Differential equation, abstract

Either a differential equation in some abstract space (a Hilbert space, a Banach space, etc.) or a differential equation with operator coefficients. The classical abstract differential equation which is most frequently encountered is the equation

$$ \tag{1 } Lu = \frac{\partial u }{\partial t } - Au = f , $$

where the unknown function $ u = u ( t) $ belongs to some function space $ X $, $ 0 \leq t \leq T \leq \infty $, and $ A: X \rightarrow X $ is an operator (usually a linear operator) acting on this space. If the operator $ A $ is a bounded operator or a constant (does not depend on $ t $), the formula

$$ u ( t) = e ^ {tA} u _ {0} + \int\limits _ { 0 } ^ { t } e ^ {( t - \tau ) A } f ( \tau ) d \tau $$

yields the unique solution of equation (1) satisfying the condition $ u ( 0) = u _ {0} $. For a variable operator $ A ( t) $, $ e ^ {( t - \tau ) A } $ is replaced by the evolution operator $ U ( t , \tau ) $( cf. also Cauchy operator). If the operator $ A $ is unbounded, the solutions of the Cauchy problem $ u ( 0) = u _ {0} $ need not exist for some $ u _ {0} $, need not be unique and may break off for $ t < T $. An exhaustive treatment of the homogeneous ( $ f \equiv 0 $) equation (1) with a constant operator is given by the theory of semi-groups, while the problems of existence and uniqueness are solved in terms of the resolvent of $ A $[1], [5]. The same method is also applicable to a variable operator, if it depends smoothly on $ t $. Another method of study of equation (1), which usually gives less accurate results, but which is applicable to wider classes of equations (even including non-linear equations in some cases), is the use of energy inequalities: $ \| u \| \leq c \| Lu \| $, which are also obtained if certain assumptions are made regarding $ A $. For a Hilbert space $ X $ one usually postulates different positivity properties of the scalar product $ ( Au, u ) $[2]. All the above can be extended, to a certain extent, to more general abstract differential equations

$$ \tag{2 } \frac{d ^ {2} u }{d t ^ {2} } + A _ {1} \frac{d u }{d t } + A _ {2} u = f , $$

studied under the condition $ u ( 0) = u _ {0} $, $ u _ {t} ^ \prime ( 0) = u _ {1} $. Very often, the study of equation (2) by various methods (reduction to a set of equations of the first order, a substitution $ u = \int _ {0} ^ {t} v ( \tau ) d \tau $, subdivision of the left-hand side into a product of two operators of the first order, etc.) really amounts to the study of equation (1). A principal reason for the existing interest in abstract differential equations is that the so-called mixed problems in cylindrical domains for classical parabolic and hyperbolic equations of the second order can be reduced to equations of the form (1) or (2): The function $ u ( t , x _ {1} \dots x _ {n} ) $ is regarded as a function of $ t $ with values in the corresponding space of functions in $ x $, while the operators $ A $, $ A _ {k} $ are generated by differentiations with respect to $ x $, subject to the boundary conditions on the side surfaces of the cylinder (the generatrices of which are parallel to the $ t $- axis). Equations (1), (2), in which the postulated properties of the operators $ A $, $ A _ {k} $ coincide with those obtained in the situation described above, are known as parabolic or hyperbolic. Abstract elliptic operators are considered less often.

Problems in scattering theory [3] in the interval $ - \infty < t < + \infty $ are often formulated in terms of semi-groups and equation (1). The reduction of problems in partial differential equations to problems (1) and (2) in abstract differential equations are very convenient in developing approximate (e.g. difference [4]) methods of solution and in the study of asymptotic methods ( "small" and "large" parameters). General abstract differential equations with operator coefficients

$$ \sum _ {k= 0 } ^ { m } A _ {k} \frac{d ^ {k} }{d t ^ {k} } $$

and boundary conditions on both ends of the interval $ ( 0 , T ) $ for unbounded operators $ A _ {k} $ can be meaningfully studied only if very special assumptions concerning $ A _ {k} $ are made. For bounded $ A _ {k} $ there is no difficulty in extending the theory of ordinary differential equations in an appropriate manner.

References

[1]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2]	J.L. Lions, "Equations différentielles operationelles et problèmes aux limites" , Springer (1961)
[3]	P.D. Lax, R.S. Philips, "Scattering theory" , Acad. Press (1967)
[4]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[5]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)

Comments

For elliptic problems see [a3].

References

[a1]	H. Tanabe, "Equations of evolution" , Pitman (1979) (Translated from Japanese)
[a2]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[a3]	S. Agmon, "Unicité convexité dans problèmes différentiels" , Univ. Montréal (1966)