Ohms law implies specific condition for the charge density

GarageDweller

We have all seen Ohms law, J=σE. This approximations makes sense in simple electric fields in which the charges are accelerated in parallel.

However as I will demonstrate, this implies a few conditions on the charge density (ρ) associated with the current density (J).

Now, from the continuity equation, ∂ρ/∂t=-DivJ
=-σDivE (using the approximation)
=-σ(ρ/ε) (From maxwells equations)
This is just a simple partial differential equation, solving...
ρ=A(x,y,z)exp(-σt/ε)+B(x,y,z)
This is a partial differential equation, so we have to allow the coefficients and constants as functions of position.

This implies that at time 0 (when the electric field is set up), the charge density is A+b, after some time the charge density reduces to B(x,y,z).
Has this been observed in ohmic resistors?

I find this result quite strange, is this a physical result?