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Usually, a differential equation

\begin{equation} \tag{a1} \frac { d u ( t ) } { d t } = A ( t ) u ( t ) + f ( t ) , \quad 0 &lt; t \leq T, \end{equation}

in a [[Banach space|Banach space]] $X$ (cf. also [[Qualitative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]]). Here, $A ( t )$ is the infinitesimal generator of a $C _ { 0 }$-semi-group for each $t \in [ 0 , T]$ (cf. also [[Semi-group|Semi-group]]; [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) and the given (known) function $f ( . )$ is usually a strongly continuous function with values in $X$. The first systematic study of this type of equations was made by T. Kato [[#References|[a4]]]. Under the assumptions

i) the domain $D ( A ( t ) )$ of $A ( t )$ is dense in $X$ and is independent of $t$;

ii) $A ( t )$ generates a contraction semi-group for each $t \in [ 0 , T]$;

iii) the bounded operator-valued function $t \mapsto ( I - A ( t ) ) ( I - A ( 0 ) ) ^ { - 1 }$ is continuously differentiable he constructed the fundamental solution (or evolution operator) $U ( t , s )$, $0 \leq s \leq t \leq T$. He required this fundamental solution to be a bounded operator-valued function with the following properties:

a) $U ( t , s )$ is strongly continuous in $0 \leq s \leq t \leq T$;

b) $U ( s , s ) = I$ for $s \in [ 0 , T]$;

c) $U ( t , r ) U ( r , s ) = U ( t , s )$ for $0 \leq s \leq r \leq t \leq T$;

d) a solution of (a1) satisfying the initial condition

\begin{equation} \tag{a2} u ( 0 ) = u _ { 0 }, \end{equation}

if it exists, can be expressed as

\begin{equation} \tag{a3} u ( t ) = U ( t , 0 ) u _ { 0 } + \int _ { 0 } ^ { t } U ( t , s ) f ( s ) d s; \end{equation}

e) if $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { 1 } ( [ 0 , T ] ; X )$ or $f \in C ( [ 0 , T ] ; D ( A ( 0 ) )$), then (a3) is the unique solution of (a1), (a2). Since Kato's paper, efforts have been made to relax the restrictions, especially the independence of the domain of $A ( t )$ and the semi-group generated by $A ( t )$ being a contraction. Typical general results are the following.

==Parabolic equations.==
"Parabolic"  means that the semi-group generated by $A ( t )$ is analytic for each $t \in [ 0 , T]$. In this case the domain of $A ( t )$ is not supposed to be dense. Consequently, property b) should be replaced by

\begin{equation*} \operatorname { lim } _ { t \rightarrow s } U ( t , s ) u _ { 0 } = u _ { 0 }\; \text { for } u _ { 0 } \in \overline { D ( A ( s ) ) }. \end{equation*}

P. Acquistapace and B. Terreni [[#References|[a1]]], [[#References|[a2]]] proved the following result: Suppose that

I) there exist an angle $\theta _ { 0 } \in ( \pi / 2 , \pi )$ and a positive constant $M$ such that:

i) $\rho ( A ( t ) )$ (the resolvent set of $A ( t )$) contains the set $S _ { \theta _ { 0 } } = \{ z \in \mathbf{C} : |\operatorname { arg } z | \leq \theta _ { 0 } \} \cup \{ 0 \}$, $t \in [ 0 , T]$;

ii) $\| ( \lambda - A ( t ) ) ^ { - 1 } \| \leq M / ( 1 + | \lambda | )$, $\lambda \in S _ { \theta _ { 0 } }$, $t \in [ 0 , T]$;

II) there exist a constant $B &gt; 0$ and a set of real numbers $\alpha _ { 1 } , \ldots , \alpha _ { k } , \beta _ { 1 } , \ldots , \beta _ { k }$ with $0 \leq \beta _ { i } &lt; \alpha _ { i } \leq 2$, $i = 1 , \ldots , k$, such that

\begin{equation*} | A ( t ) ( \lambda - A ( t ) ) ^ { - 1 } ( A ( t ) ^ { - 1 } - A ( s ) ^ { - 1 } ) \| \leq \end{equation*}

\begin{equation*} \leq B \sum _ { i = 1 } ^ { k } ( t - s ) ^ { \alpha _ { i } } | \lambda | ^ { \beta _ { i } - 1 } , \lambda \in S _ { \theta _ { 0 } } \backslash \{ 0 \} , \quad 0 \leq s \leq t \leq T. \end{equation*}

Then the fundamental solution $U ( t , s )$ exists, is differentiable in $t \in ( 0 , T ]$ and there exists a constant $C$ such that

\begin{equation*} \| \frac { \partial } { \partial t } U ( t , s ) \| \leq \frac { C } { t - s } , \quad 0 \leq s &lt; t \leq T. \end{equation*}

