To the uniqueness problem for nonlinear parabolic equations

We prove a priori estimates in $L^2(0,T,W^{1,2}(\Omega)) \cap L^{\infty}(Q)$, existence and uniqueness of solutions to Cauchy-Dirichlet problems for parabolic equations $$ \frac{\partial \sigma(u)}{\partial t} - \sum_{i=1}^n \frac{\partial}{\partial x_i} \Big \{ \rho(u) b_i \Big (t,x,\frac{\partial u}{\partial x} \Big ) \Big \} + a \Big (t,x,u,\frac{\partial u}{\partial x} \Big ) = 0, $$ $(t,x) \in Q = (0,T) \times \Omega$, where $\rho(u) = \frac{d}{du}\sigma(u)$. We consider solutions $u$ such that $\rho^{\frac{1}{2}}(u) \left \frac{\partial u}{\partial x} \right \in L^2 ( 0,T,L^2 (\Omega)), \frac{\partial}{\partial t}\sigma(u) \in L^2 (0,T, [ {\cir{W}}\hspace*{-0.1mm}^{1,2} (\Omega) ]^{\ast} )$. Our nonstandard assumption is that $\log \rho (u)$ is concave. Such assumption is natural in view of drift diffusion processes for example in semiconductors and binary alloys, where $u$ has to be interpreted as chemical potential and $\sigma$ is a distribution function like $\sigma=e^{u}$ or $\sigma=\frac{1}{1+e^{u}}$