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Usually, a differential equation
 
Usually, a differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\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|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200502.png" /> (cf. also [[Qualitative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]]). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200503.png" /> is the infinitesimal generator of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200505.png" />-semi-group for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200506.png" /> (cf. also [[Semi-group|Semi-group]]; [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) and the given (known) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200507.png" /> is usually a strongly continuous function with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200508.png" />. The first systematic study of this type of equations was made by T. Kato [[#References|[a4]]]. Under the assumptions
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a1200509.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005010.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005011.png" /> and is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005012.png" />;
+
i) the domain $D ( A ( t ) )$ of $A ( t )$ is dense in $X$ and is independent of $t$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005013.png" /> generates a contraction semi-group for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005014.png" />;
+
ii) $A ( t )$ generates a contraction semi-group for each $t \in [ 0 , T]$;
  
iii) the bounded operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005015.png" /> is continuously differentiable he constructed the fundamental solution (or evolution operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005017.png" />. He required this fundamental solution to be a bounded operator-valued function with the following properties:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005018.png" /> is strongly continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005019.png" />;
+
a) $U ( t , s )$ is strongly continuous in $0 \leq s \leq t \leq T$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005021.png" />;
+
b) $U ( s , s ) = I$ for $s \in [ 0 , T]$;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005023.png" />;
+
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
 
d) a solution of (a1) satisfying the initial condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} u ( 0 ) = u _ { 0 }, \end{equation}
  
 
if it exists, can be expressed as
 
if it exists, can be expressed as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005027.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005028.png" />), 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005029.png" /> and the semi-group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005030.png" /> being a contraction. Typical general results are the following.
+
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 equations.==
"Parabolic"  means that the semi-group generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005031.png" /> is analytic for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005032.png" />. In this case the domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005033.png" /> is not supposed to be dense. Consequently, property b) should be replaced by
+
"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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005034.png" /></td> </tr></table>
+
\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
 
P. Acquistapace and B. Terreni [[#References|[a1]]], [[#References|[a2]]] proved the following result: Suppose that
  
I) there exist an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005035.png" /> and a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005036.png" /> such that:
+
I) there exist an angle $\theta _ { 0 } \in ( \pi / 2 , \pi )$ and a positive constant $M$ such that:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005037.png" /> (the resolvent set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005038.png" />) contains the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005040.png" />;
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005043.png" />;
+
ii) $\| ( \lambda - A ( t ) ) ^ { - 1 } \| \leq M / ( 1 + | \lambda | )$, $\lambda \in S _ { \theta _ { 0 } }$, $t \in [ 0 , T]$;
  
II) there exist a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005044.png" /> and a set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005045.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005047.png" />, such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005048.png" /></td> </tr></table>
+
\begin{equation*} | A ( t ) ( \lambda - A ( t ) ) ^ { - 1 } ( A ( t ) ^ { - 1 } - A ( s ) ^ { - 1 } ) \| \leq \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005049.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005050.png" /> exists, is differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005051.png" /> and there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005052.png" /> such that
+
Then the fundamental solution $U ( t , s )$ exists, is differentiable in $t \in ( 0 , T ]$ and there exists a constant $C$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005053.png" /></td> </tr></table>
+
\begin{equation*} \| \frac { \partial } { \partial t } U ( t , s ) \| \leq \frac { C } { t - s } , \quad 0 \leq s < t \leq T. \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005055.png" /> is Hölder continuous (cf. also [[Hölder condition|Hölder condition]]), i.e. for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005056.png" />,
+
If $u _ { 0 } \in \overline { D ( A ( 0 ) ) }$ and $f$ is Hölder continuous (cf. also [[Hölder condition|Hölder condition]]), i.e. for some constant $\alpha \in ( 0,1 ]$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005057.png" /></td> </tr></table>
+
\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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005059.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005061.png" />, (a1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005062.png" /> and (a2) holds. A solution in this sense is usually called a classical solution. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005064.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005066.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005068.png" /> and (a1) holds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005069.png" />. Such a solution is usually called a strict solution.
+
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]]].
 
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]]].
Line 61: Line 69:
 
Here, equations of hyperbolic type are written as
 
Here, equations of hyperbolic type are written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005071.png" /> generates a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005072.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005073.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005074.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005075.png" /> be another Banach space embedded continuously and densely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005076.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005077.png" /> be an [[Isomorphism|isomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005078.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005079.png" />. Suppose that
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005080.png" /> is stable with stability constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005082.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005084.png" />, and for every finite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005086.png" /> the following inequality holds:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005087.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005088.png" /> stands to the left of all those with smaller <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005089.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005090.png" /> of bounded linear operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005091.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005092.png" /> is strongly measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005094.png" />, and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005095.png" /></td> </tr></table>
+
\begin{equation*} S A ( t ) S ^ { - 1 } = A ( t ) + B ( t ) , \quad t \in [ 0 , T ], \end{equation*}
  
 
with exact domain relation;
 
with exact domain relation;
  
C) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005097.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005098.png" /> is strongly continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a12005099.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050100.png" />, i.e. to the set of bounded linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050101.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050102.png" />. Then there exists a unique evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050104.png" />, having the following properties:
+
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:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050105.png" />;
+
$\| U ( t , s ) \| _ { X } \leq M e ^ { \beta ( t - s ) } , \quad ( t , s ) \in \Delta$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050106.png" /> is strongly continuous from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050107.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050108.png" /> with
+
$U ( t , s )$ is strongly continuous from $\Delta$ to $\mathcal{L} ( Y ) = \mathcal{L} ( Y , Y )$ with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050109.png" /></td> </tr></table>
+
\begin{equation*} \| U ( t , s ) \| _ { Y } \leq \overline { M } e ^ { \overline { \beta } ( t - s ) } , \quad ( t , s ) \in \Delta, \end{equation*}
  
for certain constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050111.png" />;
+
for certain constants $\overline{M}$ and $\beta$;
  
for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050113.png" /> and
+
for each $v \in Y$, $U ( ., . ) v \in C ^ { 1 } ( \Delta ; X )$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050114.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial } { \partial t } U ( t , s ) v = - A ( t ) U ( t , s ) v, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050115.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial } { \partial s } U ( t , s ) v = U ( t , s ) A ( s ) v. \end{equation*}
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050117.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050118.png" /> defined by (a3) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050119.png" /> and is the unique solution of (a4), (a2).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050120.png" /> is norm continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050121.png" />.
+
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.
 
For equations in Hilbert spaces, N. Okazawa [[#References|[a9]]] obtained a related result which is convenient in applications to concrete problems.
Line 101: Line 109:
 
Hyperbolic quasi-linear equations
 
Hyperbolic quasi-linear equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050122.png" /></td> </tr></table>
+
\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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050123.png" /></td> </tr></table>
+
\begin{equation*} Se  ^ { - s A ( t , u ) } \supset e ^ { - s \hat{A} ( t , u ) } S, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050124.png" /> is considered to be a perturbation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050125.png" /> by a bounded operator in some sense and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050126.png" /> is a closed linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050127.png" /> to a third Banach space such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050128.png" /> (see [[#References|[a7]]] for the details). The result can be applied to a system of quasi-linear partial differential equations
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050129.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050130.png" /> is a function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050131.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050132.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050133.png" /> are simultaneously diagonalizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120050/a120050134.png" />-matrix valued functions.
+
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]].
 
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====
 
====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>
+
<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>

Latest revision as of 07:42, 4 February 2024

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
How to Cite This Entry:
Abstract evolution equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_evolution_equation&oldid=12504
This article was adapted from an original article by H. Tanabe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article