Where is the thermostat on a 2003 mitsubishi galant my heater is not working but my ac is and car is not overheating and heater core hoses are cold never warm so i think it maybe the thermostat does this sound correct.

Some cars/trucks have a heater control valve that cuts off the circulation from the heater core inside the car, look for the 2 rubber hose going into the firewall, follow them and check for an inline valve. If not your coolant might be low or a stuck thermostat. Make sure the water pump is working. I just replaced one on a camry, it was doing pretty much the same thing.

Radiator is either clogged,or it's bad. You could also check the heater core.

There is probably some kind of damper that opens and closes under the dash that allows the air from the blower to be blow across the heater core under the dash. Sometimes these are controlled electronically or by vacuum tubes. Look underneath the dash and look for small vacuum hoses that might be dangling loose or possible little wire cables that might have come off the sides.
Possibly check Haynes Manuals (www.haynes.com) to learn how to change a Heater Control Valve.

heater core

you need a new heater core

Not likely.. Check the bolts at the thermostat... most likely didn't get them tight enough..tip of you replace a thermostat always buy a fail open thermostat .. Fail close ones kill more cars then you would believe.

I would suggest double checking the heater core connection, make sure every thing is connected properly. check your t-stat make sure its not stuck open. Do so by feeling the upper hose, it should be a warm-hot feel, if its cool or cold replacing the t-stat would fix that problem.

plugged heater core need to flush it out

try to back flush the heater core. and if that dont work check the blend door it see if it is working.

A thermostat that is stuck open will cause there to be no heat. If its closed is when you will run hot or overheat.