Large-time behavior of solutions to bipolar Euler-Poisson equations with space-dependent damping

RESUMO

This paper is concerned with the Cauchy problem for the one-dimensional bipolar Euler-Poisson equations with space-dependent damping, which arise in the hydrodynamic model of semiconductors. Notably, the two pressure functions are allowed to be different, and the doping profile is nonzero. By means of the time-weighted energy method with carefully chosen weights, we prove that if the initial perturbation around a constant steady-state is small enough, the Cauchy problem admits a unique global smooth solution, which converges to the constant steady-state at algebraic decay rates. We show that the decay rates coincide with those established for the constant-coefficients damping case, thereby overcoming the influence of spatially dependent damping on the decay rates of the solutions. A key feature is that no smallness of the space-dependent damping coefficients is required.