HYDRODYNAMIC MODEL OF SEMICONDUCTORS WITH BOUNDARY EFFECT
FEIMIN HUANG∗, MING MEI†, YONG WANG‡, AND TONG YANG§
Abstract. For a bipolar hydrodynamic model of semiconductors in the form of Euler-Poisson equations with Dirichlet or Neumann boundary conditions, in this paper, we first heuristically analyze the most probable asymptotic profile (the so-called diffusion waves) and then prove this long-time behavior rigorously. For this, we construct correction functions to show the convergence of the original solution to the diffusion wave with optimal convergence rates by the energy method. Moreover, in the case with Dirichlet boundary condition, when the initial perturbation is in some weighted L1 space, a faster and optimal convergence rate is also given.
Key words. bipolar hydrodynamic model, semiconductor, nonlinear damping, nonlinear diffu- sion waves, asymptotic behavior, convergence rates.
AMS subject classifications. 35L50, 35L60, 35L65, 76R50
1. Introduction. This is a continuation of our study on the stability of diffusion waves for the bipolar hydrodynamic system of semiconductors. Based on the work [15] on the initial value problem (IVP), in this paper, we consider the initial-boundary value problem (IBVP) of the5×5 Euler-Poisson system in the half-space
n1t+J1x= 0, J1t+ (Jn12
1 +p(n1))x=n1E−J1, n2t+J2x= 0,
J2t+ (Jn22
2 +q(n2))x=−n2E−J2, Ex=n1−n2,
(x, t)∈R+×R+, (1.1)
with the initial data
(n1, n2, J1, J2)|t=0= (n10, n20, J10, J20)(x), (1.2) and either the Dirichlet boundary condition
J1|x=0=J2|x=0=E|x=0= 0, (1.3) or the Neumann boundary condition
J1x|x=0=J2x|x=0= 0, and E(+∞, t) =a(t). (1.4) Here, for the later case, two situations will be considered, that is, either
|a(t)|=O(1)e−ηt forη >0, (1.5)
∗Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China ([email protected]).
†Corresponding author. Department of Mathematics, Champlain College, Saint-Lambert, Quebec J4P 3P2, Canada, and Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada ([email protected]).
‡Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China ([email protected]).
§Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China ([email protected]).
1
or
|a(t)|=O(1)(1 +t)−θ forθ > 3
2. (1.6)
In the above system, n1(x, t), n2(x, t), J1(x, t), J2(x, t) and E(x, t) represent the electron density, the hole density, the current density of electrons, the current density of holes and the electric field, respectively. The nonlinear functions p(s) and q(s) denote the pressures of the electrons and the holes which are assumed to be identical and smooth, and satisfy
p(s) =q(s)≥0, p0(s) =q0(s)>0 fors >0. (1.7) Moreover, we assume the initial data satisfying
x→+∞lim (n10(x), J10(x), n20(x), J20(x), E(x,0)) = (n+, J+, n+, J+, E+), (1.8) together with the compatibility condition with the Dirichlet boundary condition:
J10(0) =J20(0) = 0, and E+= Z
R+
[n01(x)−n02(x)]dx, (1.9) or the compatibility condition with the Neumann boundary condition:
Ji00 (0) = 0, ni|x=0=ni0(0) fori= 1,2, anda(0) =E+. (1.10) Here(n+, J+, n+, J+, E+)are some constant states. Please refer to (3.3) for deriving the expression ofE+ in (1.9).
According to the Darcy’s law, in long-time, the solutions to the Euler-Poisson system behave similar to those solutions to the porous media equations because of the frictional damping, like Euler equation with damping that has been extensively studied, cf. [10]. In fact, this can be verified by variables scaling [14, 26, 33, 35]:
x→x/ε, t→t/ε2, ni→n¯i, Ji→εJ¯i, E→E,¯
fori= 1,2 with an arbitrarily small numberε >0, then the system (1.1) becomes
¯
n1t+ ¯J1x= 0, ε3J¯1t+ε3(Jn¯¯12
1)x+ε(p(¯n1))x= ¯n1E¯−εJ¯1,
¯
n2t+ ¯J2x= 0, ε3J¯2t+ε3(Jn¯¯22
2)x+ε(p(¯n2))x=−¯n2E¯−εJ¯2, εE¯x= ¯n1−n¯2.
(1.11)
Neglecting the small terms withε, (1.11) gives
¯
nit+ ¯Jix= 0, i= 1,2 0 = ¯n1E,¯
0 = ¯n2E,¯ 0 = ¯n1−n¯2, which implies
¯
n1= ¯n2=: ¯n, J¯1= ¯J2=: ¯J , E¯ = 0. (1.12)
Thus, substituting (1.12) to (1.11) yields (n¯t+ ¯Jx= 0,
ε2J¯t+ε2(J¯n¯2)x+p(¯n)x=−J .¯
Again, by omitting the small terms with ε, we obtain the following equations which can be used to describe the asymptotic profiles of (1.1):
(n¯t+ ¯Jx= 0,
p(¯n)x=−J ,¯ or equivalently,
(n¯t−p(¯n)xx= 0, porous media equation, p(¯n)x=−J ,¯ Darcy’s law.
