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An equation of the form
 
An equation of the form
  
<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/a120070/a1200701.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}
  
where for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200703.png" /> is the infinitesimal generator of an analytic semi-group (cf. also [[Semi-group of operators|Semi-group of operators]]; [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) in some [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200704.png" />. Hence, without loss of generality it is always assumed that
+
where for each $t \in [ 0 , T]$, $A ( t )$ is the infinitesimal generator of an analytic semi-group (cf. also [[Semi-group of operators|Semi-group of operators]]; [[Strongly-continuous semi-group|Strongly-continuous semi-group]]) in some [[Banach space|Banach space]] $X$. Hence, without loss of generality it is always assumed that
  
I) there exist an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200705.png" /> and a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200706.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/a120070/a1200707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200708.png" />;
+
i) $\rho ( A ( t ) ) \supset S _ { \theta _ { 0 } } = \{ z \in {\bf C} : | \operatorname { arg } z | \leq \theta _ { 0 } \} \cup \{ 0 \}$, $t \in [ 0 , T]$;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a1200709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007011.png" />. The domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007013.png" /> is not necessarily dense. Various results on the solvability of the initial value problem for (a1) have been published. The main object is to construct the fundamental solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007014.png" /> (cf. also [[Fundamental solution|Fundamental solution]]), which is an operator-valued function satisfying
+
ii) $\| ( \lambda - A ( t ) ) ^ { - 1 } \| \leq M / ( 1 + | \lambda | )$, $\lambda \in S _ { \theta _ { 0 } }$, $t \in [ 0 , T]$. The domain $D ( A ( t ) )$ of $A ( t )$ is not necessarily dense. Various results on the solvability of the initial value problem for (a1) have been published. The main object is to construct the fundamental solution $U ( t , s ) , 0 \leq s \leq t \leq T$ (cf. also [[Fundamental solution|Fundamental solution]]), which is an operator-valued function satisfying
  
<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/a120070/a12007015.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial } { \partial t } U ( t , s ) - A ( t ) U ( t , s ) = 0 \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/a120070/a12007016.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial } { \partial s } U ( t , s ) + U ( t , s ) A ( s ) = 0 , \operatorname { lim } _ { t \rightarrow s } U ( t , s ) x = x \text { for } x \in \overline { D ( A ( s ) ) }. \end{equation*}
  
 
The solution of (a1) satisfying the initial condition
 
The 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/a120070/a12007017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} u ( 0 ) = u _ { 0 } \in \overline { D ( A ( 0 ) ) }, \end{equation}
  
 
if it exists, is given by
 
if it exists, is given 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/a120070/a12007018.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}
  
 
In parabolic cases, the fundamental solution usually satisfies the inequality
 
In parabolic cases, the fundamental solution usually satisfies the inequality
  
<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/a120070/a12007019.png" /></td> </tr></table>
+
\begin{equation*} \| \frac { \partial U ( t , s ) } { \partial t } \| \leq \frac { C } { t - s } , \quad s , t \in [ 0 , T ],  \end{equation*}
  
for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007020.png" />.
+
for some constant $C > 0$.
  
 
One of the most general result is due to P. Acquistapace and B. Terreni [[#References|[a3]]], [[#References|[a4]]]. Suppose that
 
One of the most general result is due to P. Acquistapace and B. Terreni [[#References|[a3]]], [[#References|[a4]]]. Suppose that
  
II) there exist a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007021.png" /> and a set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007024.png" />, such that
+
II) there exist a constant $K _ { 0 } > 0$ 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/a120070/a12007025.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/a120070/a12007026.png" /></td> </tr></table>
+
\begin{equation*} \leq K _ { 0 } \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 exists, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007027.png" /> satisfies (a2) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007028.png" /> is Hölder continuous (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007029.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007030.png" />, i.e.
+
Then the fundamental solution exists, and if $u_{0}$ satisfies (a2) and $f ( . )$ is Hölder continuous (i.e., $f \in C ^ { \alpha } ( [ 0 , T ] ; X )$ for some $\alpha \in ( 0,1 ]$, i.e.
  
