Uniqueness for the three-dimensional time dependent drift diffusion semiconductor equations with \(L^2\) initial data (Q1290948)

The paper under review is concerned with van Roosbroeck's system \[ \begin{aligned} n_{t} - \nabla \cdot (D_{1}\nabla n - \mu_{1}n \nabla \psi) & = r (n,p)(1-np),\tag{1} \\ p_{t} - \nabla \cdot (D_{2}\nabla\psi + \mu_{2} p \nabla \psi) & = r (n,p)(1-np),\tag{2} \\ - \Delta\psi &= p-n+f \tag{3} \end{aligned} \] in the cylinder \(Q=\Omega \times (0,T)\), \(\Omega \subset \mathbb R^{3}\) a bounded domain. System (1) - (3) is completed by the standard mixed boundary conditions of semiconductor theory on \( n,p\) and \(\psi\), and initial conditions on \(n,p\). The author proves the uniqueness of weak solutions to (1) - (3) with initial conditions \(\{ n_{0},p_{0}\} \in L^{2}_{+}(\Omega) \times L^{2}_{+}(\Omega)\). The proof makes use of the estimate \(\int^{T}_{0} \int_{\Omega}(n+p)^{2}|\nabla \psi|^{2} dx dt \leq C\).