The exact same thing happens to my new Lilliput (I got a new one since the touchscreen was broken on the old one).
If iI just leave it on for 30minutes or more the picture will get all weird (too dark, too bright, interlaced etc.) sometimes it just poweres off. Then as you said if I power it on again it will work for a few minutes before breaking down again.
I have tried this on 3 different computers with different hardware and software configurations, so I´m fairly sure that the Lilliput is to blaim. Also, my "old" one did not have this problem at all.
I´m have RMA´d this new monitor and will send it back for (yet another) replacement.