<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/a120070/a12007031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} \| f ( t ) - f ( s ) \| \leq C _ { 1 } | t - s | ^ { \alpha } , \quad 0 \leq s \leq t \leq T; \end{equation}
  
cf. also [[Hölder condition|Hölder condition]]), then the function (a3) is the unique solution of (a1), (a2) in the following sense: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007035.png" />, (a1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007036.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/a120070/a12007037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007040.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007042.png" /> and (a1) is satisfied in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007043.png" />. Such a solution is usually called a strict solution.
+
cf. also [[Hölder condition|Hölder condition]]), then the function (a3) is the unique solution of (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) is satisfied in $[0 , T]$. Such a solution is usually called a strict solution.
  
 
The following results on maximal regularity are well known.
 
The following results on maximal regularity are well known.
  
 
===Time regularity.===
 
===Time regularity.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007045.png" />. Then
+
Let $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { \alpha } ( [ 0 , T ] ; X )$. Then
  
<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/a120070/a12007046.png" /></td> </tr></table>
+
\begin{equation*} u ^ { \prime } \in C ^ { \alpha } ( [ 0 , T ] ; X ) \bigcap B ( D _ { A } ( \alpha , \infty ) ), \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/a120070/a12007047.png" /></td> </tr></table>
+
\begin{equation*} A u \in C ^ { \alpha } ( [ 0 , T ] ; X ) \end{equation*}
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007049.png" /> is the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007050.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007051.png" /> (an interpolation space between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007053.png" />; cf. also [[Interpolation of operators|Interpolation of operators]]) for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007054.png" /> and the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007055.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007056.png" /> is essentially bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007057.png" />.
+
if and only if $A ( 0 ) u _ { 0 } + f ( 0 ) \in D _ { A ( 0 ) } ( \alpha , \infty )$, where $B ( D _ { A } ( \alpha , \infty ) )$ is the set of all functions $f \in L ^ { \infty } ( 0 , T ; X )$ such that $f ( t ) \in D _ { A ( t ) } ( \alpha , \infty )$ (an interpolation space between $D ( A ( t ) )$ and $X$; cf. also [[Interpolation of operators|Interpolation of operators]]) for almost all $t \in [ 0 , T]$ and the norm of $f ( t )$ on $D _ { A ( t ) } ( \alpha , \infty )$ is essentially bounded in $[0 , T]$.
  
 
===Space regularity.===
 
===Space regularity.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007059.png" />. Then
+
Let $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in B ( D _ { A } ( \alpha , \infty ) )$. Then
  
<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/a120070/a12007060.png" /></td> </tr></table>
+
\begin{equation*} u ^ { \prime } \in B ( D _ { A } ( \alpha , \infty ) ), \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/a120070/a12007061.png" /></td> </tr></table>
+
\begin{equation*} A u \in B ( D _ { A } ( \alpha , \infty ) ) \bigcap C ^ { \alpha } ( [ 0 , T ] ; X ) \end{equation*}
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007062.png" />.
+
if and only if $A ( 0 ) u _ { 0 } \in D _ { A ( 0 ) } ( \alpha , \infty )$.
  
Hypothesis II) holds if the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007063.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007065.png" /> is Hölder continuous, i.e. there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007067.png" /> such that
+
Hypothesis II) holds if the domain $D ( A ( t ) )$ is independent of $t$ and $t \mapsto A ( t )$ is Hölder continuous, i.e. there exist constants $C _ { 2 } > 0$ and $\alpha \in ( 0,1 ]$ 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/a120070/a12007068.png" /></td> </tr></table>
+
\begin{equation*} | ( A ( t ) - A ( s ) ) A ( 0 ) ^ { - 1 } \| \leq C _ { 2 } | t - s | ^ { \alpha } , \quad t , s \in [ 0 , T ]. \end{equation*}
  
 
Another main result by Acquistapace and Terreni is the following ([[#References|[a2]]]):
 
Another main result by Acquistapace and Terreni is the following ([[#References|[a2]]]):
  
III.i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007069.png" /> is differentiable and there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007071.png" /> such that
+
III.i) $t \mapsto A ( t ) ^ { - 1 }$ is differentiable and there exist constants $K _ { 1 } > 0$ and $\rho \in ( 0,1 ]$ 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/a120070/a12007072.png" /></td> </tr></table>
+
\begin{equation*} \left\| \frac { \partial } { \partial t } ( \lambda - A ( t ) ) ^ { - 1 } \right\| \leq \frac { K _ { 1 } } { ( 1 + | \lambda | ) ^ { \rho } }, \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/a120070/a12007073.png" /></td> </tr></table>
+
\begin{equation*} \lambda \in S _ { \theta _ { 0 } } , t \in [ 0 , T ]; \end{equation*}
  
