Abstract differential equation
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 where the unknown function formula_2 belongs to some function space formula_3, formula_4 and formula_5 is an operator (usually a linear operator) acting on this space. An exhaustive treatment of the homogeneous (formula_6) case with a constant operator is given by the theory of C-semigroups. Very often, the study of other abstract differential equations amounts (by e.g. reduction to a set of equations of the first order) to the study of this equation. The theory of abstract differential equations has been  founded by professor Einar Hille in several papers and in his book "Functional Analysis and Semi-Groups." Other main contributors were Kōsaku Yosida, Ralph Phillips, Isao Miyadera and Selim Grigorievich Krein. Abstract Cauchy problem. Definition. Let formula_7 and formula_8 be two linear operators, with domains formula_9 and formula_10, acting in a Banach space formula_3. A function formula_12 is said to have strong derivative (or to be Frechet differentiable or simply differentiable) at the point formula_13 if there exists an element formula_14 such that and its derivative is formula_16. A solution of the equation is a function formula_18 such that: The  Cauchy problem consists in finding a solution of the equation, satisfying the initial condition formula_25. Well posedness. According to the definition of well-posed problem by Hadamard, the Cauchy problem is said to be well posed (or correct) on formula_26 if: A well posed Cauchy problem is said to be uniformly well posed if formula_28 implies formula_30 uniformly in formula_23 on each finite interval formula_35. Semigroup of operators associated to a Cauchy problem. To an abstract Cauchy problem one can associate a semigroup of operators formula_36, i.e. a family of bounded linear operators depending on a parameter formula_23 (formula_38) such  that Consider the operator formula_36 which assigns to the element formula_29 the value of the solution formula_42 of the Cauchy problem (formula_43) at the moment of time formula_44. If the Cauchy problem is well posed, then the operator formula_36 is defined on formula_46 and forms a semigroup. Additionally, if formula_46 is dense in formula_3, the operator formula_36 can be extended to a bounded linear operator defined on the entire space formula_3. In this case one can associate to any formula_51 the function formula_52, for any formula_44. Such a function is called generalized solution of the Cauchy problem. If formula_46 is  dense in formula_3 and the Cauchy problem is uniformly well posed, then the associated semigroup formula_36 is a C-semigroup in formula_3. Conversely, if formula_7 is the infinitesimal generator of a C-semigroup formula_36, then the Cauchy problem is uniformly well posed and the solution is given by Nonhomogeneous problem. The Cauchy problem with formula_63, is called nonhomogeneous when formula_64. The following theorem gives some sufficient conditions for the existence of the solution: Theorem. If formula_7 is an infinitesimal generator of a C-semigroup formula_66 and formula_67 is continuously differentiable, then the function is the unique solution to the (abstract ) nonhomogeneous Cauchy problem. The integral on the right-hand side as to be intended as a Bochner integral. Time-dependent problem. The problem of finding a solution to the initial value problem where the unknown is a function formula_70, formula_71 is given and, for each formula_72, formula_73 is a given, closed, linear operator in formula_3 with domain formula_75, independent of formula_23 and dense in formula_3, is called time-dependent Cauchy problem. An operator valued function formula_78 with values in formula_79 (the space of all bounded linear operators from formula_3 to formula_3), defined and strongly continuous jointly in formula_82 for formula_83 , is called a fundamental solution of the time-dependent problem if: formula_94 is also called evolution operator, propagator, solution operator or Green's function. A function formula_70 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 formula_78. In practically all cases considered in the literature formula_98 is assumed to be the infinitesimal generator of a C-semigroup on formula_3. Roughly speaking, if formula_98 is the infinitesimal generator of a contraction semigroup the equation is said to be of "hyperbolic type ; if formula_98 is the infinitesimal generator of an analytic semigroup the equation is said to be of parabolic type". Non linear problem. The problem of finding a solution to either where formula_103 is given, or where formula_7 is a nonlinear operator with domain formula_106, is called nonlinear Cauchy problem.