2003 manuel, idling/RPM problem

#1
Hello,

I bought a used Corolla 2003 which his in near perfect condition considering it's age but it as one weird problem.

When I start the car, it initially idles around 2000 rpm. After 5 minutes of running the car it goes progressively down to 800. Also, once it's reach around 800-1000, it sometimes gives short burst of RPM, roughly 200-300 more for a second or two max. The car is also hard to shift from N to 1 (get it going) once the RPM is down at 800. I counter this by giving it alot more gas but then again, it's not normal. What is going on here? o_O
 
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