You may have power supply problem due to bad component or bad solder joints that expanded when warm up, or problem in the backlights inverter circuits, or bad T-CON board.
Try this when the screen black out.
First, look at the screen in darkened room to see if you can see any slight glow from the screen, if it does, then the problem can be the T-CON (not getting the 12vdc) or the main logic board not feeding the signals to the T-CON board.
if not, next place to check will be the power supply module, make sure it has 24vdc which is required by the backlight inverter to work, the reason for not having 24vdc can be from bad DC filter caps (leaking/bulging top, high ESR) or short circuits in the inverter board which will cause the 24vdc supply to go into shutdown mode, so you may have to disconnect the power cable that feed the inverter board to see if the 24vdc will work without the inverter board hookup, you will need to apply about 3~4 amp of load to the 24vdc to make sure it can handle that. There are fuses on the backlight inverter boards also. So there is some troubleshooting you will have to do to find out the causes of the problem you are having.

If you are going to DIY and have proper tools, electronics knowledge, and know safety precaution then please read on:

Basic LCD monitor and TV troubleshooting guide:
http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
http://www.fixya.com/support/r7406380-tv_training_manuals
Failed TV and Monitors pictures: http://s807.photobucket.com/albums/yy352/budm
Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague
Parts: www.shopjimmy.com
Capacitors kit: http://lcdalternatives.auctivacommerce.com/Default.aspx he can make you a set of caps.
Or www.digikey.com, just make sure to use caps with low ESR, 105c, high ripple current and long life rating such as PANASONIC FM or FC series.