I was wondering what happens if I want to solve the heat equation in a sphere surface, neglecting its thickness. I have one initial condition for T(t=0), in particular this initial profile can depend on azimuth and zenith angles, it is not uniform. Perhaps I have saying something stupid but I think for large times the temperature would be uniform in all over the surface. The question is I have not any boundary condition, except those of angular periodicity. Or do you think the solution is precisely the initial profile?.