Let suppose that I have a egg at $T=20ºC$ and I assume it's almost an sphere of radius $R$, let's call it surface $S$. I put the egg inside a bath with water at $T_w=100ºC$. I want to know the temperature of the egg as a function of time and position. I already know that have to solve diffusion equation for the heat in spherical coordinates
$$ \left( \frac{\partial}{\partial t}-\chi\cdot \nabla^2 \right) T(t,r,\theta,\varphi)=0 $$
The problem is that I'm not sure about the initial conditions, because when I put the egg inside the bath, in the boundary are at two temperatures, witch one should I use? $T(t=0,\vec{r} \in S)=20 ºC$, $T(t=0,\vec{r}\in S)=100 ºC$ or some superposition?
From the point of view of physics the egg surface initially have to be at $T(t=0,\vec{r} \in S)=20 ºC$, but I'm not sure from the point of view of mathematics and PDE theory