1999 no heat?

My son has a 1999 galant. The heat will only blow warm for a couple seconds,then goes cold or lukewarm. I replaced the thermostat and that didn't help at all. Would it be the heater core? I noticed one hose is hot and the other is barely warm on the heater core after the car has been running for a long time. Would it help to flush the heater core? How hard would that be? the heater core isn't leaking but I guess it still could be not working properly? This is my son's first car. He loves it besides the fact it's getting cold here and heat would be GREAT Thanks for any help!!