Confusion on Continuity of Current and Free Charge in Conductor

I know there must be something wrong with the following derivation based on Maxwell's equations but could not figure out what is wrong. The derivation deals with continuity of current and free charge in a conductor in general.

Continuity of current says that,

[itex]\nabla\cdot \textbf{J}=-\frac{\partial \rho_v}{\partial t}[/itex]. (1)

However, for currents in conductor, the current density and the electric field is related as

[itex]\textbf{J}=\sigma\textbf{E}[/itex].

Using the relation between [itex]\textbf{E}[/itex] and [itex]\textbf{D}[/itex] ,

[itex]\textbf{E}=\textbf{D}/\epsilon[/itex],

we have

[itex]\nabla\cdot \textbf{J}=\sigma\nabla\cdot\textbf{E}=\frac{\sigma}{\epsilon} \nabla\cdot\textbf{D}=\frac{\sigma}{\epsilon}\rho_v[/itex]. (2)

Comparing (1) and (2) gives an equation on free charge [itex]\rho_v[/itex],

[itex]-\frac{\partial \rho_v}{\partial t}=\frac{\sigma}{\epsilon} \rho_v[/itex]

which means that

[itex]\rho_v=\rho_{v0}e^{-\frac{\sigma}{\epsilon}t}[/itex]. (3)

Since we are talking about general cases of Maxwell's equations, (3) looks like an unreal restriction on free charge in a conductor and does not make sense at all.

What went wrong in the above derivations? Thanks in advance!