III.ii) there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007075.png" /> such that
+
III.ii) there exist constants $K _ { 2 } > 0$ and $\eta \in ( 0,1 ]$ 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/a120070/a12007076.png" /></td> </tr></table>
+
\begin{equation*} \| \frac { d } { d t } A ( t ) ^ { - 1 } - \frac { d } { d s } A ( s ) ^ { - 1 } \| \leq K _ { 2 } | t - s | ^ { \eta }, \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/a120070/a12007077.png" /></td> </tr></table>
+
\begin{equation*} t , s \in [ 0 , T ]. \end{equation*}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007078.png" /> is densely defined, this case reduces to the one in [[#References|[a8]]]. Under the assumptions I), III) it can be shown that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007080.png" /> satisfying (a2) and (a4), a classical solution of (a1) exists and is unique. The solution is strict if, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007081.png" /> and
+
If $A ( t )$ is densely defined, this case reduces to the one in [[#References|[a8]]]. Under the assumptions I), III) it can be shown that for $u_{0}$ and $f$ satisfying (a2) and (a4), a classical solution of (a1) exists and is unique. The solution is strict if, moreover, $u _ { 0 } \in D ( A ( 0 ) )$ 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/a120070/a12007082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5} A ( 0 ) u _ { 0 } + f ( 0 ) - \frac { d } { d t } A ( t ) ^ { - 1 } | _ { t = 0 } A ( 0 ) u _ { 0 } \in \overline { D ( A ( 0 ) ) }. \end{equation}
  
The following maximal regularity result holds: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007085.png" />, then the solution of (a1) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007086.png" /> if and only if the left-hand side of (a5) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007087.png" />.
+
The following maximal regularity result holds: If $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { \delta } ( [ 0 , T ] ; X )$ for $\delta \in ( 0 , \eta ) \cap ( 0 , \rho ]$, then the solution of (a1) belongs to $C ^ { 1 + \delta } ( [ 0 , T ] ; X )$ if and only if the left-hand side of (a5) belongs to $D _ { A ( 0 ) } ( \delta , \infty )$.
  
 
Another of general results is due to A. Yagi [[#References|[a10]]], where the fundamental solution is constructed under the following assumptions:
 
Another of general results is due to A. Yagi [[#References|[a10]]], where the fundamental solution is constructed under the following assumptions:
  
IV) hypothesis III.i) is satisfied, and there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007088.png" /> and a non-empty set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007089.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007090.png" /> such that
+
IV) hypothesis III.i) is satisfied, and there exist constants $K _ { 3 }$ and a non-empty set of indices $\{ ( \alpha _ { i } , \beta _ { i } ) : i = 1 , \ldots , k \}$ satisfying $- 1 \leq \alpha _ { i } < \beta _ { i } \leq 1$ 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/a120070/a12007091.png" /></td> </tr></table>
+
\begin{equation*} \left| A ( t ) ( \lambda - A ( t ) ) ^ { - 1 } \frac { d A ( t ) ^ { - 1 } } { d t } + \right. \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/a120070/a12007092.png" /></td> </tr></table>
+
\begin{equation*} \left. - A ( s ) ( \lambda - A ( s ) ) ^ { - 1 } \frac { d A ( s ) ^ { - 1 } } { d s } \right\| \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/a120070/a12007093.png" /></td> </tr></table>
+
\begin{equation*} \leq K _ { 2 } \sum _ { i = 1 } ^ { k } | \lambda | ^ { \alpha _ { i } } | t - s | ^ { \beta _ { i } }, \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/a120070/a12007094.png" /></td> </tr></table>
+
\begin{equation*} \lambda \in S _ { \theta _ { 0 } } , \quad t , s \in [ 0 , T ]. \end{equation*}
  
 
It is shown in [[#References|[a4]]] that the above three results are independent of one another.
 
