I want to determine the cooling of the area outside a disk of radius 1 with insulated boundary at the edge and vanishing temperature at infinity for an arbitrary but axisymmetric initial temperature distribution. Specifically I'm interested in the cooling rate, e.g. the time needed until the maximum of the temperature field reaches a specified threshold.
I want to solve the following axisymmetric 2D-heat-flow problem in polar coordinates: $$\partial_t u(r,t) = \frac{1}{r}\left(\partial_r (r \partial_r u(r,t))\right)$$ in $t\in [0,\infty)$, $r\in(0,\infty)$ with boundary conditions $$\partial_r u(1,t)=0$$ and $$\lim_{r\to\infty} u(r,t) = 0$$ and arbitrary initial conditions $$u(r,0)=f(r).$$
Is this problem well defined? Or does one need to shift the boundary condition to finite $r$?
Separation of variables $$u(r,t)=V(r)W(t)$$ leads to $$W(t)=\exp(-c^2 t)$$ and $$V(r)=a J_0(cr) + b Y_0(cr)$$ with Bessel functions $J$ and $Y$, where $c$ and $a$ or $b$ should be determined or limited to certain values by the boundary conditions.
The boundary condition at $r=1$ gives a connection between $a$ and $b$ depending on $c$. Since the boundary condition at $r\to\infty$ seems to be valid, all values of $c$ seem to be valid, leading to a continuous spectrum.
If the boundary condition at $r\to\infty$ was shifted to a finite $r$, the spectrum would be finite --- only for discrete values of $c$, this condition would be fulfilled. The smallest possible value would define the slowest cooling mode $~\exp(-c^2 t)$ which would be 'good enough' for me. For the problem above, there is no such slowest cooling mode if the spectrum is continuous.
This reminds me of Fourier series vs. Fourier transformation. And indeed there is the Fourier-Bessel series vs. Hankel transformation - but they start at r=0. I found a transform with Dirichlet boundary conditions at $r=1$ in chapter 9 “The Hankel Transform” in the book "The Transforms and Applications Handbook: Second Edition." by Robert Piessens (e.g. http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/transforms/Piessens2000.pdf), which is close to my problem with the difference that I have a Neumann boundary condition.
How would one proceed solving the above problem?