Therefore, we expect that the solution(n1, J1, n2, J2, E)(x, t)converges to(¯n,J ,¯ n,¯ J ,¯ 0) (x, t), where (¯n,J¯)(x, t)satisfies the above porous media equations.
In most of the previous works, the asymptotic profiles for the damped Euler- Poisson equations (1.1) are chosen to be the self-similar solutions (¯n,J)(x/¯ √
1 +t) of the above porous media equations, called diffusion waves. On the other hand, as indicated in [26, 28] for the IBVP to the dampedp-system, originally see [33] for the IVP, a better asymptotic profile is the solution of the corresponding IBVP to the porous media equations itself, rather than the self-similar solution. Based on this thinking, for the problems considered in this paper, we expect the asymptotic profile of the Dirichlet IBVP (1.1)-(1.3) satisfying
¯
nt+ ¯Jx= 0, p(¯n)x=−J ,¯ n|¯t=0= ¯n0(x), J¯|x=0= 0,
or equivalently,
¯
nt−p(¯n)xx= 0, p(¯n)x=−J ,¯
¯n|t=0= ¯n0(x),
¯
nx|x=0= 0;
(1.13)
while the asymptotic profile of the Neumann IBVP (1.1), (1.2) and (1.4) satisfying
¯
nt+ ¯Jx= 0, p(¯n)x=−J ,¯ n|¯t=0= ¯n0(x), J¯x|x=0= 0,
or equivalently,
¯
nt−p(¯n)xx= 0, p(¯n)x=−J ,¯
¯n|t=0= ¯n0(x),
¯n|x=0= ¯n0(0).
(1.14)
Here, the initial datan¯0(x)→n+ asx→ ∞will be specified later in (3.41) for the Dirichlet boundary, and (4.32) for the Neumann boundary respectively. We also call such a particular solution (¯n,J¯)(x, t) for the IBVPs (1.13) or (1.14) as a diffusion wave without any confusion.
Let us now review some previous works on the Euler-Poisson equations. For the unipolar hydrodynamic model, the existence of steady-state solutions was studied in [4, 5, 6], and their stability was given in [7, 11, 16, 17, 21, 27, 39] in different settings.
Moreover, the global existence of classical and/or the entropy weak solutions as well as the zero relaxation limits were investigated in [1, 2, 9, 12, 11, 21, 22, 23, 31, 43, 48]. On the other hand, for the bipolar hydrodynamic model, the related research is much less.
For this, Natalini [37], and Hsiao and Zhang [12, 13] established the global entropy weak solutions in the framework of compensated compactness on the whole real line and spatial bounded domain, respectively. Hattori and Zhu [50] proved the stability of steady-state solutions for a re-combined bipolar hydrodynamic model. Y. Li [24]
studied the relaxation limit of a bipolar isentropic hydrodynamic model with small
momentum relaxation time. For the stability of diffusion waves, Gasser, Hsiao and H. Li [8], and Huang and Y. Li [14] studied it in strong and weak sense, respectively, under a restrictive condition
Z
R
[n10(x)−n20(x)]dx= 0. (1.15) Note that this together with (1.1)5 imply
E(+∞,0)−E(−∞,0) = 0.
This is called the switch-off case for the device because there is no voltage. For the switch-on case
E(+∞,0)−E(−∞,0)6= 0, or equivalently Z
R
[n10(x)−n20(x)]dx6= 0, there is some difference between the original solutions and the diffusion waves at the far fields so that the perturbation of a diffusion wave can not be inL2 space. Note that theL∞-stability of diffusion waves remained open. To overcome this difficulty, recently, by constructing some correction functions to take care of the difference, we proved in [15] the stability of diffusion waves for the IVP of (1.1).
