In $\mathbb{R}^3$, $u=u(r,t)$, $t$ for time, $r$ for $\sqrt{x^2+y^2+z^2}$ on (0, +\infty)
The equations are given as $\left\{\begin{array}{l}\partial_t u=\Delta u\\ u(R,t) = c\\ u(r,0) = c\\ u_r(r_0,t)=f(t) \end{array} \right.$
in which $c,0<r_0<R$ are all constants.
Since the equations are not homogeneous, I failed to apply method of separation of variables, the heat source is also not on the boundary. I don't know which method works now, could someone help me?