Winson,
Steve and Jay sound like they are on the right track.
May I recomend running this test. Run your processor so as to get
the message to display, then hold the reset line indefintely so the
uP has stopped dead in it's tracks. If the cause of the disappearing
display was due to noisy timing from the uP, then this test might
prove that. Then again, the problem doesn't happen right away...
Just for fun, add a .1uF cap to decouple the 5volts right on the LCD.
Also, please describe how you're connecting the LCD? Is it on
the Data bus or as Jay suspected directly to I/O pins.

My $.02 worth.
Josh