First, subtracting $T_e$ gives for $v=u-T_e$ a problem with homogeneous initial and boundary conditions. The eigenfunctions for the Laplace operator in polar coordinates are the Bessel functions of the first kind $J_0\left( \frac{\mu_n}R r \right)$, where $\mu_n$ is the $n$-th positive root of $J_0$. They are orthogonal on $[0,R]$ with the weight function $r$. Denote $ j_n(r)=\frac{J_0 \left( \frac{\mu_{n}}R x \right) }{\sqrt{\frac12} RJ_{1}(\mu_n)} $ the normalized eigenfunctions, $\int_0^Rrj_m(r)j_n(r)\,dr=\delta_{mn}$.
The corresponding Green function for the heat equation in the polar coordinates is $ G(r,r',t)=\sum_{n=1}^\infty e^{-a^2\frac{\mu_{n}^2}{R^2}t}j_n(r)j_n(r'). $ Solution $v$ can be written out as a volume potential: $ v(r,t)=\int_0^t\int_0^RG(r,r',t-\tau)r'e^{-r'^2}\,dr'd\tau. $ Expanding $e^{-r^2}$ into the Fourier-Bessel series $ e^{-r^2}=\sum_{n=1}^\infty c_nj_n(r),\quad r\in[0,R], $ where $ c_n=\int_0^Rrj_n(r)e^{-r^2}\,dr, $ we have $ v(r,t)= \int_0^t\int_0^Rr'\sum_{n=1}^\infty e^{-a^2\frac{\mu_{n}^2}{R^2}(t-\tau)}j_n(r)j_n(r')e^{-r'^2}\,dr'd\tau= $ $ \sum_{n=1}^\infty j_n(r)\left(\int_0^t e^{-a^2\frac{\mu_{n}^2}{R^2}(t-\tau)}d\tau\right) \left(\int_0^Rr'j_n(r')e^{-r'^2}\,dr'\right)= $ $ \frac {R^2}{a^2}\sum_{n=1}^\infty c_n\frac{1-e^{-a^2\frac{\mu_{n}^2}{R^2}t}}{\mu_n^2}j_n(r). $