Linear differential equation in a Banach space

An equation of the form

$$ \tag{1 } A _ {0} ( t) \dot{u} = A _ {1} ( t) u + g ( t) , $$

where $ A _ {0} ( t) $ and $ A _ {1} ( t) $, for every $ t $, are linear operators in a Banach space $ E $, $ g ( t) $ is a given function and $ u ( t) $ an unknown function, both with values in $ E $; the derivative $ \dot{u} $ is understood to be the limit of the difference quotient with respect to the norm of $ E $.

1. Linear differential equations with a bounded operator.

Suppose that $ A _ {0} ( t) $ and $ A _ {1} ( t) $, for every $ t $, are bounded operators acting in $ E $. If $ A _ {0} ( t) $ has a bounded inverse for every $ t $, then (1) can be solved for the derivative and takes the form

$$ \tag{2 } \dot{u} = A ( t) u + f ( t) , $$

where $ A ( t) $ is a bounded operator in $ E $, and $ f ( t) $ and $ u ( t) $ are functions with values in $ E $. If the functions $ A ( t) $ and $ f ( t) $ are continuous (or, more generally, are measurable and integrable on every finite interval), then the solution of the Cauchy problem

$$ \tag{3 } \dot{u} = A ( t) u ,\ u ( s) = u _ {0} , $$

exists for any $ u _ {0} \in E $ and is given by the formula

$$ u ( t) = U ( t , s ) u _ {0} , $$

where

$$ \tag{4 } U ( t , s ) = $$

$$ = \ I + \int\limits _ { s } ^ { t } A ( t _ {1} ) dt _ {1} + \sum _ { n= } 2 ^ \infty \int\limits _ { s } ^ { t } \int\limits _ { s } ^ { {t _ n} } \dots \int\limits _ { s } ^ { {t } _ {2} } A ( t _ {n} ) \dots A ( t _ {1} ) d t _ {1} \dots d t _ {n} $$

is the evolution operator of the equation $ \dot{u} = A ( t) u $. The solution of the Cauchy problem for equation (2) is determined by the formula

$$ u ( t) = U ( t , s ) u _ {0} + \int\limits _ { s } ^ { t } U ( t , \tau ) f ( \tau ) d \tau . $$

From (4) one obtains the estimate

$$ \tag{5 } \| U ( t , s ) \| \leq \mathop{\rm exp} \left \{ \int\limits _ { s } ^ { t } \| A ( \tau ) \| d \tau \right \} ; $$

a refinement of it is:

$$ \tag{5'} \| U ( t , s ) \| \leq \mathop{\rm exp} \left \{ \int\limits _ { s } ^ { t } r _ {A} ( \tau ) d \tau \right \} , $$

where $ r _ {A} ( \tau ) $ is the spectral radius of the operator $ A ( \tau ) $. The evolution operator has the properties

$$ U ( s , s ) = I ,\ U ( t , \tau ) U ( \tau , s ) = U ( t , s), $$

$$ U ( t , \tau ) = [ U ( \tau , t ) ] ^ {-} 1 . $$

In the study of (2) the main attention has been focused on the behaviour of its solutions at infinity, in dependence on the behaviour of $ A ( t) $ and $ f ( t) $. An important characteristic of the equation is the general (or singular) exponent

$$ \kappa = \overline{\lim\limits}\; _ {\tau , s \rightarrow \infty } \frac{1} \tau \mathop{\rm ln} \| U ( \tau + s , s ) \| . $$

Equations with periodic and almost-periodic coefficients have been studied in detail (see Qualitative theory of differential equations in Banach spaces).

Equation (2) can also be considered in the complex plane. If the functions $ A ( t) $ and $ f ( t) $ are holomorphic in a simply-connected domain containing the point $ s $, then the formulas (3), (4), (5), (5'}) remain valid if the integrals are understood to be integrals over a rectifiable arc joining $ s $ and $ t $.

A number of other questions arises in the case when the original linear equation is not solvable for the derivative. If the operator $ A _ {0} ( t) $ is boundedly invertible everywhere except at one point, say $ t = 0 $, then in the space $ E $ the equation reduces to the form

$$ \tag{6 } a ( t) \dot{u} = A ( t) u + f ( t) , $$

where $ a ( t) $ is a scalar function and $ a ( 0) = 0 $. Here the main attention is focused on the study of the behaviour of the solutions in a neighbourhood of the origin, and the analytic and non-analytic cases are distinguished.

