Does nuclear fusion happen at earth's core?

No, it's not hot enough (6000°C) . But it is kept hot by decay of uranium and potassium isotopes. For fusion, even of hydrogen, you would need tens of millions of degrees as in the center of the Sun. The other thing to consider is that the core is iron which cannot undergo fusion, except during the explosion of a supernova. read more

Nuclear fission powers the movement of Earth's continents and crust, a consortium of physicists and other scientists is now reporting, confirming long-standing thinking on this topic. read more

The hottest part of the earth is the core, with some heat coming up through the crust in areas of volcanic activity, e.g. subduction zones. But we see no evidence of fission, and certainly not fusion. read more