Your system is shutting down because the processor is most likely getting too hot without the cooling provided by the fan. It sounds like the fan may have died or be in the process of doing so. A replacement fan can be purchased, but we'd need to know your system specs (particularly the motherboard and processor type) before a recommendation can be made on what fan to purchase.
Since the computer is only 8 months old, you might also check if the CPU fan is covered under warranty as well. Then you might be able to get it replaced for free.
If it comes down to you having to pay for and replace the fan, make sure you get a decent fan. Remember that in most cases when purchasing cheap computer components... you get what you paid for.