It is shown in [[#References|[a4]]] that the above three results are independent of one another.
Line 103: Line 111:
 
The above results are applied to initial-boundary value problems for parabolic partial differential equations:
 
The above results are applied to initial-boundary value problems for parabolic 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/a120070/a12007095.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial u } { \partial t } = L ( t , x , D _ { x } ) u + f ( t , x ) \text { in } [ 0 , T ] \times \Omega, \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/a120070/a12007096.png" /></td> </tr></table>
+
\begin{equation*} B _ { j } ( t , x , D _ { x } ) u = 0 , \text { on } [ 0 , T ] \times \partial \Omega ,\quad j = 1 , \ldots , m, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007097.png" /> is an elliptic operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007098.png" /> (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a12007099.png" /> are operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070100.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070102.png" /> is a usually bounded open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070104.png" />, with smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070105.png" />. Under some algebraic assumptions on the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070107.png" /> and smoothness hypotheses of the coefficients, it is shown in [[#References|[a1]]] that the operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070108.png" /> defined by
+
where $L ( t , x , D _ { x } )$ is an elliptic operator of order $2 m$ (cf. also [[Elliptic partial differential equation|Elliptic partial differential equation]]), $\{ B _ { j } ( t , x , D _ { x } ) \} _ { j = 1 } ^ { m }$ are operators of order $< 2 m$ for each $t \in [ 0 , T]$, and $\Omega$ is a usually bounded open set in ${\bf R} ^ { n }$, $n \geq 1$, with smooth boundary $\partial \Omega$. Under some algebraic assumptions on the operators $L ( t , x , D _ { x } )$, $\{ B _ { j } ( t , x , D _ { x } ) \} _ { j = 1 } ^ { m }$ and smoothness hypotheses of the coefficients, it is shown in [[#References|[a1]]] that the operator-valued function $A ( t )$ defined 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/a120070/a120070109.png" /></td> </tr></table>
+
\begin{equation*} D ( A ( t ) ) = \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/a120070/a120070110.png" /></td> </tr></table>
+
\begin{equation*} \left\{ u \in \cap _ { q \in ( n , \infty ) } W ^ { 2 m , q } ( \Omega ) : \begin{array}{l} {  L(t, \cdot , D_x) u \in C ( \overline { \Omega } ),  } \\ {B _ { j } ( t , \cdot , D _ { x } ) u=0 \ \text{ on }  \partial \Omega,} \\ {j=1, \dots , m} \end{array} \right\},\; A(t)u=L(\cdot , t , D_x)u \ \text{ for } \ u \in D(A(t)), \end{equation*}
  
satisfies the assumptions I) and II) in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070111.png" /> if some negative constant is added to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070112.png" /> if necessary (this is not an essential restriction). The regularity of the coefficients here is Hölder continuity with some exponent. An analogous result holds for the operator defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070114.png" /> by
+
satisfies the assumptions I) and II) in the space $C ( \overline { \Omega } )$ if some negative constant is added to $L ( x , t , D _ { x } )$ if necessary (this is not an essential restriction). The regularity of the coefficients here is Hölder continuity with some exponent. An analogous result holds for the operator defined in $L ^ { p } ( \Omega )$, $1 < p < \infty$ 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/a120070/a120070115.png" /></td> </tr></table>
+
\begin{equation*} D ( A ( t ) ) = \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/a120070/a120070116.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070116.png"/></td> </tr></table>
  
<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/a120070/a120070117.png" /></td> </tr></table>
+
\begin{equation*} A ( t ) u = L ( . , t , D _ { x } ) u\text { for } u \in D ( A ( t ) ). \end{equation*}
  
 
There is also extensive literature on non-linear equations; see [[#References|[a7]]] and [[#References|[a9]]] for details. The following result on the quasi-linear partial differential equation
 
There is also extensive literature on non-linear equations; see [[#References|[a7]]] and [[#References|[a9]]] for details. The following result on the quasi-linear partial 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/a120070/a120070118.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\begin{equation} \tag{a6} \frac { d u } { d t } = A ( t , u ) u + f ( t , u ) \end{equation}
  
is due to H. Amann [[#References|[a5]]], [[#References|[a6]]]: For a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070119.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070120.png" /> be the solution of the linear problem
+
is due to H. Amann [[#References|[a5]]], [[#References|[a6]]]: For a given function $v$, let $u ( v )$ be the solution of the linear problem
  
