On the uniqueness of solutions for nonlinear elliptic-parabolic equations

Abstract

We prove a priori estimates in L ²(0,T;W ^1,2(Ω)) and L^∞(Q) _T existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems

$$ \frac{{\partial \sigma (u)}}{{\partial t}} - \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}\left\{ {\rho (u){b_{i}}\left( {t,x,\frac{{\partial (u - \upsilon )}}{{\partial x}}} \right)} \right\}} + a(t,x,\upsilon ,u) = 0, $$

$$ - \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}\left[ {\kappa (x)\frac{{\partial \upsilon }}{{\partial {x_{i}}}}} \right]} + \sigma (u) = f(t,x),(t,x) \in {Q_{T}} = (0,T) x \Omega , $$

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 ρ(u) is concave. Such assumption is natural in view of drift-diffusion models, whereahas to be specified as a probability distribution function like a Fermi integral and u resp. υ have to be interpreted as chemical resp. electrostatic potential.