Your heater core is plugged and you need a thermostat.
Check your heater hoses, that's where the water ciruclates into the heater from the water pump.
one large hose from bottom of radiator to side of water pump. other large hose from thermostat housing to top of radiator. one heater hose from water pump to heater core, I think this one has a Y in it if there is a rear heater other heater hose from heater core to somewhere on top the engine. If there is a rear heater, its return hose goes to a fitting on the radiator.
Upper and lower radiator, inlet and outlet heater core, bypass,Upper and lower radiator, inlet and outlet heater core, bypass,
the heater core has got a hole in it or your top has a hole init. If the water is green it is your heater core. if not it is your top.
With no water no heat can be transferred through the heater coils.
It could be the thermostat. Open up the radiator cap when it is hot and see if the water is flowing in the radiator. If it is not then it may be a thermostat.
Low water in the radiator will result in no heat. The water level is not sufficient to reach the heater core. The heater core will become cold.
The heater core can be flushed out and hopefully be unclogged. A water hose can be hooked to the heater core hoses, and water flushed through to clean the core out.
if you are talking about a radiator then yes it is supposed to stay on.
The heat for the interior of your car comes from the heated radiator coolant. When you are driving down the road the water pump circulates the heated engine coolant. If the engine slows and the water pump isn't real efficient you might get cooling in the heater. When you turn off the engine you aren't getting warm water pumped into the heater.
The port on top of the water pump (or on the side in some applications) is the coolant intake for the pump which coincides with the lower radiator hose. some people hook a heater hose to the intake and to the pump but doing this you are recirculating hot water from the heater core back into the engine as the water pump pushes water from front to back coming through the intake and out the upper radiator hose. so in this instance already hot water is going to the heater core and straight back to the engine without being circulated through the radiator also your water pump is essentially getting water from two sources. i normally plug the water pump and run a hose to the radiator and to the intake. there isn't anything wrong either way. one hose from intake to heater core and the second hose from the heater core to either water pump or radiator. one will take longer for the heater core to heat up but will ensure on a hot summer day all the coolant in the system is being cooled. with pump to intake setup the heater will work faster on a cold day but you may have cooling or overheating issues on a hot summer day.
The heater core is a small radiator inside the car under the dash that hot water circulates through to heat the inside of the car.