ESAIM: Control, Optimisation and Calculus of Variations
DOI:10.1051/cocv/2011191 www.esaim-cocv.org
ASYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
Ryo Kobayashi
1,2, Masakazu Yamamoto
3and Shuichi Kawashima
4Abstract. We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole spaceRnadmits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. Also, we show the sharp decay estimate for the perturbation. The stability proof is based on the time weighted Lpenergy method.
Mathematics Subject Classification. 35K45, 35B35, 82D10, 82D37.
Received September 26, 2011.
Published online 16 January 2012.
1. Introduction
We study the following drift-diffusion model arising in semiconductor device simulation and plasma physics:
⎧⎪
⎪⎨
⎪⎪
⎩
ut−Δu+∇ ·(u∇ψ) = 0, (1.1a)
vt−Δv− ∇ ·(v∇ψ) = 0, (1.1b)
−Δψ=−(u−v) +g(x), (1.1c)
with the initial conditions
u(x,0) =u0(x), v(x,0) =v0(x). (1.2)
Here u=u(x, t) andv=v(x, t) denote the electron and the hole densities, respectively, in the semiconductor, whileψ=ψ(x, t) is the electric potential andg=g(x) is the impurity doping profile.
Keywords and phrases.Drift-diffusion model, stability, decay estimates, weighted energy method.
1 Graduate School of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan.
2 Information Systems Department, Information & Communication Devision, Kyushu Electric Power Co. Inc., 810-8720 Fukuoka, Japan.
3 Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan.yamamoto@math.tohoku.ac.jp
4 Faculty of Mathematics, Kyushu University, 819-0395 Fukuoka, Japan.kawashim@math.kyushu-u.ac.jp
Article published by EDP Sciences c EDP Sciences, SMAI 2012
The initial value problem (1.1), (1.2) in the whole spaceRn was considered by Kurokiba and Ogawa in [12].
They showed the global existence of solutions for nonnegative initial data (u0, v0)(x) and forg(x) inLp spaces.
Asymptotic behavior for t → ∞ of these global solutions was studied in [1,10] in a special situation where g(x) = 0. We know that these global solutions decay to zero inLp norm at the ratet−(n/2)(1−1/p) ast → ∞, provided that the initial data are inL1∩L∞. For the details, we refer the reader to [1,10].
Our drift-diffusion model (1.1) is a parabolic-elliptic system. Similar parabolic-elliptic systems also appear in other models, such as an astrophysical model (a model of gravitating particles) and a model of chemotaxis (see, for example, [3,5,9,20] and references therein). For mathematical theory of those models, we refer the reader to [2–4,15,17] and references therein.
In this paper we study (1.1), (1.2) forg(x) satisfying g(x)→g∞
as|x| → ∞, whereg∞is a real constant state. For the initial data, we assume thatu0(x)→u∞andv0(x)→v∞ as|x| → ∞, whereu∞andv∞are nonnegative constants. In order to discuss the asymptotic behavior of solutions to (1.1), (1.2) in this situation, we need to study the corresponding stationary problem in the whole spaceRn:
⎧⎪
⎪⎨
⎪⎪
⎩
−Δu+∇ ·(u∇ψ) = 0, (1.3a)
−Δv− ∇ ·(v∇ψ) = 0, (1.3b)
−Δψ=−(u−v) +g(x), (1.3c)
with the requirements
u(x)→u∞, v(x)→v∞, ψ(x)→0
as |x| → ∞, whereu∞ andv∞ are the above nonnegative constants. It is also assumed that the derivatives of our stationary solutions tend to zero as|x| → ∞. In this situation we must have
−(u∞−v∞) +g∞= 0. (1.4)
The stationary problems for (1.3) in a bounded domain Ω with natural boundary conditions on∂Ω were considered in many papers. It is well known that these stationary problems admit unique solutions in the space L2(Ω) or Lp(Ω). Moreover, these stationary solutions are asymptotically stable in the sense that the time- dependent solutions to the corresponding initial-boundary value problem for (1.1) converge to these stationary solutions exponentially ast→ ∞. See, for example, [1,14].