The analytic case.

For the simplest equation

$$ t \dot{u} = A u $$

with a constant operator $ A $, the evolution operator $ U ( t) = U ( t , 0 ) $ has the form

$$ U ( t) = e ^ {A \mathop{\rm ln} t } , $$

and the solutions are not single-valued: as one goes round the origin in the positive direction they are multiplied by the operator $ e ^ {2 \pi i A } $.

Consider an equation with a regular singularity

$$ \tag{7 } t \dot{u} = \left ( \sum _ { k= } 0 ^ \infty A ^ {(} k) t ^ {k} \right ) u , $$

where the series on the right-hand side converges in a neighbourhood of the origin. If one looks for the operator $ U ( t) $ in the form of a series

$$ U ( t) = \left ( \sum _ { k= } 0 ^ \infty U ^ {(} k) t ^ {k} \right ) e ^ {A ^ {(} 0) \mathop{\rm ln} t } , $$

then for the determination of the coefficients $ U ^ {(} k) $ one obtains the system of equations

$$ A ^ {(} 0) U ^ {(} 0) - U ^ {(} 0) A ^ {(} 0) = 0 , $$

$$ ( A ^ {(} 0) - kI ) U ^ {(} k) - U ^ {(} k) A ^ {(} 0) = - \sum _ { j= } 1 ^ { k } A ^ {(} j) u ^ {( k- j ) } ,\ k = 1 , 2 , . . . . $$

For this system to be solvable, that is, for (7) to be formally solvable, it is sufficient that the spectra of the operators $ A ^ {(} 0) $ and $ A ^ {(} 0) - kI $ do not intersect (cf. Spectrum of an operator), or, equivalently, that there are no points differing by an integer in the spectrum of $ A ^ {(} 0) $. Under this condition the series

$$ \sum _ { k= } 0 ^ \infty U ^ {( k) } {t ^ {k} } $$

converges in the same neighbourhood of zero as the series for $ A ( t) $. Now, if there are finitely many integers representable as differences of points of the spectrum of $ A ^ {(} 0) $, and each of them is an isolated point of the spectrum of the transformer

$$ \mathfrak A X = A ^ {(} 0) X - X A ^ {(} 0) , $$

then there is a solution of the form

$$ U ( t) = \left ( I + \sum _ { k= } 1 ^ \infty U _ {k} ( \mathop{\rm ln} t ) t ^ {k} \right ) e ^ {A ^ {(} 0) \mathop{\rm ln} t } ,\ 0 < | t | < \rho , $$

where the $ U _ {k} $ are entire functions of the argument $ \mathop{\rm ln} t $, satisfying for every $ \epsilon > 0 $ the condition

$$ \| U _ {k} ( \mathop{\rm ln} t ) \| \leq C _ \epsilon e ^ {\epsilon | \mathop{\rm ln} t | } . $$

If the integer points of the spectrum of the transformer $ \mathfrak A $ are poles of its resolvent, then the functions $ U _ {k} $ are polynomials.

In the case of an irregular singularity, the differential equation

$$ t ^ {m} \dot{u} = \left ( \sum _ { k= } 0 ^ { m- } 1 A ^ {(} k) t ^ {k} \right ) u $$

has been considered in a Banach algebra $ \mathfrak B $( for example, in the algebra of bounded operators on a Banach space $ E $). Under certain restrictions on $ A ^ {(} 0) $ it reduces by means of Laplace integrals to an equation with a regular singularity $ ( m = 1 ) $ in the algebra of matrices with entries from $ \mathfrak B $.

The non-analytic case.

Suppose that in the equation

$$ t ^ {n} \dot{u} = A ( t) u + f ( t) ,\ 0 \leq t \leq T , $$

the functions $ A ( t) $ and $ f ( t) $ are infinitely differentiable. In the finite-dimensional case a complete result has been obtained: If the equation has a formal solution in the form of a power series, then it has a solution that is infinitely differentiable on $ [ 0 , T ] $ for which the formal series is the Taylor series at the point $ t = 0 $. In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions.

