  For unipolar hydrodynamic model of semiconductor device represented by
Euler-Poisson equations, when the doping profile is supersonic, the existence
of steady transonic shock solutions and C-smooth steady transonic solutions for
Euler-Poisson Equations were established in [27] and [41], respectively. In
this paper we further study the nonlinear structural stability and the linear
dynamic instability of these steady transonic solutions. When the C^1-smooth
transonic steady-states pass through the sonic line, they produce singularities
for the system, and cause some essential difficulty in the proof of structural
stability. For any relaxation time, by means of elaborate singularity analysis,
we first investigate the structural stability of the C^1-smooth transonic
steady-states, once the perturbations of the initial data and the doping
profiles are small enough. Moreover, when the relaxation time is large enough,
under the condition that the electric field is positive at the shock location,
we prove that the transonic shock steady-states are structurally stable with
respect to small perturbations of the supersonic doping profile. Furthermore,
we show the linearly dynamic instability for these transonic shock
steady-states provided that the electric field is suitable negative. The proofs
for the structural stability results are based on singularity analysis, a
monotonicity argument on the shock position and the downstream density, and the
stability analysis of supersonic and subsonic solutions. The linear dynamic
instability of the steady transonic shock for Euler-Poisson equations can be
transformed to the ill-posedness of a free boundary problem for the
Klein-Gordon equation. By using a nontrivial transformation and the shooting
method, we prove that the linearized problem has a transonic shock solution
with exponential growths. These results enrich and develop the existing
studies.
