97 Corolla Radiator Problem

Hi I am having a problem with my car the hose that runs from the radiator back to the car the bottom hose always seems to be cold for example if I drive to work 16miles it is stone cold it only seems to warm up if you drive the car for a while then let the car sit and after about 5 mins or so it warms up to the approx temp of the top hose.
Is this normal or do I have a problem and do you have any suggestions?
Thanks

well the bottom hose is where the COOLED water flows out of the radiator... so it comes in from the top and is cooled in the rad then goes out to the engine by the bottom hose... this is totally normal.

Deadring