Our stationary equations (1.3) in the whole spaceRncan be reduced to a single equation. This can be verified as follows. We rewrite the first equation (1.3a) as∇·(eψ∇(ue−ψ)) = 0. We multiply this equation byue−ψ−u∞ and integrate overRn. This givesue−ψ =u∞. Similarly, we get from (1.3b) thatveψ =v∞. Consequently, we have
u=u∞eψ, v=v∞e−ψ. (1.5)
Substituting these relations in (1.3c) and subtracting (1.4), we obtain
−Δψ=−u∞(eψ−1)−v∞(1−e−ψ) +g(x)−g∞. (1.6) This is the reduced stationary equation in which u∞ andv∞ can be considered as nonnegative parameters of the problem.
In this paper we first show the existence and uniqueness of stationary solutions to (1.6) withu∞, v∞ > 0 in the whole space Rn. This is an improvement on our previous result obtained in [11] under the restrictions u∞ =v∞ > 0 andg∞ = 0. As in [11], our existence proof is based on a fixed point theorem of the Leray- Schauder type (called the Browder-Potter fixed point theorem [18]). A crucial point of the proof is to derive the a priori estimate of stationary solutions and this can be done by using the weightedLp energy method.
SYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
The second purpose of this paper is to show the asymptotic stability of the above stationary solution when n ≥ 3. We prove that a unique solution to the initial value problem (1.1), (1.2) exists globally in time and converges to the corresponding stationary solution as t → ∞, provided that the amplitude of the stationary solution and the initial perturbation are suitably small. This stability result is based on theLp energy method.
Moreover, by employing the time weightedLpenergy method (which is a modification of the one used in [8,10]), we obtain the rate of convergence toward the stationary solution. When the initial perturbation is inL2∩Lq forqwith 2≤q < n, the convergence rate obtained inLpnorm ist−(n/2)(1/2−1/p)for 2≤p≤qand this rate is just equal to the optimalLp-L2 decay rate for the heat equation (see Thm.4.3). On the other hand, we could not prove such a sharp decay estimate forn≤q <∞andn≤p≤q (see Thm.4.5). This provides a striking contrast to the corresponding result for g(x) = 0 in [10]. Our result may suggest that p = n is the critical exponentin showing the optimal decay inLp for our problem.
This paper consists of five sections. In Section 2, we introduce several inequalities which are used in this paper. We show the existence and uniqueness of stationary solutions in Section3. In Section 4, we discuss the asymptotic stability of stationary solutions inLp (2≤p <∞). Finally in Section 5, we consider the decay of the derivative of perturbations in order to derive the asymptotic stability of stationary solutions inL∞. Notations. For x = (x1, . . . , xn) ∈ Rn, we denote by ∂xi the differentiation with respect to xi. Also, for a nonnegative integerk,∂kx denotes the totality of all thek-th order differentiations with respect tox∈Rn. The symbols∇=∂x= (∂x1, . . . , ∂xn) andΔ=n
i=1∂x2idenote the gradient and the Laplacian in then-dimensional space, respectively.
For 1 ≤ p ≤ ∞, Lp = Lp(Rn) denotes the usual Lebesgue space on Rn with the norm · Lp. Let s be a nonnegative integer. Then the corresponding Sobolev space Ws,p = Ws,p(Rn) is defined by Ws,p = {f ∈ Lp; ∂xkf ∈ Lp for k ≤ s}. When p = 2, we write Hs = Ws,2. For α ∈ R, Lpα = Lpα(Rn) denotes the weighted Lp space onRn, which consists of functions f satisfying (1 +|x|)αf ∈ Lp, equipped with the norm
f Lp
α = (1 +|x|)αf Lp. In particular, for 1≤p <∞, we have f Lpα=
Rn(1 +|x|)αp|f(x)|pdx 1/p.
The corresponding weighted Sobolev space Wαs,p = Wαs,p(Rn) is defined asWαs,p = {f ∈ Lpα; ∂xkf ∈ Lpα for k≤s}. We writeHαs=Wαs,2 forp= 2.
In this paper, various positive constants are denoted byC orcwithout confusion.
2. Preliminaries
In this section we give several preliminary inequalities used in the paper. First, we consider the Poisson equation
−Δψ=f. (2.1)
The corresponding fundamental solutionK(x) is given by
K(x) =
⎧⎨
⎩
(n−2)|S1 n−1||x|−(n−2)(n≥3),
2π1 log|x| (n= 2),
where|Sn−1|= 2πn/2/Γ(n/2) is the surface integral of the (n−1)-dimensional unit ball. Then the solution to (2.1) is given formally asψ(x) = (K∗f)(x) so that we have formally
∇ψ(x) = (∇K∗f)(x). (2.2)
In this paper we always define the gradient of the solution to the Poisson equation (2.1) by the formula (2.2).