Suppose that $ m > 1 $. If the spectrum of the operator $ A ( 0) $ does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element $ g ^ {-} $ belonging to the invariant subspace of $ A ( 0) $ corresponding to the part of the spectrum of $ A ( 0) $ lying in the left half-plane. Any solution that is continuous on $ [ 0 , T] $ appears in this family. If the whole spectrum of $ A ( 0) $ lies in the left half-plane, then there is only one infinitely-differentiable solution.

Suppose that $ m = 1 $. If there are no negative integers in the spectrum of $ A ( 0) $, then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator $ A ( 0) $, equations of the form (6) have been considered in which $ a ( t) $ and $ f ( t) $ have finite smoothness, and the solutions have the same smoothness.

A rather different picture emerges when the differential equation is unsolvable for the derivative for all $ t $, for example when $ A $ is a constant non-invertible operator. Suppose that in the equation

$$ \tag{8 } A \dot{u} = B u $$

the operators $ A $ and $ B $ are bounded in the space $ E $ and $ A $ is a non-invertible Fredholm operator. Suppose that the operator $ A + \epsilon B $ is continuously invertible for sufficiently small $ \epsilon $. Then there are decompositions into direct sums $ E= N ^ {(} 1) + M ^ {(} 1) $ and $ E = N ^ {(} 2) + M ^ {(} 2) $ such that $ A $ and $ B $ map $ N ^ {(} 1) $ into $ N ^ {(} 2) $ and $ M ^ {(} 1) $ into $ M ^ {(} 2) $. The operator $ A $ is invertible on $ M ^ {(} 1) $ and maps onto $ M ^ {(} 2) $. The subspace $ N ^ {(} 1) $ is finite-dimensional. All solutions of (8) lie in the subspace $ M ^ {(} 1) $ and have the form $ \mathop{\rm exp} ( \widetilde{A} {} ^ {-} 1 Bt ) u _ {0} $, where $ \widetilde{A} $ is the restriction of $ A $ to $ M ^ {(} 1) $ and $ u _ {0} \in M ^ {(} 1) $. For an inhomogeneous equation $ A \dot{u} = Bu + f ( t) $, a solution exists only if $ f ( t) $ has a certain smoothness and under certain compatibility conditions for the values of $ f( t) $ and its derivatives with the initial data. The number of derivatives that certain components of $ f ( t) $ must have and the number of compatibility conditions are equal to the maximal length of $ B $- adjoint chains of the operator $ A $. If these conditions are satisfied, the solution of the Cauchy problem is unique.

If the operator $ A + \epsilon B $ is non-invertible for all $ \epsilon $, then all solutions of (8) lie in a subspace that has, generally speaking, infinite deficiency (cf. also Deficiency subspace). The solution of the Cauchy problem for it is not unique. For the function $ f( t) $ in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required.

2. Linear differential equations with an unbounded operator.

Suppose that $ A _ {0} ( t) $ is invertible for every $ t $, so that (1) can be solved for the derivative and takes the form

$$ \tag{9 } \dot{u} = A ( t) u + f ( t) , $$

and suppose that here $ A ( t) $ is an unbounded operator in a space $ E $, with dense domain of definition $ D ( A ( t) ) $ in $ E $ and with non-empty resolvent set, and suppose that $ f ( t) $ is a given function and $ u ( t) $ an unknown function, both with values in $ E $.

Even for the simplest equation $ \dot{u} = Au $ with an unbounded operator, solutions of the Cauchy problem $ u ( 0) = u _ {0} $ need not exist, they may be non-unique, and they may be non-extendable to the whole semi-axis, so the main investigations are devoted to the questions of existence and uniqueness of the solutions. A solution of the equation $ \dot{u} = Au $ on the interval $ [ 0, T ] $ is understood to be a function that takes values in $ D ( A) $, is differentiable on $ [ 0, T ] $ and satisfies the equation. Sometimes this definition is too rigid and one introduces the concept of a weak solution as a function that has the same properties on $ ( 0 , T ] $ and is only continuous at $ 0 $.

Suppose that the operator $ A $ has a resolvent

$$ R ( \lambda , A ) = ( A - \lambda I ) ^ {-} 1 $$

for all sufficiently large positive $ \lambda $ and that

$$ \overline{\lim\limits}\; _ {\lambda \rightarrow \infty } \lambda ^ {-} 1 \mathop{\rm ln} \| R ( \lambda , A ) \| = h < T . $$

Then the weak solution of the problem

$$ \tag{10 } \dot{u} = Au ,\ u ( 0) = u _ {0} $$

is unique on $ [ 0 , T - h ] $ and can be branched for $ t = T - h $. If $ h = 0 $, then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of $ R ( \lambda , A ) $ as $ \lambda \rightarrow \infty $.

