Are we talking about a cheap Chinese 2-stroke?
It sounds to me like fuel starvation, a bad electrical connection in the harness, a bad mag, or a bad CDI.
If it is fuel starvation, for the first 60 seconds the engine is burning up all the available fuel in the carburetor, then it stumbles on it's face.
Check for good flow out of the fuel tank, a clogged fuel filter, or debris in the float needle valve.
Check the connections between the mag wires coming from the engine and the CDI box.
If you soldered them origionally you should be ok, however a cold solder joint can fail over time.
If you are still using the factory plug in connectors, cut them off and solder and heat shrink the connections together.
If it is a cold soldered problem, re-solder the connections only this time around, get the actual wires hot enough to melt the solder, don't just melt the solder around them.
If it is a bad mag, it is heat related. As it heats up the copper windings expand and short out internally.
It also could be a bad solder joint on the mag. They can get thermally sensitive too.
(once again, a cold soldered joint)
Check where the blue wire attaches to the coil. The solder should be relatively smooth and a dull gray color. If it has a wrinkly surface and a whitish color to it, it needs to be re-soldered.
Also check the black wire where it is soldered to the ring lug (a replaceable terminal with a bolt going through the center of it).
If it is the CDI, once again it is thermally related, you can only replace it with a known good one.
