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Abstract parabolic differential equation

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An equation of the form

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=55329
This article was adapted from an original article by H. Tanabe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article