Abstract parabolic differential equation

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.[edit]

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.[edit]

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[edit]