Then, applying Hardy-Littlewood-Sobolev inequality, we have:
Lemma 2.1 (special Hardy-Littlewood-Sobolev inequality ([19,21]). Let n ≥ 2, 1 < r < n and 1/r∗ = 1/r−1/n. Ifψ is a solution to the Poisson equation(2.1), then we have
∇ψ Lr∗ ≤C f Lr. (2.3)
Also we have the following elliptic estimate in the weighted spaceLpα.
Lemma 2.2 ([6,13]). Let n≥1,1< p <∞ and−n/p < α < n(1−1/p). Then, forψ∈L1loc withΔψ∈Lpα, we have
∂x2ψ Lpα≤C Δψ Lpα. (2.4)
Next, we list up several interpolation inequalities which are frequently used in this paper.
Lemma 2.3 (Gagliardo-Nirenberg inequality [16]). Let n ≥1. Let 1 ≤ p, q, r≤ ∞, and let k be a positive integer. Then for any integerj with 0≤j < k, we have
∂xju Lp≤C ∂xku aLq u 1−aLr , (2.5) where
1 p = j
n+a 1
q −k
n + (1−a)1 r for asatisfying j/k≤a≤1; there are the following exceptional cases:
(i) if j = 0,qk < nandr=∞, then we made the additional assumption that either u(x)→0 as|x| → ∞ or u∈Lq for some 0< q <∞;
(ii) if1< r <∞, andk−j−n/r is a nonnegative integer, then (2.5)holds only for asatisfyingj/k≤a <1.
As a special case of (2.5), we have the following estimate forn≥2 and 1< r < n:
u Lr∗ ≤C ∂xu Lr, (2.6)
where 1/r∗= 1/r−1/n.
Lemma 2.4 ([7,8]). Let n≥1. Then we have
u Lp≤C ∇(|u|p/2) 2γ/(1+γp)L2 u 1/(1+γp)Lq (2.7) for 2≤p <∞and1≤q≤p, whereγ= (n/2)(1/q−1/p).
Lemma 2.5 ([8]). Let n≥1. Then we have
∂u Lp≤C ∂(|∂u|p/2) 2/(p+2)L2 u 2/(p+2)Lp (2.8) for 2≤p <∞and
∂u p/2Lp ≤C ∂(|∂u|p/2) 1−2/pL2 ∂(|u|p/2) 2/pL2
for p= 2and4≤p <∞, where∂=∂xi fori= 1, . . . , n.
SYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
3. Stationary solutions
We study the stationary equation (1.6),i.e.,
−Δψ=−u∞(eψ−1)−v∞(1−e−ψ) + ˜g(x) (3.1) in the whole space Rn, where ˜g(x) = g(x)−g∞; g(x) is a given function, and u∞, v∞ and g∞ are constants satisfyingu∞, v∞>0 and (1.4). It is known from [11] that this equation has a unique solution in the special case whereu∞=v∞>0 andg∞= 0. Here we prove the existence and uniqueness of solutions to (3.1) without this restriction. We put
F[ψ] =u∞(eψ−1) +v∞(1−e−ψ) (3.2)
and rewrite (3.1) as
−Δψ+F[ψ] = ˜g(x). (3.3)
First, similarly to [11], we show the uniqueness of solutions.
Theorem 3.1 (uniqueness). Let u∞>0 and v∞ >0. Let n≥1 and 2≤p <∞, and suppose that g˜∈Lp. Then the solutions to equation (3.1)are unique in the space W1,p∩L∞.