If for every $ u _ {0} \in D ( A) $ there is a unique solution of the problem (10) that is continuously differentiable on $ [ 0 , T ] $, then this solution can be extended to the whole semi-axis and can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is a strongly-continuous semi-group of bounded operators on $ [ 0 , \infty ) $, $ U ( 0) = I $, for which the estimate $ \| U ( t) \| \leq M e ^ {\omega t } $ holds. For the equation to have this property it is necessary and sufficient that

$$ \tag{11 } \| ( \lambda - \omega ) ^ {m} R ^ {m} ( \lambda , A ) \| \leq M $$

for all $ \lambda > \omega $ and $ m = 1 , 2 \dots $ where $ M $ does not depend on $ \lambda $ and $ m $. These conditions are difficult to verify. They are satisfied if $ \| ( \lambda - \omega ) R ( \lambda , A ) \| \leq 1 $, and then $ \| U ( t) \| \leq e ^ {\omega t } $. If $ \omega = 0 $, then $ U ( t) $ is a contraction semi-group. This is so if and only if $ A $ is a maximal dissipative operator. If $ u _ {0} \notin D ( A) $, then the function $ U ( t) u _ {0} $ is not differentiable (in any case for $ t = 0 $); it is often called the generalized solution of (10). Solutions of the equation $ \dot{u} = Au $ can be constructed as the limit, as $ n \rightarrow \infty $, of solutions of the equation $ \dot{u} = A _ {n} u $ with bounded operators, under the same initial conditions. For this it is sufficient that the operators $ A _ {n} $ commute, converge strongly to $ A $ on $ D ( A) $ and that

$$ \| e ^ {t A _ {n} } \| \leq M e ^ {\omega t } . $$

If the conditions (11) are satisfied, then the operators $ A _ {n} = - nI - n ^ {2} R ( \lambda , A ) $( Yosida operators) have these properties.

Another method for constructing solutions of the equation $ \dot{u} = A u $ is based on Laplace transformation. If the resolvent of $ A $ is defined on some contour $ \Gamma $, then the function

$$ \tag{12 } u ( t) = - \frac{1}{2 \pi i } \int\limits _ \Gamma e ^ {\lambda t } R ( \lambda , A ) u _ {0} d \lambda $$

formally satisfies the equation

$$ \dot{u} = A u + \frac{1}{2 \pi i } \int\limits _ \Gamma e ^ {\lambda t } \ d \lambda u _ {0} . $$

If the convergence of the integrals, the validity of differentiation under the integral sign and the vanishing of the last integral are ensured, then $ u ( t) $ satisfies the equation. The difficulty lies in the fact that the norm of the resolvent cannot decrease faster than $ | \lambda | ^ {-} 1 $ at infinity. However, on some elements it does decrease faster. For example, if $ R ( \lambda , A ) $ is defined for $ \mathop{\rm Re} \lambda \geq \alpha $ and if

$$ \| R ( \lambda , A ) \| \leq M | \lambda | ^ {k} ,\ k \geq - 1 , $$

for sufficiently large $ | \lambda | $, then for $ \Gamma = ( - i \infty , i \infty ) $ formula (12) gives a solution for any $ u _ {0} \in D ( A ^ {[ k ] + 3 } ) $. In a "less good" case, when the previous inequality is satisfied only in the domain

$$ \mathop{\rm Re} \lambda \geq \alpha | \mathop{\rm Im} \lambda | ^ {a} ,\ 0 < a < 1 $$

(weakly hyperbolic equations), and $ \Gamma $ is the boundary of this domain, one obtains a solution only for an $ u _ {0} $ belonging to the intersection of the domains of definition of all powers of $ A $, with definite behaviour of $ \| A ^ {n} u _ {0} \| $ as $ n \rightarrow \infty $.

