Time translation symmetry

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A PDE is said to have time translation symmetry if for any solution $u (t, x)$ the function $u (t + t 0, x)$ is also a solution for all (or some interval of) $t_0 \in \mathbb{R}$ . 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.

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