Why does Toyota Corolla accelerate 1997 Caja Manua

My car accelerates after about 15 min of travel, I have to regulate the speed braking and the engine sounds as if it were to cut ... after a while the acceleration is normalized. If I turn off the engine when it is accelerated, it no longer turns on or starts again.
If I can't turn it on, I need someone to push the car and turn on accelerated or also after a few hours or one day, I turn on with difficulty but turn on. It has been in the mechanic thousands of times but cannot detect the fault, what does a car on?
Thank you.