<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/a120070/a120070121.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\begin{equation} \tag{a7} \frac { d u } { d t } = A ( t , v ) u + f ( t , v ) , 0 < t \leq T , u ( 0 ) = u_0. \end{equation}
  
If the problem is extended to a larger space so that the domains of the extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070122.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070123.png" />, then, under a weak regularity hypothesis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070124.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070125.png" />, the fundamental solution for the equation (a7) can be constructed, and a fixed-point theorem can be applied to the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120070/a120070126.png" /> to solve the equation (a6). The result has been applied to quasi-linear parabolic partial differential equations with quasi-linear boundary conditions.
+
If the problem is extended to a larger space so that the domains of the extensions of $A ( t , v )$ are independent of $( t , v )$, then, under a weak regularity hypothesis for $A ( t , v )$ on $( t , v )$, the fundamental solution for the equation (a7) can be constructed, and a fixed-point theorem can be applied to the mapping $v \mapsto u ( v )$ to solve the equation (a6). The result has been applied to quasi-linear parabolic partial differential equations with quasi-linear boundary conditions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Acquistapace,  "Evolution operators and strong solutions of abstract linear parabolic equations"  ''Diff. and Integral Eq.'' , '''1'''  (1988)  pp. 433–457</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Acquistapace,  B. Terreni,  "Some existence and regularity results for abstract non-autonomous parabolic equations"  ''J. Math. Anal. Appl.'' , '''99'''  (1984)  pp. 9–64</TD></TR><TR><TD valign="top">[a3]</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">[a4]</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">[a5]</TD> <TD valign="top">  H. Amann,  "Quasilinear parabolic systems under nonlinear boundary conditions"  ''Arch. Rat. Mech. Anal.'' , '''92'''  (1986)  pp. 153–192</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Amann,  "On abstract parabolic fundamental solutions"  ''J. Math. Soc. Japan'' , '''39'''  (1987)  pp. 93–116</TD></TR><TR><TD valign="top">[a7]</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">[a8]</TD> <TD valign="top">  T. Kato,  H. Tanabe,  "On the abstract evolution equation"  ''Osaka Math. J.'' , '''14'''  (1962)  pp. 107–133</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A. Lunardi,  "Analytic semigroups and optimal regularity in parabolic problems" , ''Progr. Nonlinear Diff. Eqns. Appl.'' , '''16''' , Birkhäuser  (1995)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Yagi,  "On the abstract evolution equation of parablic type"  ''Osaka J. Math.'' , '''14'''  (1977)  pp. 557–568</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Acquistapace,  "Evolution operators and strong solutions of abstract linear parabolic equations"  ''Diff. and Integral Eq.'' , '''1'''  (1988)  pp. 433–457</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P. Acquistapace,  B. Terreni,  "Some existence and regularity results for abstract non-autonomous parabolic equations"  ''J. Math. Anal. Appl.'' , '''99'''  (1984)  pp. 9–64</td></tr><tr><td valign="top">[a3]</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">[a4]</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">[a5]</td> <td valign="top">  H. Amann,  "Quasilinear parabolic systems under nonlinear boundary conditions"  ''Arch. Rat. Mech. Anal.'' , '''92'''  (1986)  pp. 153–192</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  H. Amann,  "On abstract parabolic fundamental solutions"  ''J. Math. Soc. Japan'' , '''39'''  (1987)  pp. 93–116</td></tr><tr><td valign="top">[a7]</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">[a8]</td> <td valign="top">  T. Kato,  H. Tanabe,  "On the abstract evolution equation"  ''Osaka Math. J.'' , '''14'''  (1962)  pp. 107–133</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  A. Lunardi,  "Analytic semigroups and optimal regularity in parabolic problems" , ''Progr. Nonlinear Diff. Eqns. Appl.'' , '''16''' , Birkhäuser  (1995)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  A. Yagi,  "On the abstract evolution equation of parablic type"  ''Osaka J. Math.'' , '''14'''  (1977)  pp. 557–568</td></tr></table>

Latest revision as of 18:47, 26 January 2024

An equation of the form

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

where for each $t \in [ 0 , T]$, $A ( t )$ is the infinitesimal generator of an analytic semi-group (cf. also Semi-group of operators; Strongly-continuous semi-group) in some Banach space $X$. Hence, without loss of generality it is always assumed that

