\[k \nabla^2 U =\frac{\partial U}{\partial t}\]
becomes \[\frac{\partial}{\partial r}(r^2 \frac{\partial U}{\partial r}) + \frac{1}{r^2 sin \theta} \frac{\partial}{\partial \theta} (sin \theta \frac{\partial U}{\partial \theta}) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 U}{\partial \phi^2} = \frac{1}{k} \frac{\partial U}{\partial t} \]
If
\[U=U(\theta , \phi) \]
then \[\frac{\partial U}{\partial r} = \frac{\partial U}{\partial t} =0\]
The above equation simplifies to
\[ \frac{\partial}{\partial \theta} (sin \theta \frac{\partial U}{\partial \theta}) + \frac{1}{ sin \theta} \frac{\partial^2 U}{\partial \phi^2} = 0 \]
If
\[U=U(r ,t) \]
then \[\frac{\partial U}{\partial \theta} = \frac{\partial U}{\partial \phi} =0\]
The equation simplifies to
\[\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial U}{\partial r}) = \frac{1}{k} \frac{\partial U}{\partial t} \]