In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.).
Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privileged position (e.g. time, in heat or wave equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained.
Adding boundary conditions can often be translated in terms of considering solutions in some convenient function spaces.
The classical abstract differential equation which is most frequently encountered is the equation[1] where the unknown function
belongs to some function space
) case with a constant operator is given by the theory of C0-semigroups.
The theory of abstract differential equations has been founded by Einar Hille in several papers and in his book Functional Analysis and Semi-Groups.
Other main contributors were[2] Kōsaku Yosida, Ralph Phillips, Isao Miyadera, and Selim Grigorievich Krein.
such that: The Cauchy problem consists in finding a solution of the equation, satisfying the initial condition
According to the definition of well-posed problem by Hadamard, the Cauchy problem is said to be well posed (or correct) on
To an abstract Cauchy problem one can associate a semigroup of operators
, i.e. a family of bounded linear operators depending on a parameter
If the Cauchy problem is well posed, then the operator
can be extended to a bounded linear operator defined on the entire space
Such a function is called generalized solution of the Cauchy problem.
and the Cauchy problem is uniformly well posed, then the associated semigroup
, then the Cauchy problem is uniformly well posed and the solution is given by The Cauchy problem with
is continuously differentiable, then the function is the unique solution to the (abstract) nonhomogeneous Cauchy problem.
The problem[7] of finding a solution to the initial value problem where the unknown is a function
, is called time-dependent Cauchy problem.
(the space of all bounded linear operators from
), defined and strongly continuous jointly in
, is called a fundamental solution of the time-dependent problem if:
is also called evolution operator, propagator, solution operator or Green's function.
is called a mild solution of the time-dependent problem if it admits the integral representation There are various known sufficient conditions for the existence of the evolution operator
In practically all cases considered in the literature
is the infinitesimal generator of a contraction semigroup the equation is said to be of hyperbolic type; if
is the infinitesimal generator of an analytic semigroup the equation is said to be of parabolic type.
The problem[7] of finding a solution to either where
, is called nonlinear Cauchy problem.