Local well-posedness of an evolution problem

An evolution problem is locally well posed near a datum $u_0$ in a data space $X$ if there are a time $T>t_0$, a neighborhood $V\subseteq X$ of $u_0$, and a specified solution space $Y_T$ such that:

1. Every datum in $V$ has a solution on $[t_0,T]$.
2. The solution is unique in the asserted class.
3. The map from the datum to its solution in $Y_T$ is continuous.

The equation, coefficients, and forcing are held fixed unless the data space explicitly includes them.