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On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. (English) Zbl 0801.35133
The paper concerns the following system of semiconductor equations: ${\partial u_ i\over \partial t}+ e_ i\text{ div }J_ i+ r_ i= 0\quad (i=n,p),\;e_ n=- 1,\;e_ p= +1,\tag{1}$ $- \text{div}(\varepsilon\nabla v_ 0)= D+ u_ p- u_ n\tag{2}$ where $$v_ 0$$ is the electrostatic potential, $$u_ i= f_ i(v_ i)$$ the carrier density, $$v_ i$$ the chemical potential, $$J_ i= J(x,v_ i, z_ i)$$, $$z_ i= \nabla w_ i$$ the current density, $$w_ i= v_ 0+ e_ i v_ i$$ the quasi-Fermi potential, $$\varepsilon$$ the dielectric permittivity, $$D$$ the density of impurities, $$r_ i$$ the recombination/generation rate.
System (1), (2) is completed by mixed boundary conditions and initial conditions on $$u_ n$$, $$u_ p$$. The author then introduces appropriate structural conditions on $$J_ i$$ and various smoothness conditions on $$f_ i$$ and $$r_ i$$.
The main result of the paper is a uniqueness theorem for bounded weak solutions to (1), (2) which satisfy an additional regularity condition. The proof makes essential use of a kind of distance between two weak solutions which is constructed by an energy functional.
Reviewer: J.Naumann (Berlin)

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 78A35 Motion of charged particles 35K55 Nonlinear parabolic equations 35K57 Reaction-diffusion equations
##### Keywords:
semiconductor equations; energy functional; uniqueness
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