Note that, for the IVP studied in [15] the far field layer due to the fact that Ji|x=∞−Ji|x=−∞6= 0andE|x=∞−E|x=−∞6= 0was analyzed. With the boundary effect (Dirichlet or Neumann) in the half space considered in this paper, we also need to take care of the boundary layer. For the one-side layer at the far field x= ∞, the correction function for the IBVPs in some sense can be constructed in a more straightforward way than the IVP in the full space. In fact, we can linearizeni(x, t) around the constant staten+, and derive the explicit integral equations with Green functions to the corresponding IBVPs (see (3.66) and (3.71), and (4.39) and (4.40), respectively). With this advantage, by using the energy estimates for the elementary solutions to the linear damped wave equations introduced by Matsumura [32] and Ikehata [18] (see also the improvements in [26, 28, 30, 44] for nonlinear damped p- system), together with the new techniques in [15, 16, 17, 36] on the construction of correction functions, we succeed in proving the following optimal convergence results:
1. In the Dirichlet boundary case (1.3), the solution (n1, J1, n2, J2, E)(x, t) of Dirichlet IBVP (1.1)-(1.3) converges to its corresponding diffusion wave (¯n,J ,¯ n,¯ J ,¯ 0)(x, t)of IBVP (1.13) with the optimal convergence rates:
k(n1−n, J¯ 1−J , n¯ 2−¯n, J2−J , E)(t)k¯ L∞(R+)
=O(1)(t−34, t−54, t−34, t−54, e−νt), (1.16) for the initial perturbations inH3(R+)×H2(R+), where 0< ν < 12; and
k(n1−n, J¯ 1−J , n¯ 2−¯n, J2−J , E)(t)k¯ L∞(R+)
=O(1)(t−1, t−32, t−1, t−32, e−νt), (1.17) for the initial perturbation in(H2(R+)×H1(R+))∩L1(R+). Furthermore,
k(n1−n, J¯ 1−J , n¯ 2−¯n, J2−J , E)(t)k¯ L∞(R+)
=O(1)(t−1−γ2, t−32, t−1−γ2, t−32, e−νt), (1.18) for the initial perturbation in the weighted space (H2(R+)×H1(R+))∩ L1,γ(R+)with0< γ≤14.
2. In the Neumann boundary case (1.4), the solution(n1, J1, n2, J2, E)(x, t)of Neumann IBVP (1.1), (1.2) and (1.4) converges to its corresponding diffusion wave(¯n,J ,¯n,¯ J ,¯0)(x, t)of IBVP (1.14) with the optimal convergence rates:
k(n1−¯n, J1−J , n¯ 2−¯n, J2−J , E)(t)k¯ L∞(R+)
=
(O(1)(t−34, t−54, t−34, t−54, e−νt), as |a(t)|=O(e−ηt), η >0, O(1)(t−34, t−54, t−34, t−54, t−θ), as |a(t)|=O(t−θ), θ >32, (1.19) for the initial perturbations inH3(R+)×H2(R+), where0< ν <min{γ,12};
and
k(n1−n, J¯ 1−J , n¯ 2−n, J¯ 2−J , E¯ )(t)kL∞(R+)
=
(O(1)(t−1, t−32, t−1, t−32, e−νt), as |a(t)|=O(e−ηt), η >0, O(1)(t−1, t−32, t−1, t−32, t−θ), as |a(t)|=O(t−θ), θ > 32, (1.20) for the initial perturbation in (H2(R+)×H1(R+))∩L1(R+). Note that different from the Dirichlet IVBP, there is no weighted L1,γ-decay obtained for this case.
Finally, for the study on diffusion phenomena to damped p-system or relaxation models, we refer to [3, 10, 25, 26, 33, 34, 40, 41, 42, 47, 49] and the references therein.
The rest of the paper is organized as follows. Firstly of all, at the end of this section, we introduce some notations for readers’ convenience. In section 2, we will introduce some known results about the diffusion waves, and the properties of solu- tions to the Dirichlet/Neumann IBVP of linear damped wave equations. In section 3, inspired by [15], we first heuristically study the behavior of solution for the Dirichlet IBVP (1.1)-(1.3) at the far fieldx=∞, and the difference between the original solu- tion and the possible asymptotic profiles. We then construct the correction function to take care of the difference. Here, the correction function depends on the original initial data (1.2). Later on, after we show how to find the most suitable asymptotic profiles by carefully choosing the initial data for the IBVP of porous media equa- tions (1.13), we prove the convergence of the original IBVP solution (1.1)-(1.3) to the selected asymptotic profile (1.13). In Section 4, instead of the Dirichlet boundary condition (1.3), we consider the Neumann boundary caseJix|x=0 withE|x=∞= 0for the system (1.1). After constructing some correction function and determining the asymptotic profile of (1.14) with some specified initial datan¯0(x), we prove the con- vergence of the solutions for the Neumann IBVP (1.1), (1.2) and (1.4) to the chosen diffusion wave satisfying (1.14).
Notations. Throughout the paper, (¯n,J ,¯E)(x, t)¯ denotes the diffusion wave, that is, the solution of the IBVP (1.13) or (1.14), and (ˆn1,Jˆ1,nˆ2,Jˆ2,E)(x, t)ˆ de- notes the correction function. C > 0 denotes a generic constant, while Ci > 0 (i= 0,1,2,· · ·) represents some specific constant. The derivatives of f are denoted byfx,fxxor ∂xkf, k= 0,1,2,· · ·. Lp(R+) (1 ≤p≤ ∞)is the usual Lebesque space with the norm
kfkLp=Z
R+
|f(x)|pdx1/p
for 1≤p <∞, and kfkL∞ = sup
x∈R+
|f(x)|.
Sometimes, the integral regionR+ will be omitted for brevity. Lp,γ(R+)withγ >0 and1≤p≤ ∞is a weightedLp(R+)space with the weight(1 +x)γ. Its norm is
kfkLp,γ(R+)=Z
R+
(1 +x)γ|f(x)|pdx1/p
, 1≤p≤ ∞.