Significantly weaker solutions are obtained in the case when $ \Gamma $ goes into the left half-plane, and one can use the decrease of the function $ | e ^ {\lambda t } | $ on it. As a rule, the solutions have increased smoothness for $ t > 0 $. If the resolvent is bounded on the contour $ \Gamma $: $ \mathop{\rm Re} \lambda = - \psi ( | \mathop{\rm Im} \lambda | ) $, where $ \psi ( \tau ) $ is a smooth non-decreasing concave function that increases like $ \mathop{\rm ln} \tau $ at $ \infty $, then for any $ u _ {0} \in E $ the function (12) is differentiable and satisfies the equation, beginning with some $ t _ {0} $; as $ t $ increases further, its smoothness increases. If $ \psi ( \tau ) $ increases like a power of $ \tau $ with exponent less than one, then the function (12) is infinitely differentiable for $ t > 0 $; if $ \psi ( \tau ) $ increases like $ \tau / \mathop{\rm ln} \tau $, then $ u ( t) $ belongs to a quasi-analytic class of functions; if it increases like a linear function, then $ u ( t) $ is analytic. In all these cases it satisfies the equation $ \dot{u} = A u $.

The existence of the resolvent on contours that go into the left half-plane may be obtained, by using series expansion, from the corresponding estimates on vertical lines. If for $ \mathop{\rm Re} \lambda \geq \gamma $,

$$ \tag{13 } \| R ( \lambda , A ) \| \leq M ( 1 + | \mathop{\rm Im} \lambda | ) ^ {- \beta } ,\ 0 < \beta < 1 , $$

then for every $ u _ {0} \in D ( A) $ there is a solution of problem (10). All these solutions are infinitely differentiable for $ t > 0 $. They can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is an infinitely-differentiable semi-group for $ t > 0 $ having, generally speaking, a singularity at $ t = 0 $. For its derivatives one has the estimates

$$ \| U ( k) ( t) \| \leq M _ {k} t ^ {1 - ( k+ 1 ) / \beta } e ^ { \omega t } . $$

If the estimate (13) is satisfied for $ \beta = 1 $, then all generalized solutions of the equation $ \dot{u} = Au $ are analytic in some sector containing the positive semi-axis.

The equation $ \dot{u} = Au $ is called an abstract parabolic equation if there is a unique weak solution on $ [ 0 , \infty ] $ satisfying the initial condition $ u ( 0) = u _ {0} $ for any $ u _ {0} \in E $. If

$$ \tag{14 } \| R ( \lambda , A ) \| \leq M | \lambda - \omega | ^ {-} 1 \ \ \textrm{ for } \mathop{\rm Re} \lambda > \omega , $$

then the equation is an abstract parabolic equation. All its generalized solutions are analytic in some sector containing the positive semi-axis, and

$$ \| \dot{u} ( t) \| \leq t ^ {-} 1 C e ^ {\omega t } \| u _ {0} \| , $$

where $ C $ does not depend on $ u _ {0} $. Conversely, if the equation has the listed properties, then (14) is satisfied for the operator $ A $.

If problem (10) has a unique weak solution for any $ u _ {0} \in D ( A) $ for which the derivative is integrable on every finite interval, then these solutions can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is a strongly-continuous semi-group on $ ( 0 , \infty ) $, and every weak solution of the inhomogeneous equation $ \dot{v} = Av + f ( t) $ with initial condition $ v ( 0) = 0 $ can be represented in the form

$$ \tag{15 } v ( t) = \int\limits _ { 0 } ^ { t } U ( t- s ) f ( s) ds . $$

The function $ v ( t) $ is defined for any continuous $ f ( t) $, hence it is called a generalized solution of the inhomogeneous equation. To ensure that it is differentiable, one imposes smoothness conditions on $ f ( t) $, and the "worse" the semi-group $ U ( t) $, the "higher" these should be. Thus, under the previous conditions, (15) is a weak solution of the inhomogeneous equation if $ f ( t) $ is twice continuously differentiable; if (11) is satisfied, then (15) is a solution if $ f ( t) $ is continuously differentiable; if (13) is satisfied with $ \beta > 2/3 $, then $ v ( t) $ is a weak solution if $ f ( t) $ satisfies a Hölder condition with exponent $ \gamma > 2 ( 1 - 1/ \beta ) $. Instead of smoothness of $ f ( t) $ with respect to $ t $ one can require that the values of $ f ( t) $ belong to the domain of definition of the corresponding power of $ A $.

