I guess that would depend on the level of the AI - for instance, no matter how much good press any existing model gets, if nuclear war broke out, and all the humans died, eventually infrastructure would fail (2-6 weeks, I'd guess - just basics like the remaining power plants and water providers being taxed without maintenance), and the AI goes dark.
If the AI had progressed to the point where it's controlling manufacturing, and there are no core professions that it relies on which can't be automated, then it could definitely outlive it's creators by far, but would face some of the same crises: global warming, resource scarcity, and, ultimately, final fates like having your planet swallowed by the system's host sun (like us). The same challenges that uphold the fermi paradox would apply there too, and perhaps we're too early in the development of the universe for the millions of years required for an artificial super-intelligence to have gotten there yet.
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