Hk(R+)(k≥0) is the usual Sobolev space with the norm
kfkHk =Xk
i=0
Z
R+
|∂xif|2dx1/2 .
For simplicity, we also denotek(f, g, h)k2L2 =kfk2L2+kgk2L2+khk2L2andk(f, g, h)k2Hk= kfk2Hk +kgk2Hk+khk2Hk. Let T >0 and B be a Banach space. C0([0, T];B) is the space of B-valued continuous functions on [0, T], andL2([0, T];B) is the space ofB- valued L2-functions on [0, T]. The other spaces of B-valued functions on [0,∞)can be defined similarly.
2. Preliminaries. In this section, we state some known results which will be used for the proof of the convergence in Section 3 and 4.
Lemma 2.1 ([19, 30]). Letn¯0−n+∈L1(R+)∩Hm(R+)for some positive integer m. Then the solution(¯n,J¯)(x, t)of the IBVP (1.13)globally and uniquely exists, and satisfies
k∂xk∂tj(¯n−n+)(t)kL2(R+)=O(1)δ2(1 +t)−4j+2k+14 , 0≤2k+j≤m, (2.1) k∂xk∂tj(¯n−n+)(t)kL∞(R+)=O(1)δ2(1 +t)−2j+k+12 , 0≤2k+j≤m, (2.2) k(¯n−n+)xt(t)kL1(R+)=O(1)δ2(1 +t)−32, (2.3) whereδ2:= maxx∈R+|¯n0(x)−n+|.
Furthermore, for the linear damped wave equation on the first quadrant with null Dirichlet boundary condition
φtt+αφt−µφxx=g(x, t), (x, t)∈R+×R+, (φ, φt)|t=0= (φ0, φ1)(x), x∈R+,
φ|x=0= 0, t∈R+,
(2.4)
whereα >0andµ >0, as shown in [30] (originally, we refer to [32] for the IVP case), its solution can be expressed as
φ(x, t) = Z ∞
0
[K0(x−y, t)−K0(x+y, t)]φ0(y)dy +
Z ∞ 0
[K1(x−y, t)−K1(x+y, t)]φ1(y)dy
+ Z t
0
Z ∞ 0
[K1(x−y, t−s)−K1(x+y, t−s)]g(y, s)dyds, (2.5) whereKj(x, t)(j= 0,1) are the fundamental solutions of the homogenous equation
∂ttKj+α∂tKj−µ∂xxKj = 0, x∈R, t∈R+, with
(K0(x,0) =δ(x),
∂
∂tK0(x,0) = 0, and
(K1(x,0) = 0,
∂
∂tK1(x,0) =δ(x),
whereδ(x)is the Dirac-Delta function. The Fourier transforms ofKi(x, t)(i= 0,1), denoted byKˆi(ξ, t)(i= 0,1), are given explicitly by
Kˆ1(ξ, t) =
2e−αt/2
√
α2−4µξ2sinh√
α2−4µξ2
2 t
, |ξ|< 2√αµ te−αt/2, |ξ|= 2√αµ
2e−αt/2
√
4µξ2−α2sin√
4µξ2−α2
2 t
, |ξ|> 2√α µ, and
Kˆ0(ξ, t) =α 2
Kˆ1(ξ, t) +R2(ξ, t), where
R2(ξ, t) =
e−αt/2cosh√
α2−4µξ2
2 t
, |ξ|<2√αµ e−αt/2, |ξ|=2√αµ e−αt/2cos√
4µξ2−α2
2 t
, |ξ|>2√α µ.
For the linear damped wave equation with the null Neumann boundary condition
φtt+αφt−µφxx=g(x, t), (x, t)∈R+×R+, (φ, φt)|t=0= (φ0, φ1)(x), x∈R+,
φx|x=0= 0, t∈R+,
(2.6)
its solution can be expressed as φ(x, t) =
Z ∞ 0
[K0(x−y, t) +K0(x+y, t)]φ0(y)dy +
Z ∞ 0
[K1(x−y, t) +K1(x+y, t)]φ1(y)dy
+ Z t
0
Z ∞ 0
[K1(x−y, t−s) +K1(x+y, t−s)]g(y, s)dyds. (2.7) The decay rates for the solutions are also obtained in [32] for the Cauchy problem, and extended in [30] for the Dirichlet IBVP (2.4). Without any difficulty, the same decay rates for the solution to the Neumann IBVP (2.6) can be derived similarly. Now we summarize these decay estimates as follows.
