cooling function of the electrons/positrons which takes into account synchrotron emissions and inverse Compton scatterings during the propagation, and $Q(t,\varepsilon_e,r)$ is the injection term. Here we adopt $D_0=5.8\times 10^{28}{\rm cm}^2{\rm s}^{-1}$, $\delta=1/3$, which is consistent with the boron-to-carbon ratio according to the latest GALPROP code. Atoyan et al. (1995) showed a solution in the case of an instantaneous injection from a single point-like source, i.e. $Q(t, \varepsilon_e,r)  \approx Q_0(\varepsilon_e) \delta(t-t_i)\delta(r)$. Then the observed spectrum $G(t,r,\varepsilon_e; \tilde{t})$ would be $$G(t,r,\varepsilon_e; \tilde{t})=\frac{Q_0(\varepsilon_{e,0})P(\varepsilon_{e,0})}{\pi ^{3/2} P(\varepsilon_e)d_{\rm diff}(\varepsilon_e, \varepsilon_{e,0})^3}\exp \left(-\frac{r^2}{d_{\rm diff}(\varepsilon_e, \varepsilon_{e,0})^2} \right),$$ where $\varepsilon_{e,0}$ is the energy of electrons/positrons at the time $\tilde{t} (<t)$ and which are cooled down to $\varepsilon_e$ at the time $t$, and $d_{\rm diff}$ is the diffusion length given by $$d_{\rm diff}=2\left[ \int_{\varepsilon_e}^{\varepsilon_{e,0}} \frac{D(x)dx}{P(x)} \right]^{-1/2}. \label{d_diff}$$

The cooling function $P(\varepsilon_e)$ can be described as $$P(\varepsilon_e)=\frac{4\sigma_T \varepsilon_e^2}{3m_e^2 c^3} \left[ \frac{B^2}{8\pi} + \int d\varepsilon_{\gamma} u_{\rm tot} (\varepsilon_{\gamma}) f_{\rm KN} \left( \frac{4\varepsilon_e \varepsilon_{\gamma}}{m_e^2 c^4} \right) \right]$$ where $u_{\rm tot}(\varepsilon_{\gamma})d\varepsilon_{\gamma}$ is the energy density of interstellar radiation fields with the photon energy between $\varepsilon_{\gamma}$ and $\varepsilon_{\gamma}+d\varepsilon_{\gamma}$ (including cosmic microwave background, starlight, and dust emission; Porter et al. 2008), and $B$ is the interstellar magnetic field with we here set as $1\mu {\rm G}$. Here 