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Multi-band emission from radio to ultra-high energy gamma-rays in the CrabNebula has been detected. To explain the observed results, non-thermal photonproduction \textbf{in} the Crab Nebula is carefully studied in a spatiallydependent lepto-hadronic model. In our model, the dynamical evolution of thePWN is simulated in a spherically symmetric system. Both electrons and protonsare accelerated at the termination shock. The relevant particle propagationequations as well as the photon evolving equation are simultaneously solved.For the Crab Nebula, our results reveal that the observed multi-band photonspectra can be well reproduced with reasonable model parameters. In particular,the photons with energy $\gtrsim 200$ TeV are mainly contributed by thehadronic component via proton-proton interaction. The contribution of thehadronic component depends on both proton spectral index $\alpha_{\rm p}$ andnumber density $n_{\rm H}$ of medium within the PWN. Besides, high energyneutrino fluxes are predicted with variable proton spectral indices. Thepredicted fluxes are not only far below the sensitivities of current neutrinoobservatories, but also beneath the atmospheric neutrino background with energyless than $\sim 40$ TeV. Moreover, the calculated radial profiles of surfacebrightness and spectral index are presented.

Multi-band Emission up to PeV Energy from the Crab Nebula in a Spatially Dependent Lepto-hadronic Model