Tachyon condensation is closely related to second-order phase transitions.
Therefore, information still does not propagate faster than light,[1] and solutions grow exponentially, but not superluminally (there is no violation of causality).
A very small impulse (which will always happen due to quantum fluctuations) will lead the field to roll down with exponentially increasing amplitudes toward the local minimum.
[2] In the late 1990s, Ashoke Sen conjectured[3] that the tachyons carried by open strings attached to D-branes in string theory reflect the instability of the D-branes with respect to their complete annihilation.
The character of closed-string tachyon condensation is more subtle, though the first steps towards our understanding of their fate have been made by Adams, Polchinski, and Silverstein, in the case of twisted closed string tachyons, and by Simeon Hellerman and Ian Swanson, in a wider array of cases.