Mild solution

Summary

A mild solution of a linear evolution equation $u' = A u + f$ in a Banach space 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 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.