If I heat up a ball, is the radial temperature gradient in the middle of it going to be zero in all directions? Assume I have a uniform ball, heat transfer is solely governed by the heat equation ($\dot{u} = \alpha \Delta u$). It has an initial temperature distribution solely dependent on $r$, ie $T=T(r)$, where $T(r)$.
If I start to heat my ball up from all directions uniformly, will the radial temperature gradient in the exact center be a zero?
I believe that the overall gradient, $\nabla T$ is going to be $\underline{0}$, because there is no preferential direction. That that does however mean that $\frac{\partial}{\partial r}T$ is also going to be $0$ in the middle (because it could be that $\frac{\partial}{\partial r}T$ is nonzero but constant, making the overall gradient vector zero by opposite directions cancelling each other). It would make my life much easier if $\frac{\partial}{\partial r}T$ was also $0$.
If it is zero, how do I argue for it being $0$? (and if it isn't, why it isn't?)
 A: The temperature through the ball will have zero slope in the middle.  It will not have a sharp peak or sharp minimum.  That means it has a gradient of zero in the middle.
A: If it was heated from all sides the temperature gradient in the middle would only vanish if gravity was zero and pressure uniform.
Assuming there is gravity and you have a ball filled with air then there is going to be convection of particles inside of it as some gets warmer fast (the ones close to the edges) and some takes longer (the ones in the middle). The warm air close to the edges will expand and move upward in the balloon due to pressure gradients inside and therefore the temperature inside won't be isotropic.
Now let's assume no gravity and make the pressure inside the ball uniform and then start heating the ball equally from all sides the temperature gradient will be zero as the middle of the ball will be a stationary point (temperature minimum) in the temperature profile in any cross section of the ball. You can prove this by using symmetry.
