Trouble at start-up

Hello V8 community!

I'm asking for a remote diagnosis for the latest spinnery on my V8 3.6:
After a night in the garage, he jumps immediately at the cold start, but also goes out immediately after a second. Also in several attempts at starting the same game in a row. I can then force him to stay on with a slightly stepped accelerator pedal.
Then I drive with one foot on the gas and the other foot on the brake backwards from the garage, then about 300 meters up the road to the next intersection where I have to stop, and everything is fine. He doesn't go out anymore, and from then on, he's acting perfectly normal for the rest of the ride.
Only when he has been standing for a long time, the speed goes back into the basement after the start. Sometimes it goes out, sometimes the engine starts after a second or two and stays on. But the next morning, after a night in the garage, the same game from the front.
Does anyone have an idea?

Greetings to all those willing to help,
Jürgen