The hardening and softening features in the DAMPE proton spectrum are very likely to be originated from a nearby supernova remnant (SNR). The proton spectrum from the nearby SNR is required to be very hard below $approx10$ TeV. To reproduce this feature, we illustrate that anomalously slow-diffusion zone for cosmic rays (CRs) must be existed in the local interstellar medium (ISM) after also taking the dipole anisotropy of CRs into account. Assuming that the diffusion coefficient is homogeneous in the nearby ISM, we show that the diffusion coefficient is constrained to the magnitude of $10^{26}$ cm$^2$ s$^{-1}$ when normalized to 1 GeV, which is about 100 times smaller than the average value in the Galaxy. We further discuss the spatial distribution of the slow diffusion and find two distinct possibilities. In one case, the SNR is several hundred of parsecs away from the solar system, meanwhile both the SNR and the solar system are required to be included in a large slow-diffusion zone. The homogeneous diffusion belongs to this case. In the other case, the SNR is very close with a distance of $sim50$ pc and the slow-diffusion zone is only limited around the SNR. The required diffusion coefficient is further smaller in the latter case. This work provides a new way of studying the CR diffusion in the local ISM.
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