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Hi Im David and thanks for visiting the site. Has any shop scanned the vehicle for fault codes? if so what are the codes and what did they do to attempt to fix them? Usually when a vehicle is acting this way it is due to an ignition component thermal breakdown failure. runs great, reaches a certain temp, shuts down, cools off starts back up.... rinse and repeat... without knowing any possible codes the most likely suspects are the crankshaft position sensor, the ignition control module and the ignition coil.
the P0420 code is being generated by the rear oxygen sensor. that would not cause the condition you are reporting. to get rid of that code 50% of the time would be to replace that sensor. but before doing so, i would do a back pressure test of the exhaust system to see if the cat is clogged. if so then replacement of the cat is mandatory.
to see if the crankshaft position sensor is at fault, this is an unusual but effective way of finding out. put some ice in a sandwich baggie and take it with you while driving the car. when the stall condition occurs take the bag of ice and hold it to the sensor for 30 seconds. then start your car. if it starts right away, replace the sensor. unless you just want to replace the sensor which i wouldnt do unless i knew it was the cause.
checking in to see how its going
absolutely. if the fuel pump has some years on it, it could overheat and start to malfunction until cooled back down. its one of many possibilities that would be hard to diagnose being that its inside the tank. i am gald you have it figured it. now for the 0420, like i said its probably the rear 02 sensor, back a back pressure test is needed to confirm. all i can say is that at my shop i charge 60.00 for one. but prices vary a lot from shop to shop.
are you overheating at any times?
are you smelling coolant?
have a block test do to the radiator to see if any hydrocarbons(engine gases) are making its way into the coolant.