Not knowing the whole story but... From installing car audio for 10 yrs I would start with a new decent battery. Also check the alternator.
When the cars running you should use a meter on your battery, you should get 14.4v or so. When the fan turns on it will drop but it shouldn't drop below 12v. Most car stereos (unless you have a power hog) will cut out under 11.5v or so, this using your car stereo as a make shift volt meter and provided it is installed properly.
Also a temporary fix may be to increase the idol speed a little. I would still try a decent sized new battery. Don't cheap out get a tiny wallyworld battery.
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