Time translation symmetry

A PDE is said to have time translation symmetry if for any solution ${\displaystyle u(t,x)}$ the function ${\displaystyle u(t+t_{0},x)}$ is also a solution for all (or some interval of) ${\displaystyle t_{0}\in \mathbb {R} }$. Generally speaking, a PDE has time translation symmetry when the coefficients and data do not explicitly involve the time variable t.

A PDE which has time-translation symmetry is also called time-translation invariant or autonomous.

One can often use time-translation symmetry to bootstrap a local well-posedness result into a global well-posedness one, provided that one has some uniform control on the lifespan of the solution provided by the local well-posedness theory.