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Steady hydrodynamic model of semiconductors with sonic boundary and transonic doping profile. (English) Zbl 1509.35388

Summary: As shown in [J. Li et al., SIAM J. Math. Anal. 49, No. 6, 4767–4811 (2017; Zbl 1379.35350); ibid. 50, No. 1, 718–734 (2018; Zbl 1380.35168)], for the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary and subsonic/supersonic doping profile, the structure of stationary solutions are very complicated. It may possess various solutions like subsonic/supersonic/transonic flows. In this paper, we consider a more challenging case where the doping profile is transonic, which is categorized into two types: subsonic-dominated and supersonic-dominated. In the subsonic-dominated case, we show that the system has a unique interior subsonic solution, at least one interior supersonic solution and infinitely many transonic solutions under the suitable assumptions. However, the difference with the case of subsonic doping profile is that the interior subsonic solution and interior supersonic solution may not exist in special cases when the relaxation time is small. In the supersonic-dominated case, the non-existence and existence of all types of solutions are also obtained. The approach adopted is the technical compactness analysis combining the Green’s function method. Here, the results obtained perfectly develop the existing studies.

MSC:

35R35 Free boundary problems for PDEs
35J70 Degenerate elliptic equations
35Q35 PDEs in connection with fluid mechanics
35Q81 PDEs in connection with semiconductor devices
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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