%% R.E. Showalter: Chapter 5
\chapter{Implicit Evolution Equations}
\section{Introduction}
\setcounter{equation}{0}
\setcounter{theorem}{0} 

We shall be concerned with evolution equations in which the time-derivative 
of the solution is not given explicitly. 
This occurs, for example, in problems containing the pseudoparabolic 
equation 
\begin{equation}\label{eq511}
\partial_tu (x,t) - a\partial_x^2 \partial_t u(x,t) - \partial_x^2 u(x,t) 
= f(x,t) 
\end{equation} 
where the constant $a$ is non-zero. 
However, \eqn{511} can be reduced to the standard evolution equation (3.4) 
in an appropriate space because the operator $I-a\partial_x^2$ which acts 
on $\partial_t u(x,t)$ can be inverted. 
Thus, \eqn{511} is an example of a {\it regular\/} equation; we study 
such problems in Section~2. 
Section~3 is concerned with those regular equations of a special form 
suggested by \eqn{511}. 

Another example which motivates some of our discussion is the partial 
differential equation 
\begin{equation}\label{eq512} 
m(x)\partial_t u(x,t) - \partial_x^2 u(x,t) = f(x,t) 
\end{equation} 
where the coefficient is non-negative at each point. 
The equation \eqn{512} is parabolic at those points where $m(x)>0$ 
and elliptic where $m(x)=0$. 
For such an equation of {\it mixed type\/} some care must be taken in order 
to prescribe a well posed problem. 
If $m(x)>0$ almost everywhere, then \eqn{512} is a model of a regular 
evolution equation. 
Otherwise, it is a model of a {\it degenerate\/} equation. 
We study the Cauchy problem for degenerate equations in Section~4 and in 
Section~5 give more examples of this type. 

\section{Regular Equations}		% 2.
\setcounter{equation}{0}

Let $V_m$ be a Hilbert space with scalar-product $(\cdot,\cdot)_m$ and 
denote the corresponding Riesz map from $V_m$ onto the dual $V'_m$ by 
$\M$. That is, 
$$\M x(y) = (x,y)_m\ ,\qquad x,y\in V_m\ .$$ 
Let $D$ be a subspace of $V_m$ and $L:D\to V'_m$ a linear map. 
If $u_0\in V_m$ and $f\in C((0,\infty),V'_m)$ are given, we consider the 
problem of finding $u\in C([0,\infty),V_m)\cap C^1((0,\infty),V_m)$ such that 
\begin{equation}\label{eq521}
\M u'(t) + Lu(t) = f(t)\ ,\qquad t>0\ ,
\end{equation} 
and $u(0)=u_0$. 

Note that \eqn{521} is a generalization of the evolution 
equation IV\eqn{421}.  				%(2.1). 
If we identify $V_m$ with $V'_m$ by the Riesz map $\M$ (i.e., take $\M=I$) 
then \eqn{521} reduces to IV\eqn{421}.	%(2.1). 
In the general situation we shall solve \eqn{521} by reducing it to a 
Cauchy problem equivalent to IV\eqn{421}.	%(2.1). 

We first obtain our a-priori estimate for a solution $u(\cdot)$ of \eqn{521}, 
with $f=0$ for simplicity. 
For such a solution we have 
$$D_t (u(t),u(t))_m = -2\Re Lu (t) (u(t))$$ 
and this suggests consideration of the following. 

\definition 
The linear operator $L:D\to V'_m$ with $D\le V_m$ is {\it monotone\/} 
(or {\it non-negative\/}) if 
$$\Re Lx(x) \ge 0\ ,\qquad x\in D\ .$$ 
We call $L$ {\it strictly monotone\/} (or {\it positive\/}) if 
$$\Re Lx (x) >0\ ,\qquad x\in D\ ,\ x\ne0\ .$$ 
Our computation above shows there is at most one solution of the Cauchy 
problem for \eqn{521} whenever $L$ is monotone, and it suggests that $V_m$ 
is the correct space in which to seek well-posedness results for \eqn{521}. 

To obtain an (explicit) evolution equation in $V_m$ which is equivalent 
to \eqn{521}, we need only operate on \eqn{521} with the inverse of the 
isomorphism $\M$, and this gives 
\begin{equation}\label{eq522} 
u'(t) + \M^{-1} \circ Lu (t) = \M^{-1} f(t)\ ,\qquad t>0\ .
\end{equation} 
This suggests we define $A= \M^{-1} \circ L$ with domain $D(A)=D$, 
for then \eqn{522} is equivalent to IV\eqn{421}. 
Furthermore, since $\M$ is the Riesz map determined by the scalar-product 
$(\cdot,\cdot)_m$, we have 
\begin{equation}\label{eq523} 
(Ax,y)_m = Lx(y)\ ,\qquad x\in D\ ,\ y\in V_m\ . 
\end{equation} 
This shows that $L$ is monotone if and only if $A$ is accretive. 
Thus, it follows from Theorem IV.\ref{thm4-4C} that $-A$ generates a 
contraction semigroup on $V_m$ if and only if $L$ is monotone and $I+A$ 
is surjective. Since $\M(I+A) = \M+L$, we obtain the following result 
from Theorem IV.\ref{thm4-3C}. 

\begin{theorem}\label{thm5-2A} 
Let $\M$ be the Riesz map of the Hilbert space $V_m$ with scalar product 
$(\cdot,\cdot)_m$ and let $L$ be linear from the subspace $D$ of $V_m$ 
into $V'_m$. Assume that $L$ is monotone and $\M+L:D\to V'_m$ is surjective. 
Then, for every $f\in C^1([0,\infty),V'_m)$ and $u_0\in D$ there is a unique 
solution $u(\cdot)$ of \eqn{521} with $u(0)=u_0$. 
\end{theorem} 

