---
tag: 3VBRS94V1G
kind: concept
name: Mild solution
msc: [47D06, 35K90, 34G10]
sources:
  - url: https://en.wikipedia.org/wiki/C0-semigroup
---

## Summary

A mild solution of a linear evolution equation $u' = A u + f$ in a [[Banach space|06GZ956657]] is a continuous function.
It is given by the variation of constants formula $u(t) = T(t) u_0 + \int_0^t T(t - s) f(s) \, ds$, where $T(t)$ is the [[C0-semigroup|EVF180QJXY]] generated by $A$.
It needs no differentiability in time and no assumption that $u_0$ lies in the domain of $A$.
Every classical solution is a mild solution, so mild solutions extend the notion of solution to rough data.
Many semilinear parabolic and hyperbolic equations are solved by finding a mild solution as a fixed point of this integral formula.

## Explanations

### L1 human

For a scalar equation $u' = a u + f(t)$, the solution is $u(t) = e^{at} u_0 + \int_0^t e^{a(t - s)} f(s) \, ds$.
A partial differential equation such as the heat equation can be seen as such an equation, with $a$ replaced by an operator and $e^{ta}$ by a semigroup of operators.
The same formula then makes sense even when the initial state is too rough to be differentiated.
The function it defines is called the mild solution.