Kenmore gas range model3627552800 or 72321, lower oven burner will come on for about 10 sceconds and will then ficker and go out

There is a burner control switch, which is part #12 on this webpage:
http://beta.searspartsdirect.com/partsdirect/part-model/Kenmore-Parts/Range-Parts/Model-36275528000/0582/0124002/S0080001/00001?blt=06&prst=&shdMod=
Here are diagrams of every part in your oven:
http://beta.searspartsdirect.com/partsdirect/part-model/Kenmore-Parts/Range-Parts/Model-36275528000/0582/0124002?searchedModel=3627552800&blt=06
Part #15 or 16 could be faulty, also #214...check those out. I'm not an oven tech, so can't say how to test these parts, but if you call Sears Service Center, they might know what part might be faulty just by your description of your problem. Worth a shot, anyway.
Anyway, hope this help, and good luck.

These ovens have a safety valve control which shuts off the gas supply if the oven burner flame is lost. Usually the ignitor has this sensor and should continue to flash until the burner flame is warm enough. It appears the ignitor is not detecting this and shuts of the gas.

See the parts on this site. But if you have to buy parts it is cheaper at this site.