On the uniqueness of solutions for nonlinear elliptic‐parabolic equations

Abstract.

We prove a priori estimates in $L^2(0,T;W^{1,2}(\Omega))$ and $L^{\infty}(Q_T)$, existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems¶¶$\frac {\partial \sigma(u)}{\partial t} - \sum\limits_{i=1}^n \frac {\partial}{\partial x_i} \left\{\rho(u) b_i \left (t,x,\frac {\partial (u-v)}{\partial x} \right) \right\} + a (t,x,v,u) = 0,\\- \sum\limits_{i=1}^n \frac {\partial}{\partial x_i} \left[ \kappa(x) \frac{\partial v}{\partial x_i} \right ] + \sigma(u) = f (t,x), \;(t,x) \in Q_T = (0,T) \times \Omega,$¶¶where $\rho(u) = \frac {\partial \sigma(u)}{\partial u}$. Systems of such form arise as mathematical models of various applied problems, for instance, electron transport processes in semiconductors. Our basic assumption is that $\log \rho(u)$ is concave. Such assumption is natural in view of drift-diffusion models, where $\sigma$ has to be specified as a probability distribution function like a Fermi integral and u resp. v have to be interpreted as chemical resp. electrostatic potential.