How do I fix a Samsung LN-S3241D where the brightness fades to black after 20 minutes of use

Back - light problem. Check and replace damaged component/s at its back-light inverter board, or replace this board as card basis. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts. It will be best to replace the inverter board as card basis.
http://electronicshelponline.blogspot.com/
In some models of LCD TVS; the backlight inverter and its T'Con board will be a part of the panel itself, and will be a built-in type inside the LCD panel. If so, they can't be replaced. In this case; a panel replacement is the only repair. For some other models, the backlight inverter section circuit will be a part of the main power supply regulator [SMPS]. To such sets, the power supply section circuit too should trouble shoot. Contact any service technician. The circuit will differ with models and versions.