Lemma 2.2 ([32, 30]). If f ∈L1(R+)∩Hj+k(R+), then
∂tj∂xk
Z ∞ 0
[K1(x−y, t)±K1(x+y, t)]f(y)dy L2(
R+)
≤C(1 +t)−j−2k+14 h
kfkL1(R+)+kfkHj+k−1(R+)i
, (2.8)
∂tj∂xk
Z ∞ 0
[K1(x−y, t)±K1(x+y, t)]f(y)dy L∞(
R+)
≤C(1 +t)−j−k+12 h
kfkL1+kfkHj+k(R+)
i
. (2.9)
If f ∈L1(R+)∩Hj+k+1(R+), then
∂tj∂xk
Z ∞ 0
[K0(x−y, t)±K0(x+y, t)]f(y)dy L2(
R+)
≤C(1 +t)−j−2k+14 h
kfkL1+kfkHj+k(R+)
i, (2.10)
∂tj∂xk
Z ∞ 0
[K0(x−y, t)±K0(x+y, t)]f(y)dy L∞(R+)
≤C(1 +t)−j−k+12 h
kfkL1(R+)+kfkHj+k+1(R+)
i
. (2.11) Furthermore, as shown in [18], see also [28, 44], for the Dirichlet IBVP (2.4), we have the following faster decay estimates if the initial data belongs to the weighted spaceL1,γ(R+).
Lemma 2.3 ([18, 28, 44]). Let γ∈[0,1]. Iff ∈L1,γ(R+)∩Hj+k(R+), then
∂tj∂xk
Z ∞ 0
[K1(x−y, t)−K1(x+y, t)]f(y)dy L2(
R+)
≤C(1 +t)−j−2k+14 −γ2h
kfkL1,γ(R+)+kfkHj+k−1(R+)i
, (2.12)
∂tj∂xk
Z ∞ 0
[K1(x−y, t)−K1(x+y, t)]f(y)dy L∞(
R+)
≤C(1 +t)−j−k+12 −γ2h
kfkL1,γ(R+)+kfkHj+k(R+)
i
. (2.13)
If f ∈L1,γ(R+)∩Hj+k+1(R+), then
∂tj∂xk
Z ∞ 0
[K0(x−y, t)−K0(x+y, t)]f(y)dy L2(
R+)
≤C(1 +t)−j−2k+14 −γ2h
kfkL1,γ(R+)+kfkHj+k(R+)
i
, (2.14)
∂tj∂xk
Z ∞ 0
[K0(x−y, t)−K0(x+y, t)]f(y)dy L∞(
R+)
≤C(1 +t)−j−k+12 −γ2h
kfkL1,γ(R+)+kfkHj+k+1(R+)
i. (2.15)
Remark 1. As shown in [18], for the Neumann IBVP (2.6), the solution does not possess such faster decay rates (2.12)-(2.15), no matter the initial data is(φ0, φ1)∈ L1,γ(R+)or not.
Finally, we state a known and useful auxiliary lemma as follows.
Lemma 2.4 ([45]). Let a >0,b >0. Then Z t
0
(1 +t−s)−a(1 +s)−bds≤
C(1 +t)−min(a,b), max(a, b)>1, C(1 +t)−min(a,b)ln(2 +t), max(a, b) = 1, C(1 +t)1−a−b, max(a, b)<1.
(2.16)
3. Dirichlet initial-boundary value problem. In this section, we will first investigate the exact difference between the original solution of (1.1)-(1.3) and the diffusion wave of (1.13) at the far field. Then, as shown in [15], we will construct the correction functions to take care of this difference so that the estimation can be done in aL2-framework. Finally, for a given initial data (1.2) to the IBVP (1.1)-(1.3), we will specify the solution to the IBVP for the porous media equations (1.13) as the asymptotic profile of the original IBVP (1.1)-(1.3), and then prove the convergence of the solution to the IBVP (1.1)-(1.3) to the selected diffusion wave to (1.13).
3.1. Correction functions. In order to show the convergence of the solution (n1, J1, n2, J2, E)(x, t) of the IBVP (1.1)-(1.3) to the solution (¯n,J ,¯ ¯n,J ,¯ 0)(x, t) of the IBVP for the corresponding porous media equations, we now heuristically study the difference between these two solutions at the far field.
Let us first investigate the solution(n1, J1, n2, J2, E)(x, t)of the IBVP (1.1)-(1.3) at the far fieldx= +∞. Define
(n+1, J1+, n+2, J2+, E+)(t) := (n1, J1, n2, J2, E)(+∞, t). (3.1) Integrating (1.1)5 over[0,∞)with respect tox, we have
E+(t)−E(0, t) = Z
R+
[n1(x, t)−n2(x, t)]dx, (3.2) which implies, with the boundary conditionE|x=0= 0, that,
E+=E(+∞,0) =E+|t=0= Z
R+
[n10(x)−n20(x)]dx. (3.3) Noting the boundary conditions (1.3), and differentiating (1.1)5 with respect totand using (1.1)1 and (1.1)3, we get
d
dtE+(t) = Z
R+
∂t[n1(x, t)−n2(x, t)]dx=−[J1+(t)−J2+(t)]. (3.4) So, taking limit in the equations (1.1)1-(1.1)4 as x→+∞, and combining (3.4), we have
n+1(t) =n+, d
dtJ1+(t) =n+E+(t)−J1+(t), n+2(t) =n+,
d
dtJ2+(t) =−n+E+(t)−J2+(t), d
dtE+(t) =−J1+(t) +J2+(t), (J1+, J2+, E+)|t=0= (J1+, J2+, E+).
