Stability on large non-constant steady states of semiconductor equations

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Around the large non-constant steady state, we show the existence and uniqueness of the global solution of the three-dimensional Cauchy problem of the semiconductor equations without or with the magnetic field effect. Without the magnetic field effect, we prove that the solution converges to the non-constant steady state exponentially fast as time goes to infinity. With the magnetic field effect, the optimal algebraic time-decay rates of the lower-order derivatives of the solution are obtained. Our results manifest that the degenerate dissipation of the magnetic field slows down the decay rates of the whole solution.