In order to obtain an analogue of the situation in Section IV.6, we suppose 
$L$ is obtained from a continuous sesquilinear form. 
In particular, let $V$ be a Hilbert space for which $V$ is a dense subset 
of $V_m$ and the injection is continuous; hence, we can identify $V'_m
\subset V'$.  
Let $\ell(\cdot,\cdot)$ be continuous and sesquilinear on $V$ and define the 
corresponding linear map $\L:V\to V'$ by 
$$\L x(y)  = \ell(x,y)\ ,\qquad x,y\in V\ .$$ 
Define $D\equiv \{x\in V:\L x\in V'_m\}$ and $L= \L |_D$. 
Then \eqn{523} shows that 
$$\ell (x,y) = (Ax,y)_m\ ,\qquad x\in D\ ,\ y\in V\ ,$$ 
so it follows that $A$ is the operator determined by the triple 
$\{\ell (\cdot,\cdot),V,V_m\}$ as in Theorem IV.\ref{thm4-6A}. 
Thus, from Theorems IV.\ref{thm4-6C} and IV.\ref{thm4-6E} we obtain 
the following. 

\begin{theorem}\label{thm5-2B} 
Let $\M$ be the Riesz map of the Hilbert space $V_m$ with scalar-product 
$(\cdot,\cdot)_m$. Let $\ell(\cdot,\cdot)$ be a continuous, 
sesquilinear and elliptic form on the Hilbert space $V$, which is assumed 
dense and continuously imbedded in $V_m$, and denote the corresponding 
isomorphism of $V$ onto $V'$ by $\L$. 
Then for every H\"older continuous $f:[0,\infty)\to V'_m$ and $u_0\in V_m$, 
there is a unique $u\in C([0,\infty),V_m)\cap C^1((0,\infty),V_m)$ 
such that $u(0)=u_0$, $\L u(t)\in V'_m$ for $t>0$, and 
\begin{equation}\label{eq524} 
\M u'(t) + \L u (t) = f(t)\ ,\qquad t>0\ .
\end{equation} 
\end{theorem} 

We give four elementary examples to suggest the types of initial-boundary 
value problems to which the above results can be applied. 
In the first three of these examples we let $V_m = H_0^1 (0,1)$ with 
the scalar-product 
$$(u,v)_m = \int_0^1 ( u\bar v + a\partial u\partial \bar v\,)\ ,$$ 
where $a>0$. 

\subsection{}		% 2.1 
Let $D= \{ u\in H^2 (0,1)\cap H_0^1 (0,1):u'(0)=cu'(1)\}$ where $|c|\le 1$, 
and define $LU = -\partial^3 u$. 
Then we have $Lu(\varphi) = (\partial^2 u,\partial\varphi)$ for 
$\varphi \in H_0^1 (0,1)$, and (cf., Section IV.4) 
$$2\Re Lu (u) = |u'(1)|^2 - |u'(0)|^2 \ge 0\ ,\qquad u\in D\ .$$ 
Thus, Theorem \ref{thm5-2A} shows that the initial-boundary value problem 
\begin{eqnarray*} 
&&(\partial_t-a\partial_x^2 \partial_t) U(x,t) - \partial_x^3 U(x,t)=0\ ,
\qquad 0<x<1\ ,\ t\ge 0\ ,\\
\noalign{\vskip6pt} 
&&U(0,t) = U(1,t) = 0\ ,\quad \partial_xU(0,t) = c\partial_xU(1,t)\ ,
\qquad \  t\ge 0\ ,\\
\noalign{\vskip6pt} 
&&U(x,0) = U_0(x)
\end{eqnarray*} 
has a unique solution whenever $U_0\in D$. 

\subsection{}		% 2.2 
Let $V= H_0^2 (0,1)$ and define 
$$\ell (u,v) = \int_0^1 \partial^2 u\cdot \partial^2 \bar v\ ,\qquad
u,v\in V\ .$$ 
Then $D= H_0^2 (0,1) \cap H^3 (0,1)$ and $Lu=\partial^4 u$, $u\in D$. 
Theorem \ref{thm5-2B} then asserts the existence and uniqueness of a 
solution of the problem 
\begin{eqnarray*} 
&&(\partial_t - a\partial_x^2 \partial_t) U(x,t) + \partial_x^4 U(x,t)=0\ ,
\qquad 0<x<1\ ,\ t>0\ ,\\
\noalign{\vskip6pt} 
&&U(0,t) = U(1,t) = \partial_x U(0,t) = \partial_x U(1,t)=0\ ,\qquad t>0\ ,\\
\noalign{\vskip6pt} 
&&U(x,0) = U_0(x) \ ,\qquad 0<x<1\ ,
\end{eqnarray*} 
for each $U_0 \in H_0^1 (0,1)$. 

\subsection{}		%2.3 
Let $V= H_0^1(0,1)$ and define 
$$\ell(u,v) = \int_0^1 \partial u\partial \bar v\ ,\qquad u,v\in V\ .$$ 
Then $D= V=V_m$ and $Lu = -\partial^2 u$, $u\in D$. 
From either Theorem \ref{thm5-2A} or \ref{thm5-2B} we obtain existence 
and uniqueness for the problem 
\begin{eqnarray*} 
&&(\partial_t-a\partial_x^2 \partial_t)U(x,t) - \partial_x^2U(x,t)=0\ ,
\qquad 0<x<1\ ,\ t>0\ ,\\
\noalign{\vskip6pt} 
&&U(0,t) = U(1,t) =0\ ,\qquad t>0\ ,\\
\noalign{\vskip6pt} 
&&U(x,0) = U_0(x)\ ,\qquad 0<x<1\ ,
\end{eqnarray*} 
whenever $U_0\in D=V_m$. 