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

i) $\rho ( A ( t ) ) \supset S _ { \theta _ { 0 } } = \{ z \in {\bf C} : | \operatorname { arg } z | \leq \theta _ { 0 } \} \cup \{ 0 \}$, $t \in [ 0 , T]$;

ii) $\| ( \lambda - A ( t ) ) ^ { - 1 } \| \leq M / ( 1 + | \lambda | )$, $\lambda \in S _ { \theta _ { 0 } }$, $t \in [ 0 , T]$. The domain $D ( A ( t ) )$ of $A ( t )$ is not necessarily dense. Various results on the solvability of the initial value problem for (a1) have been published. The main object is to construct the fundamental solution $U ( t , s ) , 0 \leq s \leq t \leq T$ (cf. also Fundamental solution), which is an operator-valued function satisfying

\begin{equation*} \frac { \partial } { \partial t } U ( t , s ) - A ( t ) U ( t , s ) = 0 \end{equation*}

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

The solution of (a1) satisfying the initial condition

\begin{equation} \tag{a2} u ( 0 ) = u _ { 0 } \in \overline { D ( A ( 0 ) ) }, \end{equation}

if it exists, is given by

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

In parabolic cases, the fundamental solution usually satisfies the inequality

\begin{equation*} \| \frac { \partial U ( t , s ) } { \partial t } \| \leq \frac { C } { t - s } , \quad s , t \in [ 0 , T ], \end{equation*}

for some constant $C > 0$.

One of the most general result is due to P. Acquistapace and B. Terreni [a3], [a4]. Suppose that

II) there exist a constant $K _ { 0 } > 0$ 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 K _ { 0 } \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 exists, and if $u_{0}$ satisfies (a2) and $f ( . )$ is Hölder continuous (i.e., $f \in C ^ { \alpha } ( [ 0 , T ] ; X )$ for some $\alpha \in ( 0,1 ]$, i.e.

\begin{equation} \tag{a4} \| f ( t ) - f ( s ) \| \leq C _ { 1 } | t - s | ^ { \alpha } , \quad 0 \leq s \leq t \leq T; \end{equation}

cf. also Hölder condition), then the function (a3) is the unique solution of (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) is satisfied in $[0 , T]$. Such a solution is usually called a strict solution.

The following results on maximal regularity are well known.

Time regularity.

Let $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { \alpha } ( [ 0 , T ] ; X )$. Then

\begin{equation*} u ^ { \prime } \in C ^ { \alpha } ( [ 0 , T ] ; X ) \bigcap B ( D _ { A } ( \alpha , \infty ) ), \end{equation*}

\begin{equation*} A u \in C ^ { \alpha } ( [ 0 , T ] ; X ) \end{equation*}

if and only if $A ( 0 ) u _ { 0 } + f ( 0 ) \in D _ { A ( 0 ) } ( \alpha , \infty )$, where $B ( D _ { A } ( \alpha , \infty ) )$ is the set of all functions $f \in L ^ { \infty } ( 0 , T ; X )$ such that $f ( t ) \in D _ { A ( t ) } ( \alpha , \infty )$ (an interpolation space between $D ( A ( t ) )$ and $X$; cf. also Interpolation of operators) for almost all $t \in [ 0 , T]$ and the norm of $f ( t )$ on $D _ { A ( t ) } ( \alpha , \infty )$ is essentially bounded in $[0 , T]$.

Space regularity.

Let $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in B ( D _ { A } ( \alpha , \infty ) )$. Then

\begin{equation*} u ^ { \prime } \in B ( D _ { A } ( \alpha , \infty ) ), \end{equation*}

\begin{equation*} A u \in B ( D _ { A } ( \alpha , \infty ) ) \bigcap C ^ { \alpha } ( [ 0 , T ] ; X ) \end{equation*}

if and only if $A ( 0 ) u _ { 0 } \in D _ { A ( 0 ) } ( \alpha , \infty )$.

