How do I use this interface condition?

Question

I am solving heat equation that is having a jump condition. Precisely, $$ u_t -\alpha\nabla^2 u = f(t)1_{r \leq R}(r) $$ for $r \geq 0$, and the condition is $$ u(t, R^-) = u(t, R^+) - ku_r(t, R^+) $$ where $k > 0$. I don't know how to use this. And of course $u(t, \infty) = 0$ and $u(0, r) = 0$. Should I try separation of variable, or green's function? Without the condition, I can solve this directly using the general solution of the heat equation and calculate the convolution. But with that jump condition, there will be a discontinuity at $r = R$. How do I solve it? I am trying to make the boundary into two, so we have $u(t, R^-) = a(t)$ and $u(t, R^+) = b(t)$ then I can use these as boundary condition to solve the heat equation in different region then trying to match the solution.