\subsection{}		%2.4 
For our last example we let $V_m$ be the completion of $C_0^\infty (G)$ 
with the scalar-product 
$$(u,v)_m \equiv \int_G m(x) u(x)\overline{v(x)}\,dx\ .$$ 
We assume $G$ is open in $\RR^n$ and $m\in L^\infty (G)$ is given with 
$m(x)>0$ for a.e.\ $x\in G$. 
(Thus, $V_m$ is the set of measurable functions $u$ on $G$ for which 
$m^{1/2} u\in L^2(G)$.) 
Let $V= H_0^1 (G)$ and define 
$$\ell (u,v) = \int_G \nabla u\cdot\nabla \bar v\ ,\qquad u,v\in V\ .$$ 
Then Theorem \ref{thm5-2B} implies the existence and uniqueness of a 
solution of the problem 
\begin{eqnarray*} 
&&m(x) \partial_t U(x,t) -\Delta_n U(x,t) = 0\ ,\qquad x\in G\ ,\ t>0\ ,\\
\noalign{\vskip6pt} 
&&U(s,t) = 0\ ,\qquad s\in \partial G\ ,\ t>0\ ,\\
\noalign{\vskip6pt} 
&&U(x,0) = U_0(x)\ ,\qquad x\in G\ . 
\end{eqnarray*} 
Note that the initial condition is attained in the sense that 
$$\lim_{t\to 0^+} \int_G m(x)|U(x,t) - U_0(x)|^2\,dx = 0\ .$$ 

The first two of the preceding examples illustrate the use of 
Theorems \ref{thm5-2A} and \ref{thm5-2B} when $\M$ and $L$ are both 
differential operators with the order of $L$ strictly higher than the order 
of $M$. The equation in \eqn{522} is called {\it metaparabolic\/} and 
arises in special models of diffusion or fluid flow. 
The equation in \eqn{523} arises similarly and is called 
{\it pseudoparabolic\/}. 
We shall discuss this class of problems in Section~3. 
The last example \eqn{524} contains a {\it weakly degenerate\/} 
parabolic equation. 
We shall study such problems in Section~4 where we shall assume only 
that $m(x)\ge0$, $x\in G$. 
This allows the equation to be of {\it mixed type\/}: parabolic where 
$m(x)>0$ and elliptic where $m(x)=0$. 
Such examples will be given in Section~5. 

\section{Pseudoparabolic Equations}	% 3 
\setcounter{equation}{0} 

We shall consider some evolution equations which generalize the example 
\eqn{523}. Two types of solutions will be discussed, and we shall show how 
these two types differ essentially by the boundary conditions they satisfy. 

\begin{theorem}\label{thm5-3A} 
Let $V$ be a Hilbert space, suppose $m(\cdot,\cdot)$ and $\ell(\cdot,\cdot)$ 
are continuous sesquilinear forms on $V$, and denote by $\M$ and $\L$ the 
corresponding operators in $\L(V,V')$. 
(That is, $\M x(y) = m(x,y)$ and $\L x(y) =\ell(x,y)$ for $x,y\in V$.) 
Assume that $m(\cdot,\cdot)$ is $V$-coercive. 
Then for every $u_0\in V$ and $f\in C(\RR,V')$, there is a unique $u\in C^1
(\RR,V)$ for which \eqn{524} holds for all $t\in\RR$ and $u(0)=u_0$. 
\end{theorem} 

\proof 
The coerciveness assumption shows that $\M$ is an isomorphism of $V$ onto 
$V'$, so the operator $A\equiv \M^{-1}\circ \L$ belongs to $\L(V)$. 
We can define $\exp (-tA) \in \L(V)$ as in Theorem IV.\ref{thm4-2A} and 
then define 
\begin{equation}\label{eq531}
u(t) = \exp (-tA)\cdot u_0 + \int_0^t \exp (A(\tau-t)) \circ \M^{-1} 
f(\tau)\,d\tau\ ,\qquad t>0\ .
\end{equation} 
Since the integrand is continuous and appropriately bounded, it follows 
that \eqn{531} is a solution of \eqn{522}, hence of \eqn{521}. 
We leave the proof of uniqueness as an exercise. 
\qed 

We call the solution $u(\cdot)$ given by Theorem \ref{thm5-3A} a 
{\it weak solution\/} of \eqn{521}. 
Suppose we are given a Hilbert space $H$ in which $V$ is a dense subset, 
continuously imbedded. 
Thus $H\subset V'$ and we can define $D(M) = \{v\in V:\M v\in H\}$, 
$D(L) = \{v\in V:\L v\in H\}$ and corresponding operators $M=\M|_{D(M)}$ 
and $L= \L|_{D(L)}$ in $H$. 
A solution $u(\cdot)$ of \eqn{521} for which each term in \eqn{521} belongs 
to $C(\RR,H)$ (instead of $C(\RR,V'))$ is called a {\it strong solution\/}. 
Such a function satisfies 
\begin{equation}\label{eq532} 
Mu' (t) + Lu(t) = f(t)\ ,\qquad t\in \RR\ .
\end{equation} 

\begin{theorem}\label{thm5-3B} 
Let the Hilbert space $V$ and operators $\M,\L\in \L(V,V')$ be given as in 
Theorem \ref{thm5-3A}. 
Let the Hilbert space $H$ be given as above and define the domains $D(M)$ 
and $D(L)$ and operators $M$ and $L$ as above. Assume $D(M)\subset D(L)$. 
Then for every $u_0\in D(M)$ and $f\in C(\RR,H)$ there is a (unique) 
strong solution $u(\cdot)$ of \eqn{532} with $u(0)=u_0$. 
\end{theorem}

\proof 
By making the change-of-variable $v(t) = e^{-\lambda t} u(t)$ for some 
$\lambda>0$ sufficiently large, we may assume without loss of generality 
that $D(M) = D(L)$ and $\ell(\cdot,\cdot)$ is $V$-coercive. 
Then $L$ is a bijection onto $H$ so we can define a norm on $D(L)$ by 
$\|v\|_{D(L)} = \|Lv\|_H$, $v\in D(L)$, which makes $D(L)$ a Banach space. 
(Clearly, $D(L)$ is also a Hilbert space.) 
Since $\ell(\cdot,\cdot)$ is $V$-coercive, it follows that for some $c>0$ 
$$c\|v\|_V^2 \le \|Lv\|_H \|v\|_H\ ,\qquad v\in D(L)\ ,$$ 
and the continuity of the injection $V\hookrightarrow H$ shows then that 
the injection $D(L)\hookrightarrow V$ is continuous. 
The operator $A\equiv \M^{-1} \L\in \L(V)$ leaves invariant the subspace 
$D(L)$. This implies that the restriction of $A$ to $D(L)$ is a closed 
operator in the $D(L)$-norm. 
To see this, note that if $v_n\in D(L)$ and if $\|v_n-u_0\|_{D(L)} \to0$, 
$\|Av_n-v_0\|_{D(L)}\to0$, then 
\begin{eqnarray*} 
\|v_0 -Au_0\|_V &\le & \|v_0 -Av_n\|_V + \|A(v_n- u_0)\|_V\\
\noalign{\vskip6pt} 
&\le & \|v_0 - Av_n\|_V + \|A\|_{\L(V)} \|v_n- u_0\|_V\ ,
\end{eqnarray*} 
so the continuity of $D(L)\hookrightarrow V$ implies that each of these 
terms converges to zero.
Hence, $v_0=Au_0$. 

