What are the temps? Run this app to see: [https://www.tunabellysoftware.com/tgpro/|TG Pro]
Did you run diagnostics?
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=== Update (10/07/2017) ===
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Sadly I'm thinking the logic board has a cold solder joint so as the board heats up the connection is broken.
=== Update (11/04/2017) ===
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The Mac mini 2011 had two versions, the [https://everymac.com/systems/apple/mac_mini/specs/mac-mini-core-i5-2.3-mid-2011-specs.html|Mac mini 2.3 GHz i5 (Mid-2011)] which uses the integrated Intel HD Graphics 3000 services. Whereas [https://everymac.com/systems/apple/mac_mini/specs/mac-mini-core-i5-2.5-mid-2011-specs.html|Mac mini 2.5 GHz i5 & 2.7 i7 (Mid-2011)] which uses a dedicated GPU Radeon HD 6630M.
The GPU chip maybe failing and as its a hybrid chip (has the PCH in it) this could explain things. But without knowing which version of Mac Mini you have its just conjecture at this point.
What are the temps? Run this app to see: [https://www.tunabellysoftware.com/tgpro/|TG Pro]
Did you run diagnostics?
=== Update (10/07/2017) ===
Sadly I'm thinking the logic board has a cold solder joint so as the board heats up the connection is broken.
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=== Update (11/04/2017) ===
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The Mac mini 2011 had two versions, the [https://everymac.com/systems/apple/mac_mini/specs/mac-mini-core-i5-2.3-mid-2011-specs.html|Mac mini 2.3 GHz i5 (Mid-2011)] which uses the integrated Intel HD Graphics 3000 services. Whereas [https://everymac.com/systems/apple/mac_mini/specs/mac-mini-core-i5-2.5-mid-2011-specs.html|Mac mini 2.5 GHz i5 & 2.7 i7 (Mid-2011)] which uses a dedicated GPU Radeon HD 6630M.
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The GPU chip maybe failing and as its a hybrid chip (has the PCH in it) this could explain things. But without knowing which version of Mac Mini you have its just conjecture at this point.