Proof. Equation (3.3) is rewritten as
−Δψ+a1ψ+G[ψ] = ˜g(x), wherea1= 2√u∞v∞ and
G[ψ] =F[ψ]−a1ψ=u∞(eψ−1) +v∞(1−e−ψ)−2√u∞v∞ψ. (3.4) We notice thatGsatisfies (a−b)(G[a]−G[b])≥0 for alla, b∈Rsince the inequalityG[a]≥0 holds. Let ψ1 andψ2 be solutions to (3.1). Then the differenceψ=ψ1−ψ2satisfies the equation
−Δψ+a1ψ+ (G[ψ1]−G[ψ2]) = 0. (3.5)
We multiply (3.5) by|ψ|p−2ψto obtain
a1|ψ|p+c0|∇(|ψ|p/2)|2+|ψ|p−2(ψ1−ψ2)(G[ψ1]−G[ψ2])− ∇ ·(|ψ|p−2ψ∇ψ) = 0, wherec0= 4(p−1)/p2. Integrating overRn and noting that (ψ1−ψ2)(G[ψ1]−G[ψ2])≥0, we have
ψ pLp+ ∇(|ψ|p/2) 2L2 ≤0,
which shows thatψ= 0. This completes the proof.
Next, similarly to [11], we prove the existence of solutions to (3.1) by applying the fixed point theorem of the Leray-Schauder type (called the Browder-Potter fixed point theorem [18]). For lower dimensional case 1≤n≤3, we have the following existence theorem in the weightedL2spaces.
Theorem 3.2 (existence). Letu∞>0 andv∞>0. Let1≤n≤3 andα >0, and suppose thatg˜∈L2α. Then equation (3.1)has a unique solutionψ∈Hα2 (⊂L∞α) such that
ψ H2
α≤C ˜g L2
α. (3.6)
Moreover, if ˜g∈Hαs for an integer s≥1, then the solution verifies the additional regularityψ∈Hαs+2. For higher dimensional case n ≥ 2, we have a similar existence theorem in the weighted Lp spaces with n < p <∞.
Theorem 3.3 (existence). Let u∞ >0 and v∞ > 0. Let n ≥2, n < p <∞ and α > 0, and suppose that g˜∈Lpα. Then equation (3.1)has a unique solution ψ∈Wα1,p (⊂L∞α) with∇(|∂xψ|p/2)∈L2αp/2 such that
ψ p
Wα1,p+ ∇(|∂xψ|p/2) 2L2
αp/2≤C ˜g pLp
α. (3.7)
Moreover, if ˜g∈Wαs,p for an integers≥1, then the solution verifies the additional regularityψ∈Wαs+1,p with
∇(|∂xs+1ψ|p/2)∈L2αp/2.
Remark 3.4. When ˜g∈L2α∩Lpαin the above theorems, the solutionψis inHα2∩Wα1,pwith∇(|∂xψ|p/2)∈L2αp/2 and satisfies the estimates (3.6) and (3.7). Moreover, we have the regularity∂x2ψ∈Lpα, where 0≤α ≤αand α< n(1−1/p); this regularity follows from the fact thatΔψ∈Lpα and the estimate (2.4).
These existence theorems can be proved by the same method employed in [11] where the special caseu∞= v∞ >0,g∞= 0 is treated. So we will only give the outline of the proof. As in [11], we use the following fixed point theorem of the Leray-Schauder type.
Theorem 3.5 (Browder-Potter [18]). Let X be a Banach space and let S be a closed convex subset of X. Let Φλ(ψ) =Φ(ψ, λ)be a continuous mapping of (ψ, λ)∈S×[0,1]into a compact subset of X, with the following properties:
(i) Φ0(∂S)⊂S;
(ii) for eachλ∈[0,1],Φλ has no fixed point on ∂S.
Then the mappingΦ1 has a fixed point in S.
LetXp be a Banach space defined byXp=Lpα/2∩L∞α/2 with the norm · Xp= · Lpα/2+ · L∞α/2.
We takeX =X2andX =Xpwithn < p <∞to prove Theorems3.2and3.3, respectively. We choose a closed convex subsetS as S ={ψ∈Xp; ψ Xp ≤M}, whereM is a suitably large number. We need to define the corresponding mapping to apply the above fixed point theorem. For this purpose, we rewrite (3.1) as
−Δψ+ (u∞+v∞)ψ=−H[ψ] + ˜g, (3.8)
whereH[ψ] is the nolinear part ofF[ψ] in (3.2) and is given by H[ψ] =F[ψ]−(u∞+v∞)ψ
=u∞(eψ−1) +v∞(1−e−ψ)−(u∞+v∞)ψ. (3.9) We introduce a parameterλ∈[0,1] and modify the equation (3.8) as
−Δψ+ (u∞+v∞)ψ=λ(−H[ψ] + ˜g). (3.10)
This equation can be transformed to
ψ=λ{−Δ+ (u∞+v∞)}−1(−H[ψ] + ˜g).
