Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and boundary data are in subsonic region supersonic separately, system possesses shock transonic steady-states smooth steady-states. In this paper we study nonlinear structural stability linear dynamic instability these steady solutions. any relaxation time: 0 < τ ≤ + ∞ , means elaborate singularity analysis, first investigate C 1 -smooth steady-states, once perturbations initial profiles small enough. We note that, pass through sonic line, they produce singularities for system, cause some essential difficulty proof stability. Moreover, time large enough ≫ under condition that electric field positive at location, prove structurally stable with respect to profile. Furthermore, show linearly provided suitable negative. The proofs results based on a monotonicity argument position downstream density, analysis equations can be transformed ill-posedness free problem Klein-Gordon equation. By using nontrivial transformation shooting method, linearized has solution exponential growths. These enrich develop existing studies.