# Abstract evolution equation

Usually, a differential equation

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

in a Banach space $X$ (cf. also 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; 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 [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 [a1], [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 $B > 0$ and a set of real numbers $\alpha _ { 1 } , \ldots , \alpha _ { k } , \beta _ { 1 } , \ldots , \beta _ { k }$ with $0 \leq \beta _ { i } < \alpha _ { i } \leq 2$, $i = 1 , \ldots , k$, 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 < t \leq T. \end{equation*}

If $u _ { 0 } \in \overline { D ( A ( 0 ) ) }$ and $f$ is Hölder continuous (cf. also 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). The study of non-linear equations is also extensive. For details, see [a3], [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 [a5], (and extended in [a6]), by K. Kobayashi and N. Sanekata [a8], and by A. Yagi [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 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 $\lambda > \beta$ 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 $[0 , T]$, $\sup_{t \in [0,T]} ||B(t)||_X <\infty$, 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 [a5] and generalized to quasi-stability in [a6]. In [a5], [a6] it was assumed that $t \mapsto A ( t )$ is norm continuous in ${\cal{L}} ( Y , X )$.

For equations in Hilbert spaces, N. Okazawa [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 [a7] and the bibliography there). The assumption with the most distinctive feature in [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 [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, the Navier–Stokes equations and the Schrödinger equation.

#### References

[a1] | 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 |

[a2] | P. Acquistapace, B. Terreni, "A unified approach to abstract linear non-autonomous parabolic equations" Rend. Sem. Univ. Padova , 78 (1987) pp. 47–107 |

[a3] | H. Amann, "Linear and quasilinear parabolic problems I: Abstract linear theory" , Monogr. Math. , 89 , Birkhäuser (1995) |

[a4] | T. Kato, "Integration of the equation of evolution in a Banach space" J. Math. Soc. Japan , 5 (1953) pp. 208–234 |

[a5] | T. Kato, "Linear evolution equations of `hyperbolic' type" J. Fac. Sci. Univ. Tokyo , 17 (1970) pp. 241–258 |

[a6] | T. Kato, "Linear evolution equations of `hyperbolic' type II" J. Math. Soc. Japan , 25 (1973) pp. 648–666 |

[a7] | 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 |

[a8] | 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 |

[a9] | N. Okazawa, "Remarks on linear evolution equations of hyperbolic type in Hilbert space" Adv. Math. Sci. Appl. , 8 (1998) pp. 399–423 |

[a10] | A. Lunardi, "Analytic semigroups and optimal regularity in parabolic problems" , Progress in Nonlinear Diff. Eqns. Appl. , 16 , Birkhäuser (1995) |

[a11] | A. Yagi, "Remarks on proof of a theorem of Kato and Kobayashi on linear evolution equations" Osaka J. Math. , 17 (1980) pp. 233–243 |

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Abstract evolution equation.

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