The desired mappingΦλ(ψ) is then defined as
Φλ(ψ) :=λ{−Δ+ (u∞+v∞)}−1(−H[ψ] + ˜g),
so that our solutionψto equation (3.1) can be obtained as a fixed point of the mappingΦ1, that is,ψ=Φ1(ψ).
SYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
Let 0≤β < α. Then we see that the imbeddingHα2⊂L2β∩L∞β is compact for 1≤n≤3. Also, the imbedding Wα1,p ⊂ Lpβ∩L∞β is compact for n ≥ 2 andn < p < ∞. Therefore, as in [11], we can verify that our Φλ is a continuous mapping of (ψ, λ) ∈ S×[0,1] into a compact subset of X, whereX and S are defined above.
Also, our Φλ satisfies the first condition of Theorem3.5 because of Φ0 = 0. To check the second condition of Theorem 3.5, we will show the a priori estimate of solutions to (3.10) given in Proposition 3.6 below. Once this is done, our Φλ satisfies the second condition of Theorem3.5 forS with suitably large M. Consequently, Theorem3.5is applicable to our problem and we have a fixed pointψ∈Sof the mappingΦ1. This fixed pointψ is a desired stationary solution stated in Theorems3.2and3.3. For the details, we refer the reader to [11].
Therefore, for the proof of Theorems3.2and3.3, it suffices to show the followinga prioriestimate of solutions to equation (3.10).
Proposition 3.6. Let u∞ >0,v∞ >0 and λ∈[0,1]. Let n ≥1, 2 ≤p <∞ and β ≥0, and suppose that g˜∈Lpβ. Letψbe a solution to the nonlinear equation (3.10)such thatψ∈Lpβ/2∩L∞β/2. Then we have ψ∈Wβ1,p and∇(|∂xψ|p/2)∈L2βp/2. Moreover, the solution satisfies the a priori estimate
ψ p
Wβ1,p+ ∇(|∂xψ|p/2) 2L2
βp/2 ≤Cλp ˜g pLp
β, (3.11)
whereC is a positive constant independent of λ.
Proof. Let ˜g ∈ Lpβ and let ψ ∈ Lpβ/2∩L∞β/2 be a solution to the equation (3.10). We have from (3.9) that
|H[ψ]| ≤(1/2)(u∞+v∞)|ψ|2e|ψ|. ThereforeH[ψ] is inLpβ and satisfies H[ψ] Lpβ ≤(1/2)(u∞+v∞) ψ L∞
β/2 ψ Lp
β/2eψL∞.
Consequently, we see that the right hand side of (3.10) belongs toLpβ. Therefore, in the same way as in [11], we conclude thatψ∈Wβ1,pand∇(|∂xψ|p/2)∈L2βp/2.
To prove thea prioriestimate (3.11), we employ the weightedLpenergy method developed in [11]. The proof is divided into two parts.
Step 1. Let 2≤p <∞andβ ≥0. We first show that ψ pLp
β+ ∇(|ψ|p/2) 2L2
βp/2 ≤Cλp ˜g pLp
β, (3.12)
whereC is a positive constant independent ofλ. To prove this, we rewrite the equation (3.10) as
−Δψ+aλψ+λG[ψ] =λ˜g, (3.13)
where
aλ=λa1+ (1−λ)a0, a0=u∞+v∞, a1= 2√u∞v∞,
andG[ψ] is defined in (3.4). Notice that aλ ≥a1>0. We multiply (3.13) by|ψ|p−2ψand obtain c0|∇(|ψ|p/2)|2+aλ|ψ|p+λ|ψ|p−2ψG[ψ]− ∇ ·(|ψ|p−2ψ∇ψ) =λ|ψ|p−2ψ˜g, wherec0= 4(p−1)/p2. Furthermore, multiplying by (1 +|x|)βp, we have
c0(1 +|x|)βp|∇(|ψ|p/2)|2+aλ(1 +|x|)βp
|ψ|p+λ|ψ|p−2ψG[ψ]
− ∇ ·
(1 +|x|)βp|ψ|p−2ψ∇ψ
+βp(1 +|x|)βp−1|x|−1x· |ψ|p−2ψ∇ψ=λ(1 +|x|)βp|ψ|p−2ψ˜g.
