%% 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 $$\label{eq511} \partial_tu (x,t) - a\partial_x^2 \partial_t u(x,t) - \partial_x^2 u(x,t) = f(x,t)$$ 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 $$\label{eq512} m(x)\partial_t u(x,t) - \partial_x^2 u(x,t) = f(x,t)$$ 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 $$\label{eq521} \M u'(t) + Lu(t) = f(t)\ ,\qquad t>0\ ,$$ 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 $$\label{eq522} u'(t) + \M^{-1} \circ Lu (t) = \M^{-1} f(t)\ ,\qquad t>0\ .$$ 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 $$\label{eq523} (Ax,y)_m = Lx(y)\ ,\qquad x\in D\ ,\ y\in V_m\ .$$ 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 $$\label{eq524} \M u'(t) + \L u (t) = f(t)\ ,\qquad t>0\ .$$ \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 00\ ,\\ \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 00\ ,\\ \noalign{\vskip6pt} &&U(0,t) = U(1,t) =0\ ,\qquad t>0\ ,\\ \noalign{\vskip6pt} &&U(x,0) = U_0(x)\ ,\qquad 00$for a.e.\$x\in G$. (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 $$\label{eq531} u(t) = \exp (-tA)\cdot u_0 + \int_0^t \exp (A(\tau-t)) \circ \M^{-1} f(\tau)\,d\tau\ ,\qquad t>0\ .$$ 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 $$\label{eq532} Mu' (t) + Lu(t) = f(t)\ ,\qquad t\in \RR\ .$$ \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$\|v_n-u_0\|_{D(L)} \to0$,$\|Av_n-v_0\|_{D(L)}\to0, then \begin{eqnarray*} \|v_0 -Au_0\|_V &\le & \|v_0 -Av_n\|_V + \|A(v_n- u_0)\|_V\\ \noalign{\vskip6pt} &\le & \|v_0 - Av_n\|_V + \|A\|_{\L(V)} \|v_n- u_0\|_V\ , \end{eqnarray*} so the continuity ofD(L)\hookrightarrow V$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$\varep>0and define \begin{array}{rcll} m(u,v)&=&(u,v)_H +\varep a(u,v)&\\ \noalign{\vskip6pt} \ell(u,v)&=&a(u,v)\ ,&\qquad u,v\in V\ . \end{array} Thus,D(M)= D(L)=D(A)$. Let$f\in C(\RR,H)$. If$u(\cdot)is a strong solution of \eqn{532}, then we have \label{eq533} \left.\begin{array}{ll} u'(t) + \varep A_1u'(t) + A_1u(t) = f(t)\ ,&\\ \noalign{\vskip6pt} u(t) \in V\ ,\ \hbox{ and}&\\ \noalign{\vskip6pt} \partial_1u(t)+\A_2\gamma(u(t))=0\ ,&\qquad t\in \RR\ . \end{array}\right\} Suppose instead thatF\in C(\RR,H)$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$u(\cdot)of \eqn{524} can be shown by a computation similar to the proof of Theorem III.\ref{thm3-3A} to satisfy \label{eq534} \left.\begin{array}{l} u'(t)+\varep A_1 u'(t) + A_1 u(t) = F(t)\ ,\\ \noalign{\vskip6pt} u(t)\in V\ ,\ \hbox{ and}\\ \noalign{\vskip6pt} \partial_1 (\varep u'(t) +u(t)) + \A_2(\gamma(\varep u'(t)+u(t)))=g(t)\ , \qquad t\in \RR\ . \end{array}\right\} Note that \eqn{533} implies more than \eqn{534} withg\equiv 0$. 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 solutionU(\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 thatf\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)$a sesquilinear form on$Vthat 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 $$\label{eq541} |m(x,y)|^2 \le m(x,x)\cdot m(y,y)\ ,\qquad x,y\in V\ ,$$ and thatx\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 $$\label{eq542} (\M u)'(t) + Lu(t) = f(t)\ ,\qquad t>0\ .$$ (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, $$\label{eq543} q'\M_0 q= \M\ .$$ From the linear mapL: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 $$\label{eq544} q'L_0 q= L\ .$$ 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 $$\label{eq545} \M_0 v'(t) + L_0v(t) = (q')^{-1} f(t)\ ,\qquad t>0\ .$$ 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} SinceL$is monotone, this shows$\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)$,$0\le a0\}\subset \Gamma$. 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$Fis given as in Theorem IV.\ref{thm4-7C}, then there is a unique generalized solution of the problem \label{eq553} \left.\begin{array}{ll} \partial_t (m_0(x)U(x,t)) -\Delta_n U(x,t)=m_0(x)F(x,t)\ ,&\qquad x\in G\ ,\\ \noalign{\vskip6pt} U(s,t)=0\ ,\qquad s\in\Gamma\ ,\\ \noalign{\vskip6pt} \ds {\partial U(s,t)\over\partial \nu} =0\ ,\quad s\in\partial G\sim \Gamma\ , &\qquad t>0\ ,\\ \noalign{\vskip6pt} m_0(x)(U(x,0)- U_0(x)) =0\ .& \end{array}\right\} The partial differential equation in \eqn{553} is parabolic at thosex\in G$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$\}, we obtain a unique generalized solution of the initial-boundary value problem of {\it fourth type\/} (cf., Section III.4.2) \label{eq554} \left.\begin{array}{ll} \partial_t (m_0(x)U(x,t))-\Delta_n U(x,t) = m_0(x)F(x,t)\ ,&\qquad x\in G\ ,\\ \noalign{\vskip6pt} U(s,t) = h(t)\ ,&\qquad s\in \partial G\ ,\\ \noalign{\vskip6pt} \ds\biggl(\int_{\partial G} {\partial U(s,t)\over\partial\nu} \,ds\Big/ \int_{\partial G}\,ds \biggr) +h(t) =0\ ,&\qquad t>0\ ,\\ \noalign{\vskip6pt} m_0(x) (U(x,0) - U_0(x)) =0 \ .& \end{array}\right\} The dataF(\cdot,\cdot)$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