Since $A|_{D(L)}$ is closed and defined everywhere on $D(L)$, it 
follows from Theorem III.\ref{thm3-7A} 
that it is continuous on $D(L)$. 
Therefore, the restrictions of the operators $\exp (-tA)$, $t\in \RR$, 
are continuous on $D(L)$, and the formula \eqn{531} in $D(L)$ gives a 
strong solution as desired. 

\begin{corollary}\label{cor5-3C} 
In the situation of Theorem \ref{thm5-3B}, the weak solution $u(\cdot)$ 
is a strong solution if and only if $u_0\in D(M)$. 
\end{corollary}

\subsection{}		%3.1 
We consider now an abstract {\it pseudoparabolic\/} initial-boundary value 
problem. Suppose we are given the Hilbert spaces, forms and operators as in 
Theorem IV.\ref{thm4-7B}. Let $\varep>0$ and define 
$$\begin{array}{rcll}
m(u,v)&=&(u,v)_H +\varep a(u,v)&\\
\noalign{\vskip6pt}
\ell(u,v)&=&a(u,v)\ ,&\qquad u,v\in V\ .
\end{array}$$ 
Thus, $D(M)= D(L)=D(A)$. 
Let $f\in C(\RR,H)$. 
If $u(\cdot)$ is a strong solution of \eqn{532}, then we have 
\begin{equation}\label{eq533}
\left.\begin{array}{ll} 
u'(t) + \varep A_1u'(t) + A_1u(t) = f(t)\ ,&\\
\noalign{\vskip6pt}
u(t) \in V\ ,\ \hbox{ and}&\\
\noalign{\vskip6pt}
\partial_1u(t)+\A_2\gamma(u(t))=0\ ,&\qquad t\in \RR\ .
\end{array}\right\}
\end{equation} 

Suppose instead that $F\in C(\RR,H)$ and $g\in C(\RR,B')$. If we define 
$$f(t) (v) \equiv (F(t),v)_H + g(t)(\gamma (v))\ ,\qquad v\in V\ , \ 
t\in \RR\ .$$ 
then a weak solution $u(\cdot)$ of \eqn{524} can be shown by a computation 
similar to the proof of Theorem III.\ref{thm3-3A} to satisfy 
\begin{equation}\label{eq534}
\left.\begin{array}{l}
u'(t)+\varep A_1 u'(t) + A_1 u(t) = F(t)\ ,\\
\noalign{\vskip6pt}
u(t)\in V\ ,\ \hbox{ and}\\
\noalign{\vskip6pt}
\partial_1 (\varep u'(t) +u(t)) + \A_2(\gamma(\varep u'(t)+u(t)))=g(t)\ ,
\qquad t\in \RR\ .
\end{array}\right\} 
\end{equation} 
Note that \eqn{533} implies more than \eqn{534} with $g\equiv 0$. 
By taking suitable choices of the operators above, we could obtain examples 
of initial-boundary value problems from \eqn{533} and \eqn{534} as 
in Theorem IV.\ref{thm4-7C}. 

\subsection{}		%3.2 
For our second example we let $G$ be open in $\RR^n$ and choose $V=\{v\in H^1
(G): v(s)=0$. a.e.\ $s\in\Gamma\}$, where $\Gamma$ is a closed subset of 
$\partial G$. We define 
$$m(u,v) = \int_G\nabla u(x)\cdot\nabla \overline{v(x)}\, dx\ ,\qquad 
u,v\in V$$ 
and assume $m(\cdot,\cdot)$ is $V$-elliptic. 
(Sufficient conditions for this situation are given in 
Corollary III.\ref{cor3-5D}.) 
Choose $H=L^2(G)$ and $V_0=H_0^1(G)$; the corresponding partial 
differential operator $M:V\to V'_0 \le \D^*(G)$ is given by $Mu=-\Delta_nu$, 
the Laplacian (cf.\ Section III.2.2). 
Thus, from Corollary III.\ref{cor3-3B} it follows that $D(M)=\{u\in V:
\Delta_n u\in L^2(G)$, $\partial u=0\}$ where $\partial$ is the normal 
derivative $\partial_\nu$ on $\partial G\sim \Gamma$ whenever 
$\partial G$ is sufficiently smooth. 
(Cf.\ Section III.2.3.) 
Define  a second form on $V$ by 
$$\ell (u,v) = \int_G a(x)\partial_n u(x)\overline{v(x)}\,dx\ ,\qquad 
u,v\in V\ ,$$ 
and note that $L=\L: V\to H\le V'$ is given by $\L u= a(x)(\partial u/
\partial x_n)$, where  $a(\cdot)\in L^\infty (G)$ is given. 
Assume that for each $t\in \RR$ we are given $F(\cdot,t)\in L^2(G)$ and 
that the map $t\mapsto F(\cdot,t):\RR\to L^2(G)$ is continuous. 
Let $g(\cdot,t)\in L^2(\partial G)$ be given similarly, and define 
$f\in C(\RR,V')$ by 
$$f(t) (v) = \int_G F(x,t)\overline{v(x)}\, dx 
+ \int_{\partial G} g(s,t)\overline{v(s)}\, ds\ ,\qquad 
t\in \RR\ ,\ v\in V\ .$$ 
If $u_0\in V$, then Theorem \ref{thm5-3A} gives a unique weak solution 
$u(\cdot)$ of \eqn{524} with $u(0)=u_0$. 
That is 
$$m(u'(t),v)+\ell (u(t),v) = f(t)(v)\ ,\qquad v\in V\ ,\ t\in \RR\ ,$$ 
and this is equivalent to 
$$\begin{array}{ll}
Mu'(t) + Lu(t) = F(\cdot,t)\ ,&\qquad t\in \RR\\
\noalign{\vskip6pt}
u(t)\in V\ ,\quad  \partial_t(\partial u(t)) = g(\cdot,t)\ .&
\end{array}$$ 
From Theorem IV.\ref{thm4-7A} we thereby obtain a generalized solution 
$U(\cdot,\cdot)$ of the initial-boundary value problem 
$$\begin{array}{ll}
-\Delta_n\partial_t U(x,t) + a(x)\partial_n U(x,t) = F(x,t)\ ,
&\qquad x\in G\ ,\ t\in \RR\ ,\\
\noalign{\vskip6pt}
U(s,t) = 0\ ,&\qquad s\in \Gamma\ ,\\
\noalign{\vskip6pt}
\partial_\nu U(s,t) = \partial_\nu U_0(s)+ 
\ds \int_0^t g(s,\tau)\,d\tau\ ,&\qquad s\in\partial G\sim \Gamma\ ,\\
\noalign{\vskip6pt}
U(x,0)= U_0 (x)\ ,&\qquad x\in G\ .
\end{array}$$
Finally, we note that $f\in C(\RR,H)$ if and only if $g\equiv 0$, and then 
$\partial_\nu U(s,t) =\partial_\nu U_0(s)$ for $s\in\partial G\sim\Gamma$, 
$t\in \RR$; thus, $U(\cdot,t)\in D(M)$ if and only if $U_0\in D(M)$. 
This agrees with Corollary \ref{cor5-3C}. 