(3.5)
Differentiating (3.5)5 with respect to tand using (3.5)2 and (3.5)4, we obtain
d2
dt2E+(t) + d
dtE+(t) + 2n+E+(t) = 0 E+|t=0=E+,
dE+
dt |t=0=−(J1+−J2+),
(3.6)
which possesses the eigenvalues
λ1,2=−1±√
1−8n+
2 . (3.7)
By a similar computation but simpler than in [15] for the initial value problem case, we can solve the above ODEs in the following three cases.
Case 1: when1−8n+= 0, then J1+(t) =J1++J2+
2 e−t+1 2e−t2n
(J1+−J2+) +1
2[(J2+−J1+) +1 2E+]to
, (3.8) J2+(t) =J1++J2+
2 e−t−1 2e−t2n
(J1+−J2+) +1
2[(J2+−J1+) +1 2E+]to
, (3.9) E+(t) =e−t2n
E++ [J2+−J1++1 2E+]to
. (3.10)
Case 2: when1−8n+<0, then J1+(t) =1
2(J1++J2+)e−t+1 4e−t2n
2(J1+−J1+) cos
√8n+−1
2 t
+2(J2+−J1+) +E+
√8n+−1 +p
8n+−1E+
sin
√8n+−1 2 to
, (3.11) J2+(t) =1
2(J1++J2+)e−t−1 4e−t2n
2(J1+−J2+) cos
√8n+−1
2 t
+2(J2+−J1+) +E+
√8n+−1 +p
8n+−1E+
sin
√8n+−1 2 to
, (3.12) E+(t) =e−t2n
E+cos
√8n+−1
2 t+2(J2+−J1+) +E+
√8n+−1 sin
√8n+−1 2 to
.(3.13) Case 3: when1−8n+>0, then
J1+(t) =1
2(J1++J2+)e−t−1 2
n
λ1Aeλ1t+λ2Beλ2to
, (3.14)
J2+(t) =1
2(J1++J2+)e−t+1 2
n
λ1Aeλ1t+λ2Beλ2to
, (3.15)
E+(t) =Aeλ1t+Beλ2t, (3.16)
whereAandB are defined as
A= (J2+−J1+)−λ2E+
√1−8n+
, B=−(J2+−J1+)−λ1E+
√1−8n+
.
However, according to the properties of the solutions to the IBVP of porous media equations (1.13), the expected asymptotic profiles(¯n,J ,¯ n,¯ J ,¯ E)(x, t) = (¯¯ n,J ,¯n,¯ J ,¯0)(x, t) at far filedx= +∞are
(¯n,J ,¯n,¯ J ,¯E)(+∞, t) = (n¯ +,0, n+,0,0). (3.17) From (3.8)-(3.16) and (3.17), it holds
|ni(∞, t)−¯n(∞, t)|= 0, i= 1,2,
|Ji(+∞, t)−J(+∞, t)|¯ =O(1)e−ν0t6= 0, i= 1,2,
|E(+∞, t)−0|=O(1)e−ν0t6= 0,
for some constant 0 < ν0 < 12. So, there are some difference between the original solutions and their asymptotic profiles such that
Ji(x, t)−J¯(x, t) and E(x, t)6∈L2(R+).
Hence, we need to introduce some correction functions to take care of this differ-
ence. As shown in [15], we may similarly construct the correction functions(ˆn1,Jˆ1,ˆn2,Jˆ2,E)(x, t)ˆ such that
ˆ
n1t+ ˆJ1x= 0, Jˆ1t=n+Eˆ−Jˆ1, ˆ
n2t+ ˆJ2x= 0, Jˆ2t=−n+Eˆ−Jˆ2, Eˆx= ˆn1−nˆ2,
with
ˆ
ni(x, t)→0, Jˆi(x, t)→Ji+(t), E(x, t)ˆ →E+(t),
asx→ ∞, (3.18)
fori= 1,2. Let us select the initial data for the system (3.18) as follows (Jˆi(x,0) =Ji+Rx
0 m0(y)dy, i= 1,2, E(x,ˆ 0) =E+Rx
0 m0(y)dy, (3.19)
wherem0(x)is a smooth function satisfying m0(x)≥0, m0∈C0∞(R+),
Z
R+
m0(y)dy= 1.
Then the correction functions can be constructed as follows.
