In this paper we present the results of time-dependent simulations of dipolar axisymmetric magnetospheres of neutron stars carried out both within the framework of relativistic magnetohydrodynamics and within the framework of resistive force-free electrodynamics. The results of force-free simulations reveal the inability of our numerical method to accommodate the equatorial current sheets of pulsar magnetospheres and raise a question mark about the robustness of this approach. On the other hand, the MHD approach allows to make a significant progress. We start with a nonrotating magnetically dominated dipolar magnetospheres and follow its evolution as the stellar rotation is switched on. We find that the time-dependent solution gradually approaches the steady state that is very close to the stationary solution of the Pulsar Equation found by Contopoulos et al.(1999). This result suggests that other stationary solutions that have the y-point located well inside the light cylinder are unstable. The role of the particle inertia and pressure on the structure and dynamics of MHD magnetospheres is studied in details, as well as the potential implications of the dissipative processes in the equatorial current sheet. We argue that pulsars may have differentially rotating magnetospheres which develop noticeable structural oscillations and that this may help to explain the nature of the sub-pulse phenomena. Key words: stars:pulsars:general – stars:winds and outflows – relativity – MHD
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