s \geq 0$
\item For each $u\in\bX$,
the maping $t\to S(t)u$ is continuous from $[0,\infty) $ into $\bX$.
\end{enumerate}
\end{definition}
\begin{definition}
A semigroup $\{S(t)\}_{t\geq 0}$ is called a contraction semigroup if
\bes \label{contr} \| S(t)\| \leq 1 \quad\forall\, t\geq 0\ees
where $\| \cdot \|$ stands for the operatorial norm from $\bX$ to $\bX$.
\end{definition}
\begin{definition} Let $\{S(t)\}_{t\geq 0}$ be a semigroup on
a Banach space $\bX$.
Define \bq \label {domain} D(A):=\left\{u \in \bX\;;\;\lim \limits _{t\to 0+}
\frac {S(t)u-u}{t} {\rm \ exists \,\,in}\, \bX\right\}\eq
and $A:D(A)\to \bX$ by
\bq \label{opa} Au:=\lim \limits _{t\to 0+}\ff{S(t)u-u}{t} \quad\forall u\in
D(A). \eq
This linear operator $A:D(A)\to \bX$ is called the (infinitesimal) generator
of the semigroup $\{S(t)\}_{t\geq 0}$ and $D(A)$ the definition domain of $A$.
\end{definition}
\begin{example}
{\rm
Let $\lambda\in\BbR$ (or $\BbC$) be a number. Define
$S(t) u =e^{\lambda t} u$ for all $t\in\BR$ and $u\in \bX$.
Then $\{S(t)\}_{t\geq0}$ is a semigroup and its generator is $\lambda \bI$ where $\bI$ is
the identity operator.
The definition domain of the generator is the whole space $\bX$.
The semigroup is a contraction if and only if $\Re(\lambda)\leq0$.}
\end{example}
\begin{example}
{\rm Let $A$ be a bounded operator from $\bX$ to $\bX$.
Define, for any $t\in\BbR$ and $u\in \bX$,
$$ S(t) u = \sum_{k\geq 0} \frac{ t^k A^k}{k!} u. $$
Then one can verify that $\{S(t)\}_{t\geq0}$ is a semigroup with generator
$A$ whose definition domain is $\bX$. Typically, we write the operator $S(t)$
as $e^{At}$.
Now if $(\lambda,\phi)\in\BbC\times\bX$ is an eigen pair of $A$, i.e., $A\phi=\lambda\phi$,
then $e^{At} \phi = e^{\lambda t} \phi$.
Notice that for every $u_0\in \bX$, the function $u(t):=S(t) u_0$ solves the
initial value problem
$$ u_t = A u\quad \forall\, t>0, \quad u(0)=u_0. $$ }
\end{example}
We remark that in these two examples, $\{S(t)\}_{t\in\BbR}$ is indeed a group.
\begin{example}\label{4.ex3}
{\rm Let $\Omega=(0,\pi)$ and $\bX= L^2(\Omega)$. For each integer $n\geq1$ define
$\phi_n= \sqrt{\frac 2\pi} \sin (nx)$ and $\psi_n=\sqrt{\frac2\pi}\cos (nx)$ and
$\psi_0=\sqrt{\frac1\pi}$. Then both
$\{\phi_n\}_{n=1}^\infty$ and $\{\psi_n\}_{n=0}^\infty$ are orthonomal bases of $\bX$.
Let $A=\frac{d^2}{dx^2}$. Then we see that $A\phi_n = \lambda_n \phi_n$ and
$A\psi_n=\lambda_n\psi_n$ where $\lambda_n=-n^2$.
For each $t\geq 0$ and $u\in\bX$, we define
\bess
S(t) u &=& \sum_{n\geq1} e^{\lambda_n t} (u,\phi_n)\phi_n,
\\ \tilde S(t) u &=& \sum_{n\geq 0} e^{\lambda_n t} (u,\psi_n)\psi_n.\eess
Then one can verify that both $\{S(t)\}_{t\geq 0}$ and $\{\tilde S(t)\}_{t\geq 0}$ are
contraction semigroups, and both have $A$ as their generators.
Nevertheless, the definition domains of
the generators for the two semigroups are different.
For $\{S(t)\}_{t\geq0}$, the definition domain of the generator is
$H^2(\Omega)\cap H^1_0(\Omega)$, where as for $\{\tilde S(t)\}_{t\geq 0}$, it is
$\{ u\in H^2(\Omega); u'(0)=u'(\pi)=0\}$.
One can check that for any $u_0\in \bX$,
$u(t):= S(t)u_0$ and $\tilde u(t):= \tilde S(t) u_0$ for all $t\geq 0$ solve, respectively
\bess && \left\{ \begin{array}{ll} u_t=u_{xx} \quad &\hbox{in \ } (0,\pi)\times(0,\infty),
\\ u(0,t)=u(\pi,t)=0 &\hbox{on \ } \{0,\pi\}\times (0,\infty),
\\ u(x,0)=u_0(x) &\hbox{on \ } [0,\pi]\times\{0\},
\end{array}\right.