Hypothesis II) holds if the domain $D ( A ( t ) )$ is independent of $t$ and $t \mapsto A ( t )$ is Hölder continuous, i.e. there exist constants $C _ { 2 } > 0$ and $\alpha \in ( 0,1 ]$ such that

\begin{equation*} | ( A ( t ) - A ( s ) ) A ( 0 ) ^ { - 1 } \| \leq C _ { 2 } | t - s | ^ { \alpha } , \quad t , s \in [ 0 , T ]. \end{equation*}

Another main result by Acquistapace and Terreni is the following ([a2]):

III.i) $t \mapsto A ( t ) ^ { - 1 }$ is differentiable and there exist constants $K _ { 1 } > 0$ and $\rho \in ( 0,1 ]$ such that

\begin{equation*} \left\| \frac { \partial } { \partial t } ( \lambda - A ( t ) ) ^ { - 1 } \right\| \leq \frac { K _ { 1 } } { ( 1 + | \lambda | ) ^ { \rho } }, \end{equation*}

\begin{equation*} \lambda \in S _ { \theta _ { 0 } } , t \in [ 0 , T ]; \end{equation*}

III.ii) there exist constants $K _ { 2 } > 0$ and $\eta \in ( 0,1 ]$ such that

\begin{equation*} \| \frac { d } { d t } A ( t ) ^ { - 1 } - \frac { d } { d s } A ( s ) ^ { - 1 } \| \leq K _ { 2 } | t - s | ^ { \eta }, \end{equation*}

\begin{equation*} t , s \in [ 0 , T ]. \end{equation*}

If $A ( t )$ is densely defined, this case reduces to the one in [a8]. Under the assumptions I), III) it can be shown that for $u_{0}$ and $f$ satisfying (a2) and (a4), a classical solution of (a1) exists and is unique. The solution is strict if, moreover, $u _ { 0 } \in D ( A ( 0 ) )$ and

\begin{equation} \tag{a5} A ( 0 ) u _ { 0 } + f ( 0 ) - \frac { d } { d t } A ( t ) ^ { - 1 } | _ { t = 0 } A ( 0 ) u _ { 0 } \in \overline { D ( A ( 0 ) ) }. \end{equation}

The following maximal regularity result holds: If $u _ { 0 } \in D ( A ( 0 ) )$ and $f \in C ^ { \delta } ( [ 0 , T ] ; X )$ for $\delta \in ( 0 , \eta ) \cap ( 0 , \rho ]$, then the solution of (a1) belongs to $C ^ { 1 + \delta } ( [ 0 , T ] ; X )$ if and only if the left-hand side of (a5) belongs to $D _ { A ( 0 ) } ( \delta , \infty )$.

Another of general results is due to A. Yagi [a10], where the fundamental solution is constructed under the following assumptions:

IV) hypothesis III.i) is satisfied, and there exist constants $K _ { 3 }$ and a non-empty set of indices $\{ ( \alpha _ { i } , \beta _ { i } ) : i = 1 , \ldots , k \}$ satisfying $- 1 \leq \alpha _ { i } < \beta _ { i } \leq 1$ such that

\begin{equation*} \left| A ( t ) ( \lambda - A ( t ) ) ^ { - 1 } \frac { d A ( t ) ^ { - 1 } } { d t } + \right. \end{equation*}

\begin{equation*} \left. - A ( s ) ( \lambda - A ( s ) ) ^ { - 1 } \frac { d A ( s ) ^ { - 1 } } { d s } \right\| \leq \end{equation*}

\begin{equation*} \leq K _ { 2 } \sum _ { i = 1 } ^ { k } | \lambda | ^ { \alpha _ { i } } | t - s | ^ { \beta _ { i } }, \end{equation*}

\begin{equation*} \lambda \in S _ { \theta _ { 0 } } , \quad t , s \in [ 0 , T ]. \end{equation*}

It is shown in [a4] that the above three results are independent of one another.

The above results are applied to initial-boundary value problems for parabolic partial differential equations:

\begin{equation*} \frac { \partial u } { \partial t } = L ( t , x , D _ { x } ) u + f ( t , x ) \text { in } [ 0 , T ] \times \Omega, \end{equation*}

\begin{equation*} B _ { j } ( t , x , D _ { x } ) u = 0 , \text { on } [ 0 , T ] \times \partial \Omega ,\quad j = 1 , \ldots , m, \end{equation*}

