$\rho\sim 10^{-24}{\rm g}\ {\rm cm}^{-3}$, the pressure in ISM is $p\sim 10^{-13} {\rm dyn}\ {\rm cm}^{-2}$. Solving above equations, the outer shock decays at about $$t_{\rm dec}\sim 10^6 \left(\frac{B_0}{10^{8.5}{\rm G}}\right)\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2\left(\frac{R}{10^{6}{\rm cm}}\right)^3 \left(\frac{T}{10^{3}{\rm K}}\right)^{-5/4} {\rm yr}, 
\label{eq:2.3}$$ where $T$ is the temperature of ISM. The lifetime of a MSP $\tau$ can be estimated as $$\tau=\frac{E_{\rm rot}}{L_{\rm sd}}, 
\label{eq:2.4}$$ where $E_{\rm rot}$ is the rotation energy described as $$E_{\rm rot}\sim 10^{52}\left(\frac{M}{1.4M_{\odot}}\right)\left(\frac{R}{10^6{\rm cm}}\right)^2\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^2 {\rm erg},
\label{eq:2.5}$$ where $M$ is the mass of a MSP. For the fiducial parameters of MSP $$\tau\sim 5\times 10^{10} \left(\frac{M}{1.4M_{\odot}}\right)\left(\frac{B_0}{10^{8.5}{\rm G}}\right)^{-2}\left(\frac{\Omega}{10^3{\rm s^{-1}}}\right)^{-2}\left(\frac{R}{10^{6}{\rm cm}}\right)^{-4} {\rm yr}.
\label{eq:2.7}$$ We find that the outer shock decays at a very early stage of the lifetime of MSP and does not expand after $t > t_{\rm dec}$. Therefore, in the similar to the case of white dwarf pulsars (KIK11), the adiabatic cooling shall give minor contributions to the cooling process of the high-energy electrons/positrons.

Also for the radiative cooling by the synchrotron radiation in the shocked region $R_{\rm in} < r < R_{\rm out}$, we follow the discussion in KIK11 based on the diffusion in the shocked region. We take the Bohm limit, where the fluctuation of the magnetic field is comparable 