\section{Degenerate Equations}		% 4 
\setcounter{equation}{0} 

We shall consider the evolution equation \eqn{521} in the situation where 
$\M$ is permitted to degenerate, i.e., it may vanish on non-zero vectors. 
Although it is not possible to rewrite it in the form \eqn{522}, we shall 
essentially factor the equation \eqn{521} by the kernel of $\M$ and 
thereby obtain an equivalent problem which is regular. 

Let $V$ be a linear space and $m(\cdot,\cdot)$ a sesquilinear form on $V$ 
that is symmetric and non-negative: 
$$\begin{array}{ll}
m(x,y) = \overline{m(x,y)}\ ,&\qquad x,y\in V\ ,\\
\noalign{\vskip6pt}
m(x,x)\ge 0\ ,&\qquad x\in V\ .\end{array}$$ 
Then it follows that 
\begin{equation}\label{eq541}
|m(x,y)|^2 \le m(x,x)\cdot m(y,y)\ ,\qquad x,y\in V\ ,
\end{equation} 
and that $x\mapsto m(x,x)^{1/2}= \|x\|_m$ is a seminorm on $V$. 
Let $V_m$ denote this seminorm space whose dual $V'_m$ is a Hilbert space 
(cf.\ Theorem I.\ref{thm1-3E}). The identity 
$$\M x(y) = m(x,y) \ ,\qquad x,y\in V$$ 
defines $\M\in \L(V_m,V'_m)$ and it is just such an operator which we shall 
place in the leading term in our evolution equation. 
Let $D\le V$, $L\in L(D,V'_m)$, $f\in C((0,\infty),V'_m)$ and $g_0\in V'_m$.  
We consider the problem of finding a function $u(\cdot) :[0,\infty)\to V$ 
such that 
$$\M u(\cdot) \in C([0,\infty),V'_m) \cap C^1 ((0,\infty),V'_m)\ ,\qquad 
(\M u)(0)=g_0\ ,$$ 
and $u(t) \in D$ with 
\begin{equation}\label{eq542}
(\M u)'(t) + Lu(t) = f(t)\ ,\qquad t>0\ .
\end{equation} 
(Note that when $m(\cdot,\cdot)$ is a scalar product on $V_m$ and $V_m$ 
is complete then $\M$ is the Riesz map and \eqn{542} is equivalent to 
\eqn{521}.) 

Let $K$ be the kernel of the linear map $\M$ and denote the corresponding 
quotient space by $V/K$. If $q:V\to V/K$ is the canonical surjection, 
then we define by 
$$m_0 (q(x),q(y)) = m(x,y)\ ,\qquad x,y\in V$$ 
a scalar product $m_0(\cdot,\cdot)$ on $V/K$. 
The completion  of $V/K$, $m_0(\cdot,\cdot)$ is a Hilbert space $W$ 
whose scalar product is also denoted by $m_0(\cdot,\cdot)$. 
(Cf.\ Theorem I.\ref{thm1-4B}.) 
We regard $q$ as a map of $V_m$ into $W$; thus, it is norm-preserving 
and has a dense range, so its dual $q':W'\to V'_m$ is a norm-preserving 
isomorphism (Corollary I.\ref{cor1-5C}) defined by 
$$q'(f)(x) = f(q(x))\ ,\qquad f\in W'\ ,\ x\in V_m\ .$$ 
If $\M_0$ denotes the Riesz map of $W$ with the scalar product $m_0(\cdot,
\cdot)$, then we have 
$$\begin{array}{rcl}
q'\M_0 q(x)(y)&=& \M_0 q(x) (q(y)) = m_0(q(x),q(y))\\
\noalign{\vskip6pt}
&=&\M x(y)\ ,\end{array}$$ 
hence, 
\begin{equation}\label{eq543} 
q'\M_0 q= \M\ .
\end{equation} 

From the linear map $L:D\to V'_m$ we want to construct a linear map $L_0$ on 
the image $q[D]$ of $D\le V_m$ by $q$ so that it satisfies 
\begin{equation}\label{eq544} 
q'L_0 q= L\ .
\end{equation} 
This is possible if (and, in general, only if ) $K\cap D$ is a subspace of 
the kernel of $L$, $K(L)$ by Theorem I.\ref{thm1-1A}, and we shall assume 
this is so. 

