Solving the heat equation around a point of constant temperature

Question

I have a heterogeneous medium, of temperature $T=0$. Inside this medium (let's say, at the origin) is a small (let's say infinitesimal) source of thermal energy, which is kept at a constant temperature of $T=1$. How does the medium warm up over time?

Assuming spherical symmetry, we have the heat equation (ignoring constants) in spherical co-ordinates: $$\frac{\partial{q}}{\partial{t}} = \nabla^2 q = \frac{\partial{q}^2}{\partial{r}^2} + \frac{2}{r} \frac{\partial{q}}{\partial{r}},$$ with "boundary" and initial conditions $$q(r=0, t) = 1,\quad q(r\neq0, t=0) = 0.$$

I expect the solution to be roughly shaped like $e^{-r/t}$, but that's not quite correct. Separation of variables won't work. [If it did, then the boundary condition $q(r=0, t) = 1$ implies that there is no time dependence.] Nor do I know how to deal with the discontinuity at $t=1$ (at which time I imagine the source would need to provide an infinite amount of power to stay at constant temperature).

Other solutions that I've found across the internet tend to solve slightly different problems: the point source is instantaneous rather than ever present, or the source outputs a constant power rather than be kept at constant temperature.

Answer 1

The temperature held at one point in a otherwise uniform temperature field within a medium does not have enough oomph to propagate into the medium. Consider the steady state solution for a tiny spherical surface within an infinite medium $$r^2\frac{dT}{dr}=C_1$$$$T=-\frac{C_1}{r}+C_2$$$$\frac{T-T_{\infty}}{T_0-T_{\infty}}=-\frac{r_0}{r}$$$$q(r_0)=k\frac{(T_0-T_{\infty})}{r_0}$$So, the smaller $r_0$, the higher the heat flux required to maintain the surface of the spherical surface at $T_0$.