For an equation with variable operator

$$ \tag{16 } \dot{u} = A ( t) u ,\ \ 0 \leq t \leq T , $$

there are some fundamental existence and uniqueness theorems about solutions (weak solutions) of the Cauchy problem $ u ( s) = u _ {0} $ on the interval $ s \leq t \leq T $. If the domain of definition of $ A ( t) $ does not depend on $ t $,

$$ D ( A ( t) ) \equiv D ( A), $$

if the operator $ A ( t) $ is strongly continuous with respect to $ t $ on $ D ( A) $ and if

$$ \| \lambda R ( \lambda , A ( t) ) \| \leq 1 $$

for $ \lambda > 0 $, then the solution of the Cauchy problem is unique. Moreover, if $ A ( t) $ is strongly continuously differentiable on $ D ( A) $, then for every $ u _ {0} \in D ( A) $ a solution exists and can be represented in the form

$$ u ( t) = U ( t , s ) u _ {0} , $$

where $ U ( t , s ) $ is an evolution operator with the following properties:

1) $ U ( t , s ) $ is strongly continuous in the triangle $ T _ \Delta $: $ 0 \leq s \leq t \leq T $;

2) $ U ( t , s ) = U ( t , \tau ) U ( \tau , s ) $, $ 0 \leq s \leq \tau \leq t \leq T $, $ U ( s , s ) = I $;

3) $ U ( t , s ) $ maps $ D ( A) $ into itself and the operator

$$ A ( t) U ( t , s ) A ^ {-} 1 ( s) $$

is bounded and strongly continuous in $ T _ \Delta $;

4) on $ D ( A) $ the operator $ U ( t , s ) $ is strongly differentiable with respect to $ t $ and $ s $ and

$$ \frac{\partial U }{\partial t } = A ( t) U ,\ \ \frac{\partial U }{\partial s } = - U A ( s) . $$

The construction of the operator $ U ( t , s ) $ is carried out by approximating $ A ( t) $ by bounded operators $ A _ {n} ( t) $ and replacing the latter by piecewise-constant operators.

In many important problems the previous conditions on the operator $ A ( t) $ are not satisfied. Suppose that for the operator $ A ( t) $ there are constants $ M $ and $ \omega $ such that

$$ \| R ( \lambda , A ( t _ {k} ) ) \dots R ( \lambda , A ( t _ {1} ) ) \| \leq M ( \lambda - \omega ) ^ {-} k $$

for all $ \lambda > \omega $, $ 0 \leq t _ {1} \leq \dots \leq t _ {k} \leq T $, $ k = 1 , 2, . . . $. Suppose that in $ E $ there is densely imbedded a Banach space $ F $ contained in all the $ D ( A ( t) ) $ and having the following properties: a) the operator $ A ( t) $ acts boundedly from $ F $ to $ E $ and is continuous with respect to $ t $ in the norm as a bounded operator from $ F $ to $ E $; and b) there is an isomorphism $ S $ of $ F $ onto $ E $ such that

$$ S A ( t) S ^ {-} 1 = A ( t) + B ( t) , $$

where $ B ( t) $ is an operator function that is bounded in $ E $ and strongly measurable, and for which $ \| B ( t) \| $ is integrable on $ [ 0 , T ] $. Then there is an evolution operator $ U ( t , s ) $ having the properties: 1); 2); 3') $ U ( t , s ) F \subset F $ and $ U ( t , s ) $ is strongly continuous in $ F $ on $ T _ \Delta $; and 4') on $ F $ the operator $ U ( t , s ) $ is strongly differentiable in the sense of the norm of $ E $ and $ \partial U / \partial t = A ( t) U $, $ \partial U / \partial s = - U A ( s) $. This assertion makes it possible to obtain existence theorems for the fundamental quasi-linear equations of mathematical physics of hyperbolic type.

The method of frozen coefficients is used in the theory of parabolic equations. Suppose that, for every $ t _ {0} \in [ 0 , T ] $, to the equation $ \dot{u} = A ( t _ {0} ) u $ corresponds an operator semi-group $ U _ {A ( t _ {0} ) } ( t) $. The unknown evolution operator formally satisfies the integral equations

$$ U ( t , s ) = U _ {A ( t) } ( t - s ) + $$

$$ + \int\limits _ { s } ^ { t } U _ {A ( t) } ( t - s ) [ A ( \tau ) - A ( t) ] U ( \tau , s ) d \tau , $$

$$ U ( t , s ) = U _ {A ( s) } ( t - s ) + $$

$$ + \int\limits _ { s } ^ { t } U ( t , \tau ) [ A ( \tau ) - A ( s) ] U _ {A ( s) } ( \tau - s ) d \tau . $$