Case 1: when1−8n+= 0, then Jˆ1(x, t) =J1−e−t+1
2 n
(J2++J1+)e−t +e−t2h
(J1+−J2+) +1
2(J2+−J1++1
2E+)tioZ x 0
m0(y)dy, (3.20) Jˆ2(x, t) =J2−e−t+1
2 n
(J2++J1+)e−t
−e−t2h
J1+−J2++1
2[(J2+−J1+) +1
2E+]tioZ x 0
m0(y)dy, (3.21) E(x, t) =ˆ e−t2n
E++h
J2+−J1++1 2E+i
toZ x 0
m0(y)dy, (3.22)
ˆ
n1(x, t) = m0(x) 2
n(J2++J1+)e−t+e−t2
E++ (J2+−J1++E+ 2 )to
, (3.23) ˆ
n2(x, t) = m0(x) 2
n
(J2++J1+)e−t−e−t2
E++ (J2+−J1++E+
2 )to . (3.24) Case 2: when1−8n+<0, then
Jˆ1(x, t) = J2++J1+
2 e−t Z x
0
m0(y)dy+1 4e−t2n
2(J1+−J2+) cos
√8n+−1
2 t
+2(J2+−J1+) +E+
√8n+−1 +p
8n+−1E+
×sin
√8n+−1 2 toZ x
0
m0(y)dy, (3.25)
Jˆ2(x, t) = J2++J1+
2 e−t Z x
0
m0(y)dy−1 4e−t2n
2(J1+−J2+) cos
√8n+−1
2 t
+2(J2+−J1+) +E+
√8n+−1 +p
8n+−1E+
×sin
√8n+−1 2 toZ x
0
m0(y)dy, (3.26)
Eˆ+(x, t) =e−2tn E+cos
√8n+−1
2 t
+2(J2+−J1+) +E+
√8n+−1 sin
√8n+−1 2 toZ x
0
m0(y)dy, (3.27) ˆ
n1(x, t) = J2++J1+
2 m0(x)e−t+ 1
16n+m0(x)e−t2n
8n+E+cos
√8n+−1
2 t
+hp
8n+−1(E++ 2(J2+−J1+)) +2(J2+−J1+) +E+
√8n+−1 i
×sin
√8n+−1 2 to
, (3.28)
ˆ
n2(x, t) = J2++J1+
2 m0(x)e−t− 1
16n+m0(x)e−t2n
8n+E+cos
√8n+−1
2 t
+hp
8n+−1(E++ 2(J2+−J1+)) +2(J2+−J1+) +E+
√8n+−1 i
×sin
√8n+−1 2 to
. (3.29)
Case 3: when1−8n+>0, then Jˆ1(x, t) = J2++J1+
2 e−t Z x
−∞
m0(y)dy
−1 2
λ1Aeλ1t+λ2Beλ2tZ x 0
m0(y)dy, (3.30)
Jˆ2(x, t) = J2++J1+
2 e−t Z x
0
m0(y)dy +1
2
λ1Aeλ1t+λ2Beλ2tZ x 0
m0(y)dy, (3.31)
E(x, t) =ˆ
λ1Aeλ1t+λ2Beλ2tZ x 0
m0(y)dy, (3.32)
ˆ
n1(x, t) = 1
2(J2++J1+)m0(x)e−t+1
2m0(x)
λ1Aeλ1t+λ2Beλ2t
, (3.33) ˆ
n2(x, t) = 1
2(J2++J1+)m0(x)e−t−1
2m0(x)
λ1Aeλ1t+λ2Beλ2t
. (3.34) Summarizing the properties of these correction functions, we have
Lemma 3.1. It holds that
k(ˆn1,nˆ2,Jˆ1,Jˆ2,E)(t)kˆ L∞(R+)≤Cδ1e−ν0t, (3.35) (ˆn1,Jˆ1,nˆ2,Jˆ2,E)|ˆ x=0= (0,0,0,0,0), (3.36) (ˆn1,Jˆ1,nˆ2,Jˆ2,E)|ˆ x=∞= (0, J1+(t),0, J2+(t), E+(t)), (3.37) Z
R+
[ˆn1(x,0)−nˆ2(x,0)]dx= Z
R+
[n10(x)−n20(x)]dx=E+, (3.38) whereδ1:=|J1+|+|J2+|+|E+|.
3.2. Asymptotic profiles. Now we turn to show how to find the particular asymptotic profiles for the original solutions. Let us make a perturbation of the orig- inal system (1.1) around the specified diffusion waves (1.13) by adding the correction functions (3.18), then we obtain
(n1−nˆ1−n)¯ t+ (J1−Jˆ1−J)¯x= 0, (J1−Jˆ1−J)¯t+ (Jn12
1 +p(n1)−p(¯n))x
=n1E−n+Eˆ−(J1−Jˆ1−J¯) +p(¯n)xt, (n2−nˆ2−n)¯ t+ (J2−Jˆ2−J)¯x= 0,
(J2−Jˆ2−J)¯t+ (Jn22
2 +p(n2)−p(¯n))x
=−n2E+n+Eˆ−(J2−Jˆ2−J) +¯ p(¯n)xt, (E−E)ˆ x= (n1−nˆ1−n)¯ −(n2−nˆ2−n).¯
(3.39)
Integrating (3.39)1 and (3.39)3 over R+ with respect to x, and noting Ji(∞, t) = Jˆi(∞, t) = ¯J(∞, t) = 0,Ji(0, t) = ˆJi(0, t) = ¯J(0, t) = 0 fori= 1,2, we have
d dt
Z
R+
[ni(x, t)−nˆi(x, t)−¯n(x, t)]dx= 0, i= 1,2.
