That Kenmore 70 Series gas dryer probably has a gas burner system like the once that is described in the image below. Since the burner initially lights, it is likely that you have weak gas valve coils (kit part 279834 fits most Kenmore 70 Series gas dryers). When the gas valve coils are weak, the will normally be strong enough to open the gas valves on the initial ignition from a cold start. When they are heated up after that initial ignition, they are too weak to open the gas valves on subsequent ignition attempts. Replacing those gas valve coils will probably fix your problem. I recommend that you reply with the model number of your dryer so that we can verify that gas valve coil kit 279834 will fit your dryer. Using that kit is much less expensive that buying the valve coils separately. You can try to verify that the kit 279834 will fit your dryer by looking up the coil parts for your dryer through the
website. The coils may substitute to that kit for your model of Kenmore 70 Series dryer.
Here are 2 videos that show how to replace those coils on the common Kenmore 70 Series dryers:
Whirlpool/ Kenmore Gas Dryer Gas Valve Solenoid Coil Kit on 29 inch models
Whirlpool/ Kenmore Gas Dryer Solenoid Kit on 27 inch models Replacement
This information should help you resolve this heating problem in your gas dryer. Reply with additional details and the full model number of you Kenmore 70 Series gas dryer if you need more help.
If you do not feel confident repairing this problem yourself, then you can have it repaired at your home by a Sears technician. Here is a link for the website:
Sears Home Services