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
$\frac {\partial \sigma(u)}{\partial t} - \sum_{i=1}^n
\frac {\partial}{\partial x_i}${$\rho(u) b_i
(t,x,\frac{\partial u}{\partial x})$}
$ + a (t,x,u,\frac{\partial u}{\partial x}) =
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) | \frac{\partial u}{\partial x}
| \in L^2 (0,T;L^2 (\Omega ) ), \frac {\partial }{\partial
t}\sigma(u) \in L^2 ( 0,T;[ W^{1,2} (
\Omega ) ]^\star ). $
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 $\sigma=\frac
{1}{1+e^{-u}}$.
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