If $u _ { 0 } \in \overline { D ( A ( 0 ) ) }$ and $f$ is Hölder continuous (cf. also [[Hölder condition|Hölder condition]]), i.e. for some constant $\alpha \in ( 0,1 ]$,

\begin{equation*} \| f ( t ) - f ( s ) \| \leq C _ { 1 } | t - s | ^ { \alpha } , \quad s , t \in [ 0 , T ], \end{equation*}

then the function (a3) is the unique solution of the initial-value problem (a1), (a2) in the following sense: $u \in C ( [ 0 , T ] ; X ) \cap C ^ { 1 } ( ( 0 , T ] ; X )$, $u ( t ) \in D ( A ( t ) )$ for $t \in ( 0 , T ]$, $A u \in C ( ( 0 , T ] ; X )$, (a1) holds for $t \in ( 0 , T ]$ and (a2) holds. A solution in this sense is usually called a classical solution. If, moreover, $u _ { 0 } \in D ( A ( 0 ) )$ and $A ( 0 ) u_0 + f ( 0 ) \in \overline { D ( A ( 0 ) ) }$, then $u \in C ^ { 1 } ( [ 0 , T ] ; X )$, $u ( t ) \in D ( A ( t ) )$ for $t \in [ 0 , T]$, $A u \in C ( [ 0 , T ] ; X )$ and (a1) holds in $[0 , T]$. Such a solution is usually called a strict solution.

The above result can be applied to initial-boundary value problems for parabolic partial differential equations (cf. also [[Parabolic partial differential equation|Parabolic partial differential equation]]). The study of non-linear equations is also extensive. For details, see [[#References|[a3]]], [[#References|[a10]]].

==Hyperbolic equations.==
Here, equations of hyperbolic type are written as

\begin{equation} \tag{a4} \frac { d u ( t ) } { d t } + A ( t ) u ( t ) = f ( t ), \end{equation}

conforming to the notations of the papers quoted below, so that $- A ( t )$ generates a $C _ { 0 }$-semi-group. A general result on this class of equations was first established by Kato [[#References|[a5]]], (and extended in [[#References|[a6]]]), by K. Kobayashi and N. Sanekata [[#References|[a8]]], and by A. Yagi [[#References|[a11]]] and others. A typical general result is as follows. Suppose that $D ( A ( t ) )$ is dense in $X$. Let $Y$ be another Banach space embedded continuously and densely in $X$, and let $S$ be an [[Isomorphism|isomorphism]] of $Y$ onto $X$. Suppose that

A) $\{ A ( t ) \}$ is stable with stability constants $M$, $\beta$, i.e. $\rho ( A ( t ) ) \supset ( \beta , \infty )$, $t \in [ 0 , T]$, and for every finite sequence $0 \leq t _ { 1 } \leq \ldots \leq t _ { k } \leq T$ and $\lambda &gt; \beta$ the following inequality holds:

\begin{equation*} \left| \prod _ { j = 1 } ^ { k } ( \lambda - A ( t _ { j } ) ) ^ { - 1 } \right\| _ { X } \leq M ( \lambda - \beta ) ^ { - k }, \end{equation*}

where the product is time ordered, i.e. a factor with a larger $t_j$ stands to the left of all those with smaller $t_j$;

B) there is a family $\{ B ( t ) \}$ of bounded linear operators in $X$ such that $B ( . )$ is strongly measurable in $[0 , T]$, $\sup_{t \in [0,T]} ||B(t)||_X &lt;\infty$, and

\begin{equation*} S A ( t ) S ^ { - 1 } = A ( t ) + B ( t ) , \quad t \in [ 0 , T ], \end{equation*}

with exact domain relation;

C) $Y \subset D ( A ( t ) )$, $t \in [ 0 , T]$, and $A ( . )$ is strongly continuous from $[0 , T]$ to ${\cal{L}} ( Y , X )$, i.e. to the set of bounded linear operators on $Y$ to $X$. Then there exists a unique evolution operator $U ( t , s )$, $( t , s ) \in \Delta = \{ ( t , s ) : 0 \leq s \leq t \leq T \}$, having the following properties:

$\| U ( t , s ) \| _ { X } \leq M e ^ { \beta ( t - s ) } , \quad ( t , s ) \in \Delta$;

$U ( t , s )$ is strongly continuous from $\Delta$ to $\mathcal{L} ( Y ) = \mathcal{L} ( Y , Y )$ with

\begin{equation*} \| U ( t , s ) \| _ { Y } \leq \overline { M } e ^ { \overline { \beta } ( t - s ) } , \quad ( t , s ) \in \Delta, \end{equation*}

for certain constants $\overline{M}$ and $\beta$;

for each $v \in Y$, $U ( ., . ) v \in C ^ { 1 } ( \Delta ; X )$ and

\begin{equation*} \frac { \partial } { \partial t } U ( t , s ) v = - A ( t ) U ( t , s ) v, \end{equation*}

\begin{equation*} \frac { \partial } { \partial s } U ( t , s ) v = U ( t , s ) A ( s ) v. \end{equation*}

For $u _ { 0 } \in Y$ and $f \in C ( [ 0 , T ] ; X ) \cap L ^ { 1 } ( 0 , T ; Y )$, the function $u$ defined by (a3) belongs to $C ^ { 1 } ( [ 0 , T ] ; X ) \cap C ( [ 0 , T ] ; Y )$ and is the unique solution of (a4), (a2).

