General recombination-generation laws for charge transport equations (Q2777994)

The paper under review is concerned with the following system of partial differential equations in a cylinder \(\Omega \times (0,T)\) (\( \Omega \subset \mathbb{R}^3\) a bounded domain): NEWLINENEWLINENEWLINE(1) \(\displaystyle\frac{\partial p}{\partial t}- div (B(\theta)\nabla p+\mu p \nabla\varphi)=R(p,n),\) NEWLINENEWLINENEWLINE(2) \(\displaystyle\frac{\partial n}{\partial t} - div (D(\theta)\nabla n-\nu n \nabla \varphi)= R (p,n),\) NEWLINENEWLINENEWLINE(3) \(-\varepsilon \Delta \varphi = p-n,\) NEWLINENEWLINENEWLINE(4) \( \displaystyle\frac{\partial \theta} {\partial t}-div (\kappa \nabla \theta)=(\mu p +\nu n) \Big|\nabla \varphi \big|^2 (B(\theta)\nabla p - D (\theta) \nabla n) \cdot \nabla \varphi \). The initial and boundary conditions on the unknown functions \( p, n, \varphi \) and \(\theta\) are as follows: NEWLINENEWLINENEWLINE(5) \(p=p_0, \quad n=n_0,\quad \theta=\theta_0 \qquad \)in \(\Omega\), NEWLINENEWLINENEWLINE(6) \(p=p_b,\quad n=n_b,\quad \theta=\theta_b,\quad \varphi=\varphi_b\) on \(\partial\Omega \times (0,T)\).NEWLINENEWLINENEWLINEThe initial-boundary value problem (1)--(6) models the electro-diffusion of ions. If the Dirichlet boundary conditions (6) are replaced by mixed Dirichlet and (homogeneous) Neumann boundary conditions then with these boundary conditions problem (1)--(5) \( \big(\text{with a given doping function}D\) on the right in (3)\big) models also the transport of electrons and holes in a semiconductor device. The conditions on the recombination-generation term \(R(p,n)\) in (1) and (2) include many physically important cases. NEWLINENEWLINENEWLINEThe first result of the paper is an existence and uniqueness theorem which is obtained by the Leray-Schauder principle. Then the author studies the asymptotic behaviour of solutions to (1)--(6) when \(T \rightarrow +\infty\). In the last section, the existence and uniqueness of a classical solution for the stationary case of (1)--(4), is proved provided an appropriate smallness upon the data condition is satisfied.