  The question of global existence or non-existence of solution to a given
stochastic partial differential equation under some non-linear conditions
always comes to mind. To show that our weak-predictable random field solutions
do not have global existence for all time , it requires that we first establish
that the solutions exhibit continuity in time property. The results discuss the
mean-square and mean continuity in time of a class of jump-discontinuous heat
equations perturbed by compensated and non-compensated Poisson random noises
respectively; and we showed that our mild solutions are mean and mean-square
continuous in time for any time interval ; better put, our solutions have
continuous versions or modifications for any time interval.