where $L ( t , x , D _ { x } )$ is an elliptic operator of order $2 m$ (cf. also Elliptic partial differential equation), $\{ B _ { j } ( t , x , D _ { x } ) \} _ { j = 1 } ^ { m }$ are operators of order $< 2 m$ for each $t \in [ 0 , T]$, and $\Omega$ is a usually bounded open set in ${\bf R} ^ { n }$, $n \geq 1$, with smooth boundary $\partial \Omega$. Under some algebraic assumptions on the operators $L ( t , x , D _ { x } )$, $\{ B _ { j } ( t , x , D _ { x } ) \} _ { j = 1 } ^ { m }$ and smoothness hypotheses of the coefficients, it is shown in [a1] that the operator-valued function $A ( t )$ defined by

\begin{equation*} D ( A ( t ) ) = \end{equation*}

\begin{equation*} \left\{ u \in \cap _ { q \in ( n , \infty ) } W ^ { 2 m , q } ( \Omega ) : \begin{array}{l} { L(t, \cdot , D_x) u \in C ( \overline { \Omega } ), } \\ {B _ { j } ( t , \cdot , D _ { x } ) u=0 \ \text{ on } \partial \Omega,} \\ {j=1, \dots , m} \end{array} \right\},\; A(t)u=L(\cdot , t , D_x)u \ \text{ for } \ u \in D(A(t)), \end{equation*}

satisfies the assumptions I) and II) in the space $C ( \overline { \Omega } )$ if some negative constant is added to $L ( x , t , D _ { x } )$ if necessary (this is not an essential restriction). The regularity of the coefficients here is Hölder continuity with some exponent. An analogous result holds for the operator defined in $L ^ { p } ( \Omega )$, $1 < p < \infty$ by

\begin{equation*} D ( A ( t ) ) = \end{equation*}

\begin{equation*} A ( t ) u = L ( . , t , D _ { x } ) u\text { for } u \in D ( A ( t ) ). \end{equation*}

There is also extensive literature on non-linear equations; see [a7] and [a9] for details. The following result on the quasi-linear partial differential equation

\begin{equation} \tag{a6} \frac { d u } { d t } = A ( t , u ) u + f ( t , u ) \end{equation}

is due to H. Amann [a5], [a6]: For a given function $v$, let $u ( v )$ be the solution of the linear problem

\begin{equation} \tag{a7} \frac { d u } { d t } = A ( t , v ) u + f ( t , v ) , 0 < t \leq T , u ( 0 ) = u_0. \end{equation}

If the problem is extended to a larger space so that the domains of the extensions of $A ( t , v )$ are independent of $( t , v )$, then, under a weak regularity hypothesis for $A ( t , v )$ on $( t , v )$, the fundamental solution for the equation (a7) can be constructed, and a fixed-point theorem can be applied to the mapping $v \mapsto u ( v )$ to solve the equation (a6). The result has been applied to quasi-linear parabolic partial differential equations with quasi-linear boundary conditions.

References

[a1] P. Acquistapace, "Evolution operators and strong solutions of abstract linear parabolic equations" Diff. and Integral Eq. , 1 (1988) pp. 433–457
[a2] P. Acquistapace, B. Terreni, "Some existence and regularity results for abstract non-autonomous parabolic equations" J. Math. Anal. Appl. , 99 (1984) pp. 9–64
[a3] 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
[a4] P. Acquistapace, B. Terreni, "A unified approach to abstract linear non-autonomous parabolic equations" Rend. Sem. Univ. Padova , 78 (1987) pp. 47–107
[a5] H. Amann, "Quasilinear parabolic systems under nonlinear boundary conditions" Arch. Rat. Mech. Anal. , 92 (1986) pp. 153–192
[a6] H. Amann, "On abstract parabolic fundamental solutions" J. Math. Soc. Japan , 39 (1987) pp. 93–116
[a7] H. Amann, "Linear and quasilinear parabolic problems I: Abstract linear theory" , Monogr. Math. , 89 , Birkhäuser (1995)
[a8] T. Kato, H. Tanabe, "On the abstract evolution equation" Osaka Math. J. , 14 (1962) pp. 107–133
[a9] A. Lunardi, "Analytic semigroups and optimal regularity in parabolic problems" , Progr. Nonlinear Diff. Eqns. Appl. , 16 , Birkhäuser (1995)
[a10] A. Yagi, "On the abstract evolution equation of parablic type" Osaka J. Math. , 14 (1977) pp. 557–568
How to Cite This Entry:
Abstract parabolic differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_parabolic_differential_equation&oldid=11870
This article was adapted from an original article by H. Tanabe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article