When the kernels of these equations have weak singularities, one can prove that the equation has solutions and also that $ U ( t , s ) $ is an evolution operator. The following statement has the most applications: If

$$ D ( A ( t) ) \equiv D ( A) ,\ \ \| R ( \lambda , A ( t) ) \| < \ M ( 1 + | \lambda | ) ^ {-} 1 $$

for $ \mathop{\rm Re} \lambda \geq 0 $ and

$$ \| [ A ( t) - A ( s) ] A ^ {-} 1 ( 0) \| \leq C | t - s | ^ \rho $$

(a Hölder condition), then there is an evolution operator $ U ( t , s ) $ that gives a weak solution $ U ( t , s ) u _ {0} $ of the Cauchy problem for every $ u _ {0} \in E $. Uniqueness of the solution holds under the single condition that the operator $ A ( t) A ^ {-} 1 ( 0) $ is continuous (in a Hilbert space). An existence theorem similar to the one given above holds for the operator $ A ( t) $ with a condition of type (13) and for a certain relation between $ \beta $ and $ \rho $.

The assumption that $ D ( A ( t) ) $ is constant does not make it possible in applications to consider boundary value problems with boundary conditions depending on $ t $. Suppose that

$$ \| R ( \lambda , A ( t) ) \| \leq M ( 1 + | \lambda | ) ^ {-} 1 ,\ \ \mathop{\rm Re} \lambda > 0 ; $$

$$ \left \| \frac{d A ^ {-} 1 ( t) }{dt} - \frac{d A ^ {-} 1 ( s) }{ds} \right \| \leq K | t - s | ^ \alpha ,\ 0 < \alpha < 1 ; $$

$$ \left \| \frac \partial {\partial t } R ( \lambda , A ( t) ) \right \| \leq N | \lambda | ^ {\rho - 1 } ,\ 0 \leq \rho \leq 1 , $$

in the sector $ | \mathop{\rm arg} \lambda | \leq \pi - \phi $, $ \phi < \pi / 2 $; then there is an evolution operator $ U ( t , s ) $. Here it is not assumed that $ D ( A ( t) ) $ is constant. There is a version of the last statement adapted to the consideration of parabolic problems in non-cylindrical domains, in which $ D ( A ( t) ) $ for every $ t $ lies in some subspace $ E ( t) $ of $ E $.

The operator $ U ( t , s ) $ for equation (16) formally satisfies the integral equation

$$ \tag{17 } U ( t , s ) = I + \int\limits _ { s } ^ { t } A ( \tau ) U ( \tau , s ) d \tau . $$

Since $ A ( t) $ is unbounded, this equation cannot be solved by the method of successive approximation (cf. Sequential approximation, method of). Suppose that there is a family of Banach spaces $ E _ \alpha $, $ 0 \leq \alpha \leq 1 $, having the property that $ E _ \beta \subset E _ \alpha $ and $ \| x \| _ \alpha \leq \| x \| _ \beta $ for $ \alpha < \beta $. Suppose that $ A ( t) $ is bounded as an operator from $ E _ \beta $ to $ E _ \alpha $:

$$ \| A ( t) \| _ {E _ \beta \rightarrow E _ \alpha } \leq C ( \beta - \alpha ) ^ {-} 1 , $$

and that $ A ( t) $ is continuous with respect to $ t $ in the norm of the space of bounded operators from $ E _ \beta $ to $ E _ \alpha $. Then in this space the method of successive approximation for equation (17) will converge for $ | t - s | \leq ( \beta - \alpha ) ( Ce ) ^ {-} 1 $. In this way one can locally construct an operator $ U ( t , s ) $ as a bounded operator from $ E _ \beta $ to $ E _ \alpha $. In applications this approach gives theorems of Cauchy–Kovalevskaya type (cf. Cauchy–Kovalevskaya theorem).

For the inhomogeneous equation (9) with known evolution operator, for the equation $ \dot{u} = A ( t) u $ the solution of the Cauchy problem is formally written in the form

$$ u ( t) = U ( t , s ) u _ {0} + \int\limits _ { s } ^ { t } U ( t , \tau ) f ( \tau ) d \tau . $$

This formula can be justified in various cases under certain smoothness conditions on $ f ( t) $.

References

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