Integrating the above equation with respect tot, we further have Z
R+
[ni(x, t)−ˆni(x, t)−n(x, t)]dx¯ = Z
R+
[ni0(x)−nˆi(x,0)−¯n0(x)]dx= 0, (3.40) by selecting the initial data ¯n0(x)for the IBVP porous media equations (1.15) such that
Z
R+
[ni0(x)−nˆi(x,0)−¯n0(x)]dx= 0, i= 1,2, (3.41) namely,
Z
R+
[ni0(x)−n¯0(x)]dx= Z
R+
ˆ
ni(x,0)dx, i= 1,2. (3.42) Here, the determined initial datan¯0(x)in (3.42) is consistent for the pairs of(n10(x), ˆ
n1(x,0)) and (n20(x),nˆ2(x,0)), because, from (3.38), with the same selected initial datan¯0(x),
Z
R+
[n10(x)−n¯0(x)]dx= Z
R+
ˆ
n2(x,0)dx is equivalent to
Z
R+
[n20(x)−n¯0(x)]dx= Z
R+
ˆ
n1(x,0)dx.
With this preparation, we can study the perturbation around the diffusion wave in theL2-framework:
ni−n¯−nˆi, Ji−J¯−Jˆi, E−Eˆ∈L2(R+), i= 1,2.
Claim 1. The asymptotic profile for the original solution(n1, J1, n2, J2, E)(x, t) of the Dirichlet IBVP (1.1)-(1.3)is (¯n,J ,¯n,¯ J ,¯0)(x, t), where (¯n,J¯)(x, t)is the solu- tion of the IBVP (1.13)with a specified initial data n¯0(x)satisfying (3.41).
3.3. Convergence theorem. Let us define
φi(x, t) :=
Z x 0
[ni(ξ, t)−ˆni(ξ, t)−¯n(ξ, t)]dξ, ψi(x, t) :=Ji(x, t)−Jˆi(x, t)−J¯(x, t), i= 1,2, H(x, t) :=E(x, t)−E(x, t),ˆ
(3.43)
then we get the following equations for perturbation
φ1t+ψ1= 0,
ψ1t+(−ψ1t+ ˆJ1+ ¯J)2
φ1x+ ˆn1+ ¯n +p(φ1x+ ˆn1+ ¯n)−p(¯n)
x
= (φ1x+ ˆn1+ ¯n)H+ (φ1x+ ˆn1+ ¯n−n+) ˆE−ψ1+p(¯n)xt, φ2t+ψ2= 0,
ψ2t+(−ψ2t+ ˆJ2+ ¯J)2
φ2x+ ˆn2+ ¯n +p(φ2x+ ˆn2+ ¯n)−p(¯n)
x
=−(φ2x+ ˆn2+ ¯n)H −(φ2x+ ˆn2+ ¯n−n+) ˆE−ψ2+p(¯n)xt, H=φ1−φ2,
(3.44)
with the initial data
φi0(x) :=φi(x,0) = Z x
0
[ni0(ξ)−nˆi(ξ,0)−n¯0(ξ)]dξ, ψi0(x) :=ψi(x,0) =Ji0(x)−Jˆi(x,0)−J(x, t),¯ i= 1,2, H0(x) :=φ10(x)−φ20(x),
(3.45)
and the boundary conditions
((φi, ψi,H)|x=0= 0,
(φi, ψi,H)|x=∞= 0, i= 1,2. (3.46) Substituting (3.44)1 to (3.44)2 and (3.44)3 to (3.44)4 respectively, we have
φ1tt+φ1t−
p(φ1x+ ˆn1+ ¯n)−p(¯n)
x+ (φ1x+ ˆn1+ ¯n)H
=−f1+g1x−p(¯n)xt, (3.47)
φ2tt+φ2t−
p(φ2x+ ˆn2+ ¯n)−p(¯n)
x
−(φ2x+ ˆn2+ ¯n)H
=f2+g2x−p(¯n)xt, (3.48)
where
fi= [φix+ (ˆni−n+) + ¯n] ˆE, gi=(−φit+ ˆJi+ ¯J)
φix+ ˆni+ ¯n , i= 1,2, (3.49) with the initial-boundary values
φi|t=0=φi0(x), φit|t=0=−ψi0(x), φi|x=0= 0, φi|x=∞= 0, i= 1,2. (3.50) Furthermore, subtracting (3.48) from (3.47), we get
Htt+Ht+ 2¯nH=h:=h1x−h2−h3+h4x, H|t=0=:H0(x),
Ht|t=0=−[ψ10(x)−ψ20(0)] =:H1(x), H|x=0= 0,
(3.51)
