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The condensed formulation of a [[Cauchy problem|Cauchy problem]] (as phrased by J. Hadamard) in an infinite-dimensional [[Topological vector space|topological vector space]]. While it seems to have arisen between the two World Wars (F. Browder in [[#References|[a2]]], Foreword), it was apparently introduced as such by E. Hille in 1952, [[#References|[a2]]], Sec. 1.7.

Narrowly, but loosely speaking, the abstract Cauchy problem consists in solving a linear abstract differential equation (cf. also [[Differential equation, abstract|Differential equation, abstract]]) or abstract [[Evolution equation|evolution equation]] subject to an initial condition. More precise explanations slightly differ from textbook to textbook [[#References|[a2]]], [[#References|[a5]]]. Following A. Pazy [[#References|[a5]]], given a [[Linear operator|linear operator]] $A$ on a [[Banach space|Banach space]] $X$ with domain $D ( A )$ and given an element $x _ { 0 } \in X$, one tries to solve

\begin{equation*} x ^ { \prime } ( t ) = A x ( t ) , t > 0 ; \quad x ( 0 ) = x 0, \end{equation*}

i.e., one looks for a [[Continuous function|continuous function]] $x$ on $[ 0 , \infty )$ such that $x$ is differentiable on $( 0 , \infty )$, $x ( t ) \in D ( A )$ for all $t > 0$, and $( d / d t ) x ( t ) = A x ( t )$ for all $t \in ( 0 , \infty )$.

Since $x$ is required to be continuous at $0$, the Cauchy problem can only be solved for $x _ { 0 } \in \overline { D ( A ) }$.

A Cauchy problem is called correctly set if the solution $x$ is uniquely determined by the initial datum $x _ { 0 }$. It is called well-posed (properly posed) if, in addition, the solution $x$ depends continuously on the initial datum $x _ { 0 }$, i.e., for every $\tau > 0$ there exists some constant $c > 0$ (independent of $x _ { 0 }$) such that

\begin{equation*} \| x ( t ) \| \leq c \| x _ { 0 } \| \text { for all } \, t \in [ 0 , \tau ], \end{equation*}

and all $x _ { 0 }$ for which a solution exists. Sometimes it is also required that solutions exist for a subspace of initial data which is large enough in an appropriate sense, e.g., dense in $X$.

The notion of a Cauchy problem can be extended to non-autonomous evolution equations [[#References|[a2]]], [[#References|[a5]]] and to semi-linear [[#References|[a5]]], quasi-linear [[#References|[a5]]], or fully non-linear evolution equations [[#References|[a1]]], [[#References|[a4]]]. In this process it may become necessary to replace classical solutions by more general solution concepts (mild solutions [[#References|[a1]]], limit solutions [[#References|[a4]]], integral solutions (in the sense of Ph. Bénilan; [[#References|[a4]]]) in order to keep the problem meaningful. See [[#References|[a1]]] and the references therein.

Well-posedness of linear Cauchy problems is intimately linked to the existence of $C _ { 0 }$-semi-groups of linear operators (cf. also [[Semi-group of operators|Semi-group of operators]]), strongly continuous evolution families [[#References|[a2]]], [[#References|[a5]]] and related more general concepts like distribution semi-groups, integrated semi-groups, convoluted semi-groups, and regularized semi-groups, while the well-posedness of non-linear Cauchy problems is linked to the existence of non-linear semi-groups (the Crandall–Liggett theorem and its extensions) or (semi-) dynamical systems [[#References|[a1]]], [[#References|[a4]]], and to (evolutionary) processes and skew product flows [[#References|[a3]]].

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  P. Benilan,   P. Wittbold,   "Nonlinear evolution equations in Banach spaces: Basic results and open problems"  K.D. Bierstedt (ed.)  A. Pietsch (ed.)  W.M. Ruess (ed.)  D. Vogt (ed.) , ''Functional Analysis'' , ''Lecture Notes Pure Appl. Math.'' , '''150''' , M. Dekker  (1994)  pp. 1–32</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H.O. Fattorini,   "The Cauchy problem" , Addison-Wesley  (1983)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.K. Hale,   "Asymptotic behavior of dissipative systems" , Amer. Math. Soc.  (1988)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  V. Lakshmikantham,   S. Leela,   "Nonlinear differential equations in abstract spaces" , Pergamon  (1981)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Pazy,   "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</td></tr>
</table>