Local mild solution of a semilinear evolution equation

A local mild solution is a fixed point of the nonlinear variation-of-constants map in $C([0,T];X)$. Local Lipschitz continuity of the nonlinearity gives existence and uniqueness for sufficiently small $T$.

For a locally Lipschitz semilinear evolution, a maximal mild solution either exists for all positive time or its phase-space norm becomes unbounded as the finite maximal time is approached.