Could be the starter solenoid or a loose or corroded battery or starter cable.
Does it shake roughly before cutting back off
There is a signal wire connector on the solenoid that is mounted on the starter.If you cut that wire and crimp on a butt connector then bolt it directly to the solenoid you may never have that problem again.Very common with theses cars
Looking for a Used Grand Marquis in your area?
CarGurus has 1,217 nationwide Grand Marquis listings starting at $1,900.
Search Mercury Grand Marquis Questions
Mercury Grand Marquis Experts