  Conventionally, space-charge (SC) limited current density is defined as the
maximal cur- rent density allowed to traverse a diode under a DC voltage when a
time-invariant flow is injected from the cathode. In this work, we study the SC
limited current density under a time-varying injection for both classical and
relativistic regimes and determine the maxi- mal amount of limited current
density under certain conditions. Our simulations show that it is unlikely that
a time-varying injection emitted from cathode exceeds the known SC limits, in
either classical or relativistic regime.
