Asymptotic Convergence to Diffusive Wave of Bipolar Hydrodynamical Model for Semiconductors

We consider the large time behavior of solutions of the one-dimensional isentropic bipolar hydrodynamical model for semiconductors: (1) $$ {{n}_{t}} + {{J}_{x}} = 0, $$ (2) $$ {{J}_{t}} + {{\left( {\frac{{{{J}^{2}}}}{n} + p(n)} \right)}_{x}} = nE - \frac{J}{{{{\mu }_{n}}\tau }}, $$ (3) $$ {{m}_{t}} + {{I}_{x}} = 0, $$ (4) $$ {{I}_{t}} + {{\left( {\frac{{{{I}^{2}}}}{m} + q(m)} \right)}_{x}} = - mE - \frac{I}{{{{\mu }_{m}}\tau }}, $$ (5) $$ {{\lambda }^{2}}{{E}_{x}} = n - m, $$ where n > 0, m > 0, J, I and E denote the densities, current densities, and electric field respectively, p = p(n) and q = q(m) are the pressure-density functions which satisfy (6) $$ {{\left( {{{\rho }^{2}}p'(\rho )} \right)}^{\prime }} > 0,\quad {{\left( {{{\rho }^{2}}q'(\rho )} \right)}^{\prime }} > 0,\quad \rho > 0. $$ And τ n > 0, τ m > 0 are the momentum relaxation times, λ is the re-scaled Debye number. The device domain is chosen to be the whole real line. The equations (l)–(5) are used in the modelling of semiconductors device to describe the to model hot electron effects and can be derived by applying the moment method to the bipolar semiconductor Boltzmann equations for electron and hole.