In this note, we consider a bipolar hydrodynamic model (Euler-Poisson system) in one space dimension. Denoting by n
= 1, 2, and E
the charge densities, current densities, pressures and electric field, the scaled equations of the hydrodynamic model are given by
The positive constants τ
= 1,2) and λ
denote the relaxation time and the Debye length, respectively. The relaxation terms describe in a very rough manner the damping effect of a possible neutral background charge. The Debye length is related to the Coulomb screening of the charged particles. The hydrodynamic models are generally used in the description of charged particle fluids. These models take an important place in the fields of applied and computational mathematics. They can be derived from kinetic models by the moment method. For more details on the semiconductor applications, see [1
] and on the applications in plasma physics, see [1
]. To begin with, we assume in the present article that the pressure-density functions satisfy
and set τ1
to be one for simplicity In particular, we note that γ
= 1 is an important case in the applications of engineer. Hence, (1.1) can be simplifies as
Recently, many efforts were made for the bipolar isentropic hydrodynamic equations of semiconductors. More precisely, Zhou and Li [4
] and Tsuge [5
] discussed the unique existence of the stationary solutions for the one-dimensional bipolar hydrodynamic model with proper boundary conditions. Natalini [6
], and Hsiao and Zhang [7
] established the global entropic weak solutions in the framework of compensated compactness on the whole real line and spatial bounded domain respectively. Zhu and Hattori [9
] proved the stability of steady-state solutions for a recombined bipolar hydrodynamical model. Ali and Jüngel [10
] studied the global smooth solutions of Cauchy problem for multidimensional hydrodynamic models for two-carrier plasma. Lattanzio [11
] and Li [12
] studied the relaxation time limit of the weak solutions and local smooth solutions for Cauchy problems to the bipolar isentropic hydrodynamic models, respectively. Gasser and Marcati [13
] discussed the relaxation limit, quasineutral limit and the combined limit of weak solutions for the bipolar Euler-Poisson system. Gasser et al. [14
] investigated the large time behavior of solutions of Cauchy problem to the bipolar model basing on the fact that the frictional damping will cause the nonlinear diffusive phenomena of hyperbolic waves, while Huang and Li recently studied large-time behavior and quasineutral limit of L∞
solution of the Cauchy problem in [15
]. As far as we know, no results about the global existence and large time behavior to (1.2) with boundary effect can be found. In this article we will consider global existence and asymptotic behavior of smooth solutions to the initial boundary value problems for the bipolar Euler-Poisson system on the quarter plane ℝ+
. Then, we now prescribe the initial-boundary value conditions:
Moreover, we also investigate the time-asymptotic behavior of the solutions to (1.2)-(1.4). Our results discussed below show that even for the case with boundary condition, the solutions of (1.2)-(1.4) can be captured by the corresponding porous equation as in initial data case. For the sake of simplicity, we can assume j+ = 0. This assumption can be removed because of the exponential decay of the momentum at x = ±∞ induced by the linear frictional damping.
here the nonlinear diffusion waves
will be defined in Section 2, and the shift xi 0
which can be computed from the standard arguments, see [16–19].
Throughout this article C always denotes a harmless positive constant. L
(ℝ+) is the space of square integrable real valued function defined on ℝ+ with the norm and H
(ℝ+) denotes the usual Sobolev space with the norm ∥·∥
Now one of main results in this paper is stated as follows.
Theorem 1.1 Suppose that n10
) and satisfies
(2.4) for some δ0
> 0, (φ10
) ∈ (H3
) ∩ L1
)) × (H2
) ∩ L1
)) × (H3
) ∩ L1
)) × (H2
) ∩ L1
)) with x10
= x20 and that
hold. Then there exists a unique time-global solution
) of IBVP
(1.2)-(1.4), such that for i
where α > 0 and C is positive constant.
Next, with the help of Fourier analysis, we can obtain the following optimal convergence rate.
Theorem 1.2 Under the assumptions of Theorem
1.1, it holds that
Remark 1.3 The condition
and it is a technique one. As to more general case, we will discuss it in the forthcoming future. Theorems 1.1 and 1.2 show that the nonlinear diffusive phenomena is maintained in the bipolar Euler-Poisson system with the interaction of two particles and the additional electric field, which indeed implies that this diffusion effect is essentially due to the friction of momentum relaxation.
Using the energy estimates, we can establish a priori estimate, which together with local existence, leads to global existence of the smooth solutions for IBVP (1.2)-(1.4) by standard continuity arguments. In order to obtain the asymptotic behavior and optimal decay rate, noting that E = φ1 - φ2 satisfies the damping "Klein-Gordon" equation (see [14, 15]), we first obtain the exponential decay rate of the electric field E by energy methods. Then, we can establish the algebraical decay rate of the perturbed densities φ1 and φ2. Finally, from the estimates of the wave equation with damping in  and using the idea of , we show the optimal algebraical decay rates of the total perturbed density φ1 + φ2, which together with the exponential decay rate of the difference of two perturbed densities, yields the optimal decay rate. In these procedure, we have overcome the difficulty from the coupling and cancelation interaction between n1 and n2. Finally, it is worth mentioning that similar results about the Euler equations with damping have been extensively studied by many authors, i.e., the authors of [16–19, 21, 22], etc.
The rest of this article is arranged as follows. We first construct the optimal nonlinear diffusion waves and recall some inequalities in Section 2. In Section 3, we reformulate the original problem, and show the main Theorem. Section 4 is to prove an important decay estimate, which has been used to show the main theorem in Section 3.