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Global solutions of the time-dependent drift-diffusion semiconductor equations. (English) Zbl 0845.35050

The paper is concerned with the following drift-diffusion model in semiconductor theory: \[ {\partial n\over \partial t}- \text{div}(\mu_n (\nabla u)(\nabla n- n \nabla u))= R(n, p) (1- np)+ g, \tag{1} \]
\[ {\partial p\over \partial t}- \text{div}(\mu_p (\nabla u)(\nabla p+ p \nabla u))= R(n, p) (1- np)+ g,\tag{2} \]
\[ \varepsilon \Delta u= n- p- D\tag{3} \] (\(n=\) density of electrons, \(p=\) density of holes, \(u=\) electrostatic potential, \(\mu_n, \mu_p=\) field dependent mobilities, \(\varepsilon=\text{const}> 0\), \(R\), \(g\), \(D\) given functions). System (1)–(3) has to be considered in a cylinder \(\Omega\times (0, +\infty)\) (\(\Omega\subset \mathbb{R}^d\) bounded domain, \(d= 1,2\) or 3) and completed by mixed boundary conditions on \(n\), \(p\), \(u\) and initial conditions on \(n\), \(p\).
The authors prove the existence of a weak solution to (1)–(3) and, additionally, a uniform bound from below and above on \(\log n\), \(\log p\) (independent of \(t\)) for any time interval \((0, T)\). More refined estimates show that for suitably choosen initial values there exists an absorbing set for the associated dynamical system. The proofs are based on estimates on the solutions of the time-discretized system.
Reviewer: J.Naumann (Berlin)

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
78A35 Motion of charged particles
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