We investigate a non-supersymmetric $SO(10)times U(1)_{rm PQ}$ axion model in which the spontaneous breaking of $U(1)_{rm PQ}$ occurs after inflation, and the axion domain wall problem is resolved by employing the Lazarides-Shafi mechanism. This requires the introduction of two fermion 10-plets, such that the surviving discrete symmetry from the explicit $U(1)_{rm PQ}$ breaking by QCD instantons is reduced from $Z_{12}$ to $Z_4$, where $Z_4$ coincides with the center of $SO(10)$ (more precisely $Spin(10)$). An unbroken $Z_2$ subgroup of $Z_4$ yields intermediate scale topologically stable strings, as well as a stable electroweak doublet non-thermal dark matter candidate from the fermion 10-plets with mass comparable to or somewhat smaller than the axion decay constant $f_{rm a}$. We present an explicit realization with inflation taken into account and which also incorporates non-thermal leptogenesis. The fermion dark matter mass lies in the $3times 10^{8}-10^{10}~{rm GeV}$ range and its contribution to the relic dark matter abundance can be comparable to that from the axion.
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