On the semiconductor drift diffusion equations. (English) Zbl 0859.35055

The paper under review is concerned with the following system of drift-diffusion equations in semiconductor theory: \[ {\partial p\over \partial t}-\nabla\cdot(D_1\nabla p+\mu_1p\nabla\psi)=R(p,n),\tag{1} \]
\[ {\partial n\over \partial t}-\nabla\cdot(D_2\nabla n-\mu_2n\nabla\psi)=R(p,n),\tag{2} \]
\[ -\nabla\cdot(a\nabla\psi)=p-n+f\tag{3} \] in \(\Omega\times(0,+\infty)\) (\(\Omega\subset\mathbb{R}^N\) a bounded Lipschitz domain, \(N=2,3,4\)), where \(p=\) density of holes, \(n=\) density of electrons, \(\psi=\) electrostatic potential, \(D_k\), \(\mu_k=\text{const}>0\) \((k=1,2)\), \(R=\) recombination/generation of carriers, \(f=\) doping profile, \(a=\text{const}>0\). Equations (1)–(3) are completed by the usual mixed boundary conditions on \(p\), \(n\) and \(\psi\) along \(\partial\Omega\times (0,+\infty)\) modelling an operating semiconductor device, and initial conditions on \(p\) and \(n\).
First, the author proves the existence of a weak solution to (1)–(3) globally in time for initial conditions in \(L^2_+(\Omega)\). Then he proves two theorems concerning the uniqueness of weak solutions to (1)–(3) and their continuous dependence on the initial data. These results are heavily based on the integrability to a power \(q>2\) if \(N=2\), \(q=3\) if \(N=3\) of the gradient of the weak solution to the Poisson equation in \(\Omega\) under mixed boundary conditions.
Reviewer: J.Naumann (Berlin)


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs