On the uniqueness of solutions for nonlinear elliptic‐parabolic equations

We prove a priori estimates in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ L^2(0,T;W^{1,2}(\Omega)) $\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ L^{\infty}(Q_T) $\end{document}, existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \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, $\end{document}¶¶where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \rho(u) = \frac {\partial \sigma(u)}{\partial u} $\end{document}. Systems of such form arise as mathematical models of various applied problems, for instance, electron transport processes in semiconductors. Our basic assumption is that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \log \rho(u) $\end{document} is concave. Such assumption is natural in view of drift-diffusion models, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \sigma $\end{document} 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.