The notion of stability was introduced by Kato [[#References|[a5]]] and generalized to quasi-stability in [[#References|[a6]]]. In [[#References|[a5]]], [[#References|[a6]]] it was assumed that $t \mapsto A ( t )$ is norm continuous in ${\cal{L}} ( Y , X )$.

For equations in Hilbert spaces, N. Okazawa [[#References|[a9]]] obtained a related result which is convenient in applications to concrete problems.

Hyperbolic quasi-linear equations

\begin{equation*} \frac { d u ( t ) } { d t } + A ( t , u ( t ) ) u ( t ) = f ( t , u ( t ) ) \end{equation*}

have also been extensively studied. Especially deep research was carried out by Kato (see [[#References|[a7]]] and the bibliography there). The assumption with the most distinctive feature in [[#References|[a7]]] is the intertwining condition

\begin{equation*} Se  ^ { - s A ( t , u ) } \supset e ^ { - s \hat{A} ( t , u ) } S, \end{equation*}

where $\hat{A} ( t , u )$ is considered to be a perturbation of $A ( t , u )$ by a bounded operator in some sense and $S$ is a closed linear operator from $X$ to a third Banach space such that $D ( S ) = Y$ (see [[#References|[a7]]] for the details). The result can be applied to a system of quasi-linear partial differential equations

\begin{equation*} \frac { \partial u } { \partial t } + \sum _ { j = 1 } ^ { m }a _ { j } ( t , u ) \frac { \partial u } { \partial x _ { j } } = f ( t , u ), \end{equation*}

where the unknown $u$ is a function from $\mathbf{R} \times \mathbf{R} ^ { m }$ into $\mathbf{R} ^ { N }$, and $a_j ( .,. )$ are simultaneously diagonalizable $( N \times N )$-matrix valued functions.

The theory and methods for abstract evolution equations have been applied to many physical problems, such as the [[Wave equation|wave equation]], the [[Navier–Stokes equations|Navier–Stokes equations]] and the [[Schrödinger equation|Schrödinger equation]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Acquistapace,   B. Terreni,   "On fundamental solutions for abstract parabolic equations"  A. Favini (ed.)  E. Obrecht (ed.) , ''Differential equations in Banach spaces, Bologna, 1985'' , ''Lecture Notes Math.'' , '''1223''' , Springer  (1986)  pp. 1–11</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P. Acquistapace,   B. Terreni,   "A unified approach to abstract linear non-autonomous parabolic equations"  ''Rend. Sem. Univ. Padova'' , '''78'''  (1987)  pp. 47–107</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  H. Amann,   "Linear and quasilinear parabolic problems I: Abstract linear theory" , ''Monogr. Math.'' , '''89''' , Birkhäuser  (1995)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  T. Kato,   "Integration of the equation of evolution in a Banach space"  ''J. Math. Soc. Japan'' , '''5'''  (1953)  pp. 208–234</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  T. Kato,   "Linear evolution equations of `hyperbolic' type"  ''J. Fac. Sci. Univ. Tokyo'' , '''17'''  (1970)  pp. 241–258</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  T. Kato,   "Linear evolution equations of `hyperbolic' type II"  ''J. Math. Soc. Japan'' , '''25'''  (1973)  pp. 648–666</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  T. Kato,   "Abstract evolution equations, linear and quasilinear, revisited"  H. Komatsu (ed.) , ''Functional Analysis and Related Topics, 1991'' , ''Lecture Notes Math.'' , '''1540''' , Springer  (1993)  pp. 103–125</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  K. Kobayashi,   N. Sanekata,   "A method of iterations for quasi-linear evolution equations in nonreflexive Banach spaces"  ''Hiroshima Math. J.'' , '''19'''  (1989)  pp. 521–540</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  N. Okazawa,   "Remarks on linear evolution equations of hyperbolic type in Hilbert space"  ''Adv. Math. Sci. Appl.'' , '''8'''  (1998)  pp. 399–423</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A. Lunardi,   "Analytic semigroups and optimal regularity in parabolic problems" , ''Progress in Nonlinear Diff. Eqns. Appl.'' , '''16''' , Birkhäuser  (1995)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  A. Yagi,   "Remarks on proof of a theorem of Kato and Kobayashi on linear evolution equations"  ''Osaka J. Math.'' , '''17'''  (1980)  pp. 233–243</td></tr></table>