# Differential equation, abstract

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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)