TV won't stay on...

Either the capacitors in the power supply are bad, or the caps on the main board. You can test them with a capacitor tester or ohmmeter.
It is common, especially in newer devices.

Replacement Parts: Replace the original four (4) 1000µF, 16V, 85ºC capacitors on the DM Module with Radial Polarized, 1000µF, 35V, 105ºC Capacitors.
Description: Computer Grade Electrolytic Capacitors or High Temp Electrolytic Capacitors; Capacitance = 1000µF; Voltage = 35 V; Operating Temperature Range = - 20º C to + 105º C; Termination Style = Radial, Operating Hours = 10,000 Life Hours. The original DM capacitors are rated at 1000µF, 16V, 85ºC. Do not use capacitors with this rating to repair your Mitsubishi television, or else the same issue will reappear in 1-2 years.

The capacitors are available from several on-line electronics stores and possibly from a local television repair
GA_googleFillSlot("TL-IMU-Article"); shop. I recommend that you use ONLY high temp Capacitors rated at 105º C, with a voltage rating of 35 V. Refer to the "General Information" section below for additional information about purchasing the capacitors from a online website.