Hard to tell what may be happening. Would suspect a heat related problem - gets hot, something expands, shorts out, and off it goes.
Are you using the remote or directly turning on the TV? Remote could be malfunctioning - sends on followed by immediate off. Have you tried new batteries.
Do you have the User Manual? If not, here:
http/www.manualslib.com/products/Rca-14f514t-20539.html
I scanned through it and did not really note anything directly addressing the problem you are experiencing.
Without knowing where you live I will still risk the comment that diagnosis and repair is likely to cost more than a new set. Unless the problem is some replaceable and easily obtained component. And you must also have an honest repair shop.....
Just saw an LG 24" flat screen listed at $120 USD at Best Buy. (Not promoting LG or Best Buy - only that a new purchase may be a better option for a few more dollars than the cost of repair.)