This shows a plate being heated by a circular heating element (e.g. from the perimeter). White is hotter than red.
The hottest part of the plate would be around the perimeter of the circle. The reason for this is simple - Newton's law of cooling. Assume that the heating element provides constant heat flux to the plate around the circle where the two bodies contact. The temperature of that circle will rise, and it will continue to rise as more heat moves into the plate from the element. At the same time, heat will move out of the circle to the rest of the plate (including the center area surrounded by the circle) because the circle is hotter than the rest of the plate. Newton's law of cooling implies that the rest of the plate will approach the temperature of the circle, but will never reach it, because the rate of heat flow decreases at the difference in temperature between the two parts of the plate decreases. Therefore, the temperature of the circle always exceeds the temperature of every other point on the plate.
There is no general analytic solution to transient 2D conduction problems. If someone has derived one for this specific case, I have not been able to find it. Generally, the finite difference method is used to solve this sort of problem.