In the Lorenz gauge, the Maxwell equations reduce to four inhomogenous wave equations, with the charge density acting as the source for V, and the current density for A. For now, just take a static charge distribution -- say, a point charge at the origin. It is well known that a static charge distribution leads to the electrostatic field; for our point charge at the origin, V(r, t=infinity) = 1/r. However, if the charge is static, V would be constantly increasing...wouldn't it? So at t=infinity, V would be infinity everywhere? Now, running a simulation (because I suck at PDEs), it appears that while this appears to be true, E = gradV does appear to stay constant...so perhaps this is a red herring, but I would feel much more confident if I could find an analytical solution (or a logical explanation). Any takers?