Let $f(\cdot)$ and $g_0$ be given as above and consider the problem of 
finding a function $v(\cdot) \in C([0,\infty),W)\cap C^1((0,\infty),W)$ 
such that $v(0) = (q'\M_0)^{-1} g_0$ and 
\begin{equation}\label{eq545} 
\M_0 v'(t) + L_0v(t) = (q')^{-1} f(t)\ ,\qquad t>0\ .
\end{equation} 
Since the domain of $L_0$ is $q[D]$, if $v(\cdot)$ is a solution of 
\eqn{545} then for each $t>0$ we can find a $u(t)\in D$ for which $v(t)= 
q(u(t))$. But $q'\M_0 :W\to V'_m$ is an isomorphism and so from \eqn{543}, 
\eqn{544} and \eqn{545} it follows that $u(\cdot)$ is a solution of \eqn{542} 
with $\M u(0)=g_0$. 
This leads to the following results. 

\begin{theorem}\label{thm5-4A} 
Let $V_m$ be a seminorm space obtained from a symmetric and non-negative 
sesquilinear form $m(\cdot,\cdot)$, and let $\M\in \L(V_m,V'_m)$ be the 
corresponding linear operator given by $\M x(y) = m(x,y)$, $x,y\in V_m$. 
Let $D$ be a subspace of $V_m$ and $L:D\to V'_m$ be linear and monotone. 
{\rm (a)}~If $K(\M) \cap D\le K(L)$ and if $\M+L:D\to V'_m$ 
is a surjection, then 
for every $f\in C^1([0,\infty),V'_m)$ and $u_0\in D$ there exists a 
solution of \eqn{542} with $(\M u)(0)= \M u_0$. 
{\rm (b)}~If $K(\M) \cap K(L)=\{0\}$, then there is at most one solution. 
\end{theorem} 

\proof 
The existence of a solution will follow from Theorem \ref{thm5-2A} applied 
to \eqn{545} if we show $L_0 :q[D]\to W'$ is monotone and $\M_0 + L_0$ is onto. 
But \eqn{545} shows $L_0$ is monotone, and the identity 
$$q'(\M_0 +L_0) q(x) = (\M+L) (x)\ ,\qquad x\in D\ ,$$ 
implies that $\M_0 +L_0$ is surjective whenever $\M+L$ is surjective. 

To establish the uniqueness result, let $u(\cdot)$ be a solution of 
\eqn{542} with $f\equiv 0$ and $\M u(0)=0$; define $v(t)=qu(t)$, $t\ge0$. 
Then 
$$D_t m_0 (v(t),v(t)) = 2\Re (\M_0 v'(t))(v(t))\ ,\qquad t>0\ ,$$ 
and this implies by \eqn{543} that 
$$\begin{array}{rcll} 
D_t m(u(t),u(t)) &=& 2\Re (\M u)'(t)(u(t))&\\
\noalign{\vskip6pt}
&=&-2\Re Lu (t) (u(t))\ ,&\qquad t>0\ .\end{array}$$

Since $L$ is monotone, this shows $\M u(t)=0$, $t\ge0$, and \eqn{542} implies 
$Lu(t)=0$, $t>0$. Thus $u(t)\in K(\M) \cap K(L)$, $t\ge0$, and the desired 
result follows. 

We leave the proof of the following analogue of Theorem \ref{thm5-2B} 
as an exercise. 

\begin{theorem}\label{thm5-4B} 
Let $V_m$ be a seminorm space obtained from a symmetric and non-negative 
sesquilinear form $m(\cdot,\cdot)$, and let $\M\in \L(V_m,V'_m)$ denote 
the corresponding operator. 
Let $V$ be a Hilbert space which is dense and continuously imbedded in $V_m$. 
Let $\ell (\cdot,\cdot)$ be a continuous, sesquilinear and elliptic form 
on $V$, and denote the corresponding isomorphism of $V$ onto $V'$ by $\L$. 
Let $D= \{u\in V:\L u\in V'_m\}$. 
Then, for every H\"older continuous $f:[0,\infty)\to V'_m$ and every $u_0\in 
V_m$, there exists a unique solution of \eqn{542} with $(\M u)(0)=\M u_0$. 
\end{theorem}

\section{Examples}		% 5 
\setcounter{equation}{0} 

We shall illustrate the applications of Theorems \ref{thm5-4A} and 
\ref{thm5-4B} by solving some initial-boundary value problems with partial 
differential equations of mixed type. 

\subsection{}		% 5.1 
Let $V_m = L^2 (0,1)$, $0\le a<b\le 1$, and 
$$m(u,v) = \int_a^b u(x)\overline{v(x)}\,dx\ ,\qquad v\in V_m\ .$$ 
Then $V'_m=L^2(a,b)$, which we identify as that subspace of $L^2(0,1)$ 
whose elements are zero a.e.\ on $(0,a)\cup (b,1)$, and $\M$ becomes 
multiplication by the characteristic function of the interval $(a,b)$. 
Let $L=\partial$ with domain $D= \{u\in H^1(0,1):u(0)= cu(1)$, 
$\partial u\in V'_m \subset L^2(0,1)\}$. 
We assume $|c|\le1$, so $L$ is monotone (cf.\ Section IV.4(a)). 
Note that each function in $D$ is constant on $(0,a)\cup (b,1)$. 
Thus, $K(\M) \cap D=\{0\}$ and $K(\M)\cap D\le K(L)$ follows. 
Also, note that $K(L)$ is either $\{0\}$ or consists of the constant 
functions, depending  on whether or not $c\ne1$, respectively. 
Thus, $K(\M)\cap K(L)=\{0\}$. 
If $u$ is the solution of (cf.\ Section IV.4(a)) 
$$u(x) + \partial u(x) = f(x)\ ,\qquad a<x<b\ ,\ u(a) = cu(b)$$ 
and is extended to $(0,1)$ by being constant on each of the intervals, 
$[0,a]$ and $[b,1]$, then $(\M+L)u = f\in V'_m$. 
Hence $\M+L$ maps onto $V'_m$ and Theorem \ref{thm5-4A} asserts the 
existence and uniqueness of a generalized solution of the problem 
\begin{equation}\label{eq551}
\left.\begin{array}{l}
\partial_t U(x,t) +\partial_xU(x,t) = F(x,t)\ ,\qquad a<x<b\ ,\ t\ge0\ ,\\
\noalign{\vskip6pt}
\partial_x U(x,t)=0\ ,\qquad x\in (0,a)\cup (b,a)\ ,\\
\noalign{\vskip6pt} 
U(0,t) = cU(1,t)\ ,\qquad U(x,0) = U_0(x)\ ,\qquad a<x<b\ ,
\end{array}\right\}
\end{equation}
for appropriate $F(\cdot,\cdot)$ and $U_0$. 
This example is trivial (i.e., equivalent to Section IV.4(a) on the 
interval $(a,b)$) but motivates the proof-techniques of Section~4. 

