Current density constant in time implies charge density linear in time?

[davethecello]

Does current density being constant in time and the charge density NOT being constant in time imply that the charge density has to be a linear function of time? i.e of the form
$$p(r,t) = p(r,0) + \dot{p}(r,0)t }$$
Can this be derived using the continuity equation alone?

 

[Meir Achuz]

Yes. For j to be constant in time requires dp/dt to be constant in time.