Differential equations in Banach spaces

Summary

A differential equation in a Banach space 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 of operators.