\\ &&
\left\{ \begin{array}{ll} \tilde u_t=\tilde u_{xx} \quad &\hbox{in \ } (0,\pi)\times(0,\infty),
\\ \tilde u_x(0,t)=\tilde u_x(\pi,t)=0 &\hbox{on \ } \{0,\pi\}\times (0,\infty),
\\ \tilde u(x,0)=u_0(x) &\hbox{on \ } [0,\pi]\times\{0\}.\end{array}\right.\eess
}
\end{example}
From this example, one sees that the specification of
the definition domain of $A$ is very important.
%For example, the same Laplacian operator
%$\Delta$ with Dirichlet and with Neumann boundary
%considtions should be considered as different oprators.
\begin{exerse} Verify the conclusions stated in example \ref{4.ex3}.
\end{exerse}
\subsection{Properties of Generators}
\begin{theorem} Let $A$ be the generator of a semigroup $\{S(t)\}_{t\geq0}$ on a
banach space $\bX$. Then, for each $ u \in D(A)$, the following holds:
\begin{enumerate}
\item For each $t\geq 0$, $S(t)u\in D(A)$ and $AS(t)u=S(t)Au$.
\item Define $x(t)=S(t) u$ for all $t\geq 0$. Then
$x\in C^1[0,\infty;\bX)$ and
$$ \dot x(t) = A \,x(t) \quad \forall\, t\geq 0, \qquad x(0)=u. $$
\end{enumerate}\end{theorem}
\begin{theorem} Assume that $A$ is the generator
of a contraction semigroup on $\bX$. Then
\begin{enumerate}
\item $D(A)$ is dense in $\bX$, and
\item $A$ is a closed operator.
\end{enumerate}\end{theorem}
%\subsection{Resolvant of the Generator}
\begin{definition} Let $\bX$ be a banach space and
$A:D(A)\subset \bX\to \bX$ be linear.
\begin{enumerate}
\item A real number $\la$ belongs to $\rho (A)$, the resolvent set of
$A$,
if
$$\la I -A:D(A)\to X\quad\hbox{is one to one and onto}. $$
\item If $\la \in \rho(A)$, the resolvent operator $R_{\la}:\bX\to
D(A)$ is
defined by $$R_{\la}u:=(\la I-A)^{-1}u\qquad \forall u\in\bX.$$
\end{enumerate}\end{definition}
Now suppose that $A$ is a closed operator and $\lambda$ is in the resolvant set of $A$. Then
it is an easy consequence of the Closed Mapping Theorem that $R_{\la}$ is
a bounded linear operator from $\bX\to D(A)\subset \bX$.
Moreover, $A$ and $R_{\la}$ commute i.e.
$$ AR_{\la}u=R_{\la}Au\qquad\forall u\in D(A).$$
\begin{theorem} Let $A$ be a generator of a contraction semigroup
$\{S(t)\}_{t\geq0}$.
\begin{enumerate}
\item If $\la,\mu\in \rho(A)$ then
\bq \label{rez} R_{\la}-R_{\mu}=(\mu-\la)R_{\la}R_{\mu}\eq
and \bq R_{\la}R_{\mu}=R_{\mu}R_{\la}\eq
\item If $\la>0$ then $\la \in \rho(A)$ and
\bq \label{int} R_{\la}u=\int_{0}^{\infty} e^{-\la t}S(t)udt\quad\forall
u\in\bX. \eq
Consequentrly, $\|R_{\la}\| \leq {\la}^{-1}.$
\end{enumerate}\end{theorem}
\subsection{The Hille--Yosida Theorem}
\begin{theorem}{ \bf(Hille Yosida Theorem)} Let $A$ be
a closed ,densely--defined
linear operator on a Banach space $\bX$.
Then $A$ is the generator of a contraction semigroup $\{S(t)\}_{t\geq0}$
if and only if \bq \label{hile} (0,\infty) \subset \rho (A) {\rm \ \ and \ \ }
\| R_{\la}\| \leq {\la}^{-1}\ \,\,\ \forall\,\la>0.\eq
\end{theorem}
\begin{definition}A semigroup ${S(t)}_{t\geq 0}$ is called $\omega$-contractive
if
\bq \| S(t)\|\leq e^{\omega t} \quad \forall \,t\geq 0.\eq
\end{definition}
The necessary and sufficient condition for a closed, densely defined
operator $A$ to generate an $\omega$-contractive semigroup is
\bq (\omega,\infty) \subset \rho(A) \,\, \quad\hbox{and}\quad
\| R^{\la}(A)\| \leq
\ff{1}{\la-\omega}\ \ \forall \la>\omega. \eq
\subsection{Applications}
\begin{example}
{ \rm
Consider the heat equation
\begin{eqnarray}\label{nse1} \left\{\begin {array}{ll}
u_{t}=\Delta u \quad &
\hbox {on \ } \om\times (0,\infty)\\
u=0& \hbox{
on } \partial\Omega\times [0,\infty) \\
u(x,0)=u_{0}(x) &\hbox{on \ }
\Omega\times\{0\}.
