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]

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au+f}$

where the unknown function ${\displaystyle u=u(t)}$ belongs to some function space ${\displaystyle X}$, ${\displaystyle 0\leq t\leq T\leq \infty }$ and ${\displaystyle A:X\to X}$ is an operator (usually a linear operator) acting on this space. An exhaustive treatment of the homogeneous (${\displaystyle f=0}$) case with a constant operator is given by the theory of C₀-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 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.^[3]

The Cauchy problem

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au+f\quad u(0)=u_{0}\in D(A)}$

with ${\displaystyle f:[0,\infty )\to X}$, is called nonhomogeneous when ${\displaystyle f(t)\neq 0}$. The following theorem gives some sufficient conditions for the existence of the solution:

Theorem. If ${\displaystyle A}$ is an infinitesimal generator of a C₀-semigroup ${\displaystyle T(t)}$ and ${\displaystyle f}$ is continuously differentiable, then the function

${\displaystyle u(t)=T(t)u_{0}+\int _{0}^{t}T(t-s)f(s)\,ds,\quad t\geq 0}$

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.

The problem^[7] of finding a solution to the initial value problem

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=A(t)u+f\quad u(0)=u_{0}\in D(A),}$

where the unknown is a function ${\displaystyle u:[0,T]\to X}$, ${\displaystyle f:[0,T]\to X}$ is given and, for each ${\displaystyle t\in [0,T]}$, ${\displaystyle A(t)}$ is a given, closed, linear operator in ${\displaystyle X}$ with domain ${\displaystyle D[A(t)]=D}$, independent of ${\displaystyle t}$ and dense in ${\displaystyle X}$, is called time-dependent Cauchy problem.

An operator valued function ${\displaystyle U(t,\tau )}$ with values in ${\displaystyle B(X)}$ (the space of all bounded linear operators from ${\displaystyle X}$ to ${\displaystyle X}$), defined and strongly continuous jointly in ${\displaystyle t,\tau }$ for ${\displaystyle 0\leq \tau \leq t\leq T}$, is called a fundamental solution of the time-dependent problem if:

• the partial derivative ${\displaystyle {\frac {\mathrm {\delta } U(t,\tau )}{\mathrm {\delta } t}}}$ exists in the strong topology of ${\displaystyle X}$, belongs to ${\displaystyle B(X)}$ for ${\displaystyle 0\leq \tau \leq t\leq T}$, and is strongly continuous in ${\displaystyle t}$ for ${\displaystyle 0\leq \tau \leq t\leq T}$;
• the range of ${\displaystyle U(t,\tau )}$ is in ${\displaystyle D}$;
• ${\displaystyle {\frac {\mathrm {\delta } U(t,\tau )}{\mathrm {\delta } t}}+A(t)U(t,\tau )=0,\quad 0\leq \tau \leq t\leq T,}$ and
• ${\displaystyle U(\tau ,\tau )=I}$.

${\displaystyle U(\tau ,\tau )}$ is also called evolution operator, propagator, solution operator or Green's function.

A function ${\displaystyle u:[0,T]\to X}$ is called a mild solution of the time-dependent problem if it admits the integral representation

${\displaystyle u(t)=U(t,0)u_{0}+\int _{0}^{t}U(t,s)f(s)\,ds,\quad t\geq 0.}$

There are various known sufficient conditions for the existence of the evolution operator ${\displaystyle U(t,\tau )}$. In practically all cases considered in the literature ${\displaystyle -A(t)}$ is assumed to be the infinitesimal generator of a C₀-semigroup on ${\displaystyle X}$. Roughly speaking, if ${\displaystyle -A(t)}$ is the infinitesimal generator of a contraction semigroup the equation is said to be of hyperbolic type; if ${\displaystyle -A(t)}$ 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

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=f(t,u)\quad u(0)=u_{0}\in X}$

where ${\displaystyle f:[0,T]\times X\to X}$ is given, or

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=A(t)u\quad u(0)=u_{0}\in D(A)}$

where ${\displaystyle A}$ is a nonlinear operator with domain ${\displaystyle D(A)\in X}$, is called nonlinear Cauchy problem.

• Cauchy problem
• C0-semigroup