\subsection{}		% 5.2 
We consider some problems with a partial differential equation of mixed 
elliptic-parabolic type. 
Let $m_0(\cdot)\in L^\infty (G)$ with $m_0(x)\ge0$, a.e.\ $x\in G$, an open 
subset of $\RR^n$ whose boundary $\partial G$ is a $C^1$-manifold with $G$ 
on one side of $\partial G$. 
Let $V_m= L^2(G)$ and 
$$m(u,v) = \int_G m_0(x) u(x)\overline{v(x)}\, dx\ ,\qquad u,v\in V_m\ .$$ 
Then $\M$ is multiplication by $m_0(\cdot)$ and maps $L^2(G)$ into 
$V'_m \equiv \{\sqrt{m_0} \cdot g:g\in L^2 (G)\}\subset L^2(G)$. 
Let $\Gamma$ be a closed subset of $\partial G$ and define $V= \{v\in H^1(G):
\gamma_0 v=0$ on $\Gamma\}$ as in Section III.4.1. 
Let 
\begin{equation}\label{eq552}
\ell (u,v) = \int_G \nabla u\cdot\overline{\nabla v}\, dx\ ,\qquad u,v\in V 
\end{equation} 
and assume $\sum\equiv \{ s\in\partial G:\nu_n(s)>0\}\subset \Gamma$. 
Thus, Theorem III.\ref{thm3-5C} implies $\ell(\cdot,\cdot)$ is $V$-elliptic, 
so $\M+\L$ maps onto $V'$, hence, onto $V'_m$. 
Theorem \ref{thm5-4B} shows that if $U_0\in L^2(G)$ and if $F$ is given as 
in Theorem IV.\ref{thm4-7C}, then there is a unique generalized solution of 
the problem 
\begin{equation}\label{eq553}
\left.\begin{array}{ll}
\partial_t (m_0(x)U(x,t)) -\Delta_n U(x,t)=m_0(x)F(x,t)\ ,&\qquad x\in G\ ,\\
\noalign{\vskip6pt}
U(s,t)=0\ ,\qquad s\in\Gamma\ ,\\
\noalign{\vskip6pt}
\ds {\partial U(s,t)\over\partial \nu} =0\ ,\quad s\in\partial G\sim \Gamma\ ,
&\qquad t>0\ ,\\
\noalign{\vskip6pt}
m_0(x)(U(x,0)- U_0(x)) =0\ .&
\end{array}\right\}
\end{equation}
The partial differential equation in \eqn{553} is parabolic at those $x\in G$ 
for which $m_0(x)>0$ and elliptic where $m_0(x)=0$. 
The boundary conditions are of mixed Dirichlet-Neumann type 
(cf.\ Section III.4.1) and the initial value of $U(x,0)$ is prescribed only 
at those points of $G$ at which the equation  is parabolic. 

Boundary conditions of the third type may be introduced by modifying 
$\ell(\cdot,\cdot)$ as in Section III.4.2. 
Similarly, by choosing 
$$\ell (u,v) = \int_G \nabla u\cdot\overline{\nabla v}\,dx 
+ (\gamma_0 u)\overline{(\gamma_0 v)}$$ 
on $V= \{u\in H^1(G) : \gamma_0 u$ is constant$\}$, we obtain a unique 
generalized solution of the initial-boundary value problem of 
{\it fourth type\/} (cf., Section III.4.2) 
\begin{equation}\label{eq554} 
\left.\begin{array}{ll}
\partial_t (m_0(x)U(x,t))-\Delta_n U(x,t) = m_0(x)F(x,t)\ ,&\qquad x\in G\ ,\\ 
\noalign{\vskip6pt}
U(s,t) = h(t)\ ,&\qquad s\in \partial G\ ,\\
\noalign{\vskip6pt}
\ds\biggl(\int_{\partial G} {\partial U(s,t)\over\partial\nu} \,ds\Big/ 
\int_{\partial G}\,ds \biggr) +h(t) =0\ ,&\qquad t>0\ ,\\
\noalign{\vskip6pt}
m_0(x) (U(x,0) - U_0(x)) =0 \ .&
\end{array}\right\}
\end{equation}
The data $F(\cdot,\cdot)$ and $U_0$ are specified as before; $h(\cdot)$ 
is unknown and part of the problem. 

\subsection{}		%5.3 
Problems with a partial differential equation of mixed 
pseudoparabolic-parabolic type can be similarly handled. 
Let $m_0(\cdot)$ be given as above and define 
$$m(u,v) = \int_G (u(x)\overline{v(x)} + m_0 (x)\nabla u(x)\cdot 
\overline{\nabla v}\,(x))\,dx \ ,\qquad u,v\in V_m\ ,$$ 
with $V_m=H^1(G)$. 
Then $V_m \hookrightarrow L^2(G)$ is continuous so we can identify $L^2(G)
\le V'_m$. Define $\ell(\cdot,\cdot)$ by \eqn{552} where $V$ is a subspace 
of $H^1(G)$ which contains $C_0^\infty (G)$ and is to be prescribed. 
Then $K(\M) = \{0\}$ and $m(\cdot,\cdot) + \ell(\cdot,\cdot)$ is 
$V$-coercive, so Theorem \ref{thm5-4B} will apply. 
In particular, if $U_0 \in L^2 (G)$ and $F$ as in Theorem IV.\ref{thm4-7C} 
are given, then there is a unique solution of the equation 
$$\partial_t(U(x,t)-\sum_{j=1}^n \partial_j (m_0(x)\partial_j U(x,t))) 
- \Delta_n U(x,t) = F(x,t)\ ,\qquad x\in G\ ,\ t>0\ ,$$ 
with the initial condition 
$$U(x,0) = U_0 (x)\ ,\qquad x\in G\ ,$$ 
and boundary conditions which depend on our choice of $V$. 