\end{array}\right.\end{eqnarray}
Set
$$\bX=L^{2}(\om),
\quad D(A)= H^2(\Omega)\cap H^1_0(\Omega),\quad A u = \Delta u.
$$
Then $A$ is closed and $D(A)$ is dense in $\bX$. Also,
for every $\la >0$, $(\la I-A)^{-1} $ from $\bX$ to $D(A)$ exists.
To estimate $\|R_\lambda\|$ consider
the equation
$$\la u -\tri u=f \in L^{2}(\Omega),\quad u\in H^1_0(\Omega).$$
We multiply by u and we integrate by parts to get
\bq \la \vv u\vv ^{2} + \| \nabla u \|^{2}=(f,u)\leq \|f\|_{L^2(\Omega)}
\|u\|_{L^2(\Omega)}.\eq
Hence
$$ \| u\|_{L^{2}} \leq \frac {1}{\la} \|f\| _{L^{2}}$$
which implies $\| R_{\la}\| \leq \ff{1}{\la}$.
Consequently the Hille--Yosida theorem implies that $A$ generates a semigroup
$\{S(t)\}_{t\geq 0}$ on $\bX$.
Hence, for every $u_0\in L^2(\Omega)$, $u(t):= S(t) u_0$ for all $t\geq 0$ is a
solution to our problem.}
\end{example}
\bigskip
\begin{example}{\rm Consider the following
parabolic equation in non--divergence form
\begin{eqnarray}\label{nse} \left\{\begin {array}{ll}
u_{t}=-\cL u :=a^{ij}u_{x_{i}x_j} - b^i u_{x_i} - c u \quad& \hbox{in \
}\Omega\times(0,\infty),
\\
u=0& \hbox{on \ } \partial\Omega\times[0,\infty)
\\
u(x,0)=u_{0}(x) &\hbox{on \ }
\Omega\times\{0\}
\end{array}\right.\end{eqnarray}
where $a^{ij}\in C^0(\bar\Omega)$ and $b^i, c\in L^\infty(\Omega)$ are
independent of $t$, and for
some positive constant $\theta$,
$a^{ij}(x)\xi_{i}\xi_{j}\geq \theta |\xi|^{2}$, for every $\xi \in \BbR^{n}$
and $x\in\bar\Omega$.
Set $\bX=L^{2}(\Omega)$, $D(A)= H^1_0(\Omega)\cap H^2(\Omega)$ and $A=-\cL$, i.e.,
\bq A: u\in D(A) \to A(u)=a^{ij}u_{x_{i}x_{j}}-b^i u_{x_i} + cu \in \bX .\eq
It is easy to see that $D(A)$ is dense in $\bX$ and $A$ is a closed operator.
Also, by ellptic theory,
there exists $\omega\in\BbR$ such that \bq (\omega,\infty) \subset
\rho(A) \,\, \hbox{\ and \ \ } \| R^{\la}(A)\| \leq
\frac{1}{\la-\omega}\ \ \forall \la >\omega.\eq
Hence, the Hille--Yosida theorem can be applied to give solutions to the intial
value problem for each $u_0\in L^2(\Omega)$. }
\end{example}
\bigskip
\begin{example}{\rm Consider the wave equation
\begin{eqnarray}\label{nse2} \left\{\begin {array}{ll}
u_{tt}=\tri u \quad &
\textrm {on $\om\times (0,\infty)$}\\
u=0& \textrm{
on $\pa\Omega\times[0,\infty)$ }\\
u(x,0)=u_{0}(x) &\textrm{
on $\Omega\times\{0\}$ }\\
u_{t}(x,0)=v_{0}(x) &\textrm{
on $\Omega\times\{0\}$ }\\
\end{array}\right.\end{eqnarray}
To use the Hille--Yosida theorem, we
first transfer the second order (in time)
scalar equation in to a first order (in time)
system:
Set $v=u_t$ and
$U=(u,v)$. Then
$$U_{t}=(u,v)_{t}=(v,\tri u)=:AU.$$
Also $U_{0}=(u_{0},v_{0})$.
Set $\bX= H^1_0(\Omega) \times L^{2}(\Omega)$, and
$D(A)=(H^2(\Omega)\cap H^1_0(\Omega)) \times H^1_0(\Omega). $
It can be proven that $A$ satisfies the conditions of Hille-Yosida
theorem and consequently A generates a semigroup on
$\bX=H^1_0(\Omega)\times L^2(\Omega)$; namely,
the wave equation is solvable for every initial data $(u_0,v_0)\in \bX$.}
\end{example}