---
tag: JN7362WCD8
kind: concept
name: Differential equations in Banach spaces
msc: [34G10, 34G20, 47D06]
sources:
  - url: https://en.wikipedia.org/wiki/Abstract_differential_equation
same_as:
  wikidata: Q55603110
  wikipedia: Abstract differential equation
---

## Summary

A differential equation in a [[Banach space|06GZ956657]] has an unknown $u(t)$ that is a vector of an infinite-dimensional space, such as a space of functions.
The basic example is the abstract Cauchy problem $u'(t) = A u(t)$, $u(0) = u_0$, with a linear operator $A$.
Many partial differential equations of evolution type, such as the heat equation, take this form when the space variables are absorbed into the state space.
The linear theory rests on [[strongly continuous semigroups|EVF180QJXY]] of operators.

## Explanations

### L1 human

The heat in a rod is a whole temperature profile, a function of position.
As time passes, the profile changes, and its rate of change is given by its second derivative in space.
So one can view the heat equation as an ordinary differential equation for one moving point in a space of functions.
When $A$ is a bounded operator, the solution is $u(t) = e^{tA} u_0$, given by the exponential series, just as for matrices.
For the heat equation $A$ is unbounded, and this is where the theory becomes deeper than for ordinary differential equations.