\subsection{}		%5.4 
We consider a problem with a time derivative and possibly a partial 
differential equation on a boundary. 
Let $G$ be as in \eqn{552} and assume for simplicity that $\partial G$ 
intersects the hyperplane $\RR^{n-1}\times \{0\}$ in a set with 
relative interior $S$. 
Let $a_n(\cdot)$ and $b(\cdot)$ be given nonnegative, real-valued 
functions in $L^\infty (S)$. 
We define $V_m= H^1(G)$ and 
$$m(u,v) =\int_G u(x)\overline{v(x)}\,dx 
+ \int_S a(s)u(s)\overline{v(s)}\,ds\ ,\qquad u,v\in V_m\ ,$$ 
where we suppress the notation for the trace operator, i.e., 
$u(s)=(\gamma_0u) (s)$ for  $s\in \partial G$. 
Define $V$    to be the completion of $C^\infty (\bar G)$ with the norm 
given by 
$$\|v\|_V^2 \equiv \|v\|_{H^1(G)}^2 + 
\biggl( \int_S b(s) \sum_{j=1}^{n-1} |D_jv(s)|^2\,ds\biggr)\ .$$ 
Thus, $V$ consists of these $v\in H^1(G)$ for which $b^{1/2}\cdot\partial_j 
(\gamma_0 v)\in L^2(S)$ for $1\le j\le n-1$; it is a Hilbert space. 
We define 
$$\ell (u,v)=\int_G \nabla u(x)\cdot\nabla \overline{v(x)}\,dx 
+ \int_S b(s)\biggl( \sum_{j=1}^{n-1} \partial_j u(s)\partial_j 
\overline{v(s)}\biggr)\,ds \ ,\qquad u,v\in V\ .$$ 
Then $K(\M) = \{0\}$ and $m(\cdot,\cdot) +\ell(\cdot,\cdot)$ is $V$-coercive. 
If $U_0\in L^2(G)$ and $F(\cdot,\cdot)$ is given as above, then 
Theorem \ref{thm5-4B} asserts the existence and uniqueness of the solution 
$U(\cdot,\cdot)$ of the initial-boundary value problem 
$$\cases{
\partial_t U(x,t)-\Delta_n U(x,t)=F(x,t)\ ,&$x\in G\ ,\ t>0\ ,$\cr 
\noalign{\vskip6pt}
\ds \partial_t(a(s)U(s,t))+ {\partial U(s,t)\over\partial\nu} 
= \sum_{j=1}^{n-1} \partial_j(b(s)\partial_jU(s,t))\ ,&$s\in S\ ,$\cr
\noalign{\vskip6pt}
\ds {\partial U(s,t)\over \partial\nu} =0\ ,&$s\in\partial G\sim S\ ,$\cr
\noalign{\vskip6pt}
\ds b(s) {\partial U(s,t)\over\partial \nu_S} = 0\ ,&$s\in\partial S\ ,$\cr
\noalign{\vskip6pt}
U(x,0) = U_0(x)\ ,&$x\in G\ ,$\cr
\noalign{\vskip6pt}
a(s) (U(s,0)- U_0(s))=0\ ,&$s\in S\ .$\cr}$$
Similar problems with a partial differential equation of mixed type or 
other combinations of boundary conditions can be handled by the same 
technique. Also, the $(n-1)$-dimensional surface $S$ can occur inside the 
region $G$ as well as on the boundary. (Cf., Section III.4.5.) 

\exercises 

\begin{description}
\item[1.1.] 
Use the separation-of-variables technique to obtain a series representation 
for the solution of \eqn{511} with $u(0,t) = u(\pi,t)=0$ and $u(x,0)=u_0(x)$, 
$0<x<\pi$. Compare the rate of convergence of this series with that of 
Section IV.1.
\medskip
\item[2.1.] 
Provide all details in support of the claim that Theorem \ref{thm5-2A} 
follows from Theorem IV.\ref{thm4-3C}. 
Show that Theorem \ref{thm5-2B} follows from Theorems IV.\ref{thm4-6C} 
and IV.\ref{thm4-6E}. 
\item[2.2.]
Show that Theorem \ref{thm5-2B} remains true if we replace the hypothesis 
that $\L$ is $V$-elliptic by $\lambda\M +\L$ is $V$-elliptic for some 
$\lambda\in\RR$.
\item[2.3.]
Characterize $V'_m$ in each of the examples \eqn{521}--\eqn{523}. 
Construct appropriate terms for $f(t)$ in Theorems \ref{thm5-2A} and 
\ref{thm5-2B}. Write out the corresponding initial-boundary value problems 
that are solved. 
\item[2.4.] 
Show $V'_m=\{m^{1/2} v:v\in L^2(0,1)\}$ in \eqn{524}. 
Describe appropriate non-homogeneous terms for the partial differential 
equation in \eqn{524}. 
\medskip
\item[3.1.]
Verify that \eqn{531} is a solution of \eqn{522} in the situation of 
Theorem \ref{thm5-3A}. 
\item[3.2.] 
Prove uniqueness holds in Theorem \ref{thm5-3A}. 
[Hint: Show $\sigma (t)\equiv \|u(t)\|_V^2$ satisfies $|\sigma'(t)| \le 
K\sigma (t)$, $t\in\RR$, where $u$ is a solution of the homogeneous equation, 
then show that $\sigma (t) \le \exp (K|t|)\cdot\sigma (0)$.] 
\item[3.3.] 
Verify that \eqn{534} characterizes the solution of \eqn{524} in the case 
of Section 3.1. 
Discuss the regularity of the solution when $a(\cdot,\cdot)$ is $k$-regular. 
\medskip
\item[4.1.]
Prove \eqn{541}. 
\item[4.2.] 
Prove Theorem \ref{thm5-4B}. [Hint: Compare with Theorem \ref{thm5-2B}.] 
\medskip
\item[5.1.]
Give sufficient conditions on the data $F$, $u_0$ in \eqn{551} in order to 
apply Theorem \ref{thm5-4A}. 
\item[5.2.] 
Extend the discussion in Section 5.2 to include boundary conditions 
of the third type. 
\item[5.3.] 
Characterize $V'_m$ in Section 5.3. Write out the initial-boundary value 
problem solved in Section 5.3 for several choices of $V$.
\item[5.4.] 
Write out the problem solved in Section 5.4 when $S$ is an interface as in 
Section III.4.5. 
\end{description}