Integrating this equality overRn and using ψG[ψ]≥0, we have aλ ψ pLp
β+c0 ∇(|ψ|p/2) 2L2
βp/2≤A1+A2, (3.14)
where
A1=βp
Rn(1 +|x|)βp−1|ψ|p−1|∇ψ|dx, A2=λ
Rn(1 +|x|)βp|ψ|p−1|˜g|dx. (3.15) We estimate the right hand side of (3.14). For the termA1, we have
A1= (β/2)
Rn(1 +|x|)βp−1|ψ|p/2|∇(|ψ|p/2)|dx
≤(β/2) ψ p/2Lp
β ∇(|ψ|p/2) L2
βp/2−1 ≤ε ψ pLp
β+β2Cε ∇(|ψ|p/2) 2L2 βp/2−1
for any ε >0, where Cε is a positive constant depending on εbut not on β. Similarly, we can estimate the termA2 as
A2≤λ ψ p−1Lp β ˜g Lp
β ≤ε ψ pLp
β+Cελp ˜g pLp β
for anyε >0, whereCεis a positive constant depending onεbut not onβ. Substituting these estimates in (3.14) and takingε >0 suitably small, we obtain
ψ pLp
β+ ∇(|ψ|p/2) 2L2
βp/2 ≤Cλp ˜g pLp
β+β2C ∇(|ψ|p/2) 2L2
βp/2−1. (3.16)
This proves (3.12) forβ = 0. Also, (3.16) together with (3.12) forβ= 0 gives (3.12) for 0< βp≤2. Repeating this procedure, we conclude that (3.12) holds true for anyβ≥0.
Step 2. Next we show that
∂xψ pLp
β+ ∇(|∂xψ|p/2) 2L2
βp/2≤Cλp ˜g pLp
β, (3.17)
where C is a positive constant independent of λ. To prove this, we differentiate (3.13) with respect to xi (i= 1, . . . , n), obtaining
−Δψi+aλψi+λ∂xiG[ψ] =λ∂xig,˜ (3.18) where we putψi =∂xiψ. We multiply (3.18) by |ψi|p−2ψito get
c0|∇(|ψi|p/2)|2+aλ|ψi|p+λ|ψi|p−2ψi∂xiG[ψ]
− ∇ ·(|ψi|p−2ψi∇ψi) =λ∂xi(|ψi|p−2ψig)˜ −λ(p−1)|ψi|p−2∂xiψi˜g, wherec0= 4(p−1)/p2. Furthermore, multiplying by (1 +|x|)βp, we have
c0(1 +|x|)βp|∇(|ψi|p/2)|2+ (1 +|x|)βp
aλ|ψi|p+λ|ψi|p−2ψi∂xiG[ψ]
− ∇ ·
(1 +|x|)βp|ψi|p−2ψi∇ψi
+βp(1 +|x|)βp−1|x|−1x· |ψi|p−2ψi∇ψi
=λ∂xi
(1 +|x|)βp|ψi|p−2ψig˜
−λβp(1 +|x|)βp−1|x|−1xi|ψi|p−2ψi˜g−λ(p−1)(1 +|x|)βp|ψi|p−2∂xiψi˜g.
Integrating this equality overRn and noting thatψi∂xiG[ψ] =G[ψ]|ψi|2≥0, we obtain aλ ψi pLp
β+c0 ∇(|ψi|p/2) 2L2
βp/2≤B1+B2+B3, (3.19)
where
B1=βp
Rn(1 +|x|)βp−1|ψi|p−1|∇ψi|dx, B2=λβp
Rn(1 +|x|)βp−1|ψi|p−1|˜g|dx, B3=λ(p−1)
Rn(1 +|x|)βp|ψi|p−2|∂xiψi| |˜g|dx.
SYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
We estimate the right hand side of (3.19). The termB1 is just the same asA1 in (3.15), so that we have B1≤ε ψi pLp
β+β2Cε ∇(|ψi|p/2) 2L2 βp/2−1
for anyε >0, whereCεis a positive constant depending onεbut not onβ. For the termB2, we have B2≤λβp ψi p−1Lp
β ˜g Lp
β−1 ≤ε ψi pLp
β+Cελpβp ˜g pLp β−1
for anyε >0, whereCεis a positive constant depending onεbut not onβ. On the other hand, we can estimate the termB3 as
B3=λ(2/p)(p−1)
Rn(1 +|x|)βp|ψi|p/2−1|∂xi(|ψi|p/2)| |˜g|dx
≤λ(2/p)(p−1) ψi p/2−1Lp
β ∂xi(|ψi|p/2) L2βp/2 ˜g Lp
β
≤ε ψi pLp
β+ ∇(|ψi|p/2) 2L2 βp/2
+Cελp ˜g pLp β
for any ε >0, where Cε is a positive constant depending on ε but not on β. Here we have used the H¨older inequality with (p−2)/2p+ 1/2 + 1/p= 1. Substituting all these estimates in (3.19) and takingε >0 suitably small, we obtain
∂xψ pLp
β+ ∇(|∂xψ|p/2) 2L2
βp/2≤Cλp(1+βp) ˜g pLp
β+β2C ∇(|∂xψ|p/2) 2L2 βp/2−1. This gives (3.17) forβ= 0 and hence for anyβ ≥0.
Now, the desired estimate (3.11) follows from (3.12) and (3.17), and therefore the proof of Proposition3.6is
complete.
4. Asymptotic stability
In this section we discuss the asymptotic stability of stationary solutions to the drift-diffusion model in Rn with n ≥ 3. We denote by (¯u,¯v,ψ)(x) the stationary solution constructed in Theorems¯ 3.2 and 3.3 for g˜∈L2α∩Lqα0 withq0> nandα >0. This stationary solution satisfies the estimates
(¯u−u∞,v¯−v∞,ψ) ¯ W1,r ≤C ˜g for 2≤r≤ ∞,
(¯u−u∞,v¯−v∞,ψ) ¯ W2,p ≤C ˜g for 2≤p≤q0, (4.1) provided that
˜g := ˜g L2+ ˜g Lq0 (4.2)
is bounded by a constantC. In fact, it follows from (3.6) and (3.7) that ψ¯ W1,p≤C g˜ for 2≤p≤q0. Since q0 > n, this together with (2.5) shows that ψ¯ L∞ ≤C g˜ . Therefore, by virtue of equation (3.1), we know that Δψ¯ Lp ≤C ˜g and hence ∂x2ψ¯ Lp ≤ C ˜g for 2 ≤p ≤ q0 by (2.4) (see also [19]). Consequently, we have ψ¯ W2,p ≤C ˜g for 2≤p≤q0, which together with (1.5) shows that (¯u−u∞,v¯−v∞) W2,p ≤C ˜g for 2 ≤p≤q0. Thus we have proved the second estimate in (4.1). Also, this combined with (2.5) yields the first estimate in (4.1).
Now we look for solutions to the nonstationary problem (1.1), (1.2) in the form (u, v, ψ) = (¯u,¯v,ψ)(x) + (w, z, φ)(x, t),¯
where (¯u,¯v,ψ) is the above stationary solution and (w, z, φ) denotes the corresponding perturbation. The¯ problem is then reduced to
⎧⎪
⎨
⎪⎩
wt−Δw+u∞Δφ=−∇ ·f1− ∇ ·(w∇φ), (4.3a) zt−Δz−v∞Δφ=∇ ·f2+∇ ·(z∇φ), (4.3b)
−Δφ=−(w−z), (4.3c)
w(x,0) =w0(x), z(x,0) =z0(x), (4.4)
wherew0=u0−u,¯ z0=v0−¯vand
f1=w∇ψ¯+ (¯u−u∞)∇φ, f2=z∇ψ¯+ (¯v−v∞)∇φ. (4.5) First we show the global existence and uniformLp estimate of solutions to the problem (4.3), (4.4).
Theorem 4.1 (global existence). Let n ≥ 3 and q1 ≤ q < ∞, where q1 = max{2, n/2}. Let (¯u,v,¯ ψ)¯ be the stationary solution satisfying (4.1). Suppose that the initial data (w0, z0) are in L2∩Lq and put E0 = (w0, z0) L2+ (w0, z0) Lq. IfE0+ ˜g is suitably small ( g˜ is defined in (4.2)), then the problem (4.3),(4.4) has a unique global solution (w, z, φ) which satisfies the following uniformLp estimate:
(w, z)(t) pLp+ t
0 ∇(|(w, z)|p/2)(τ) 2L2+ (w−z)(τ) pLpdτ ≤CE0p (4.6) for p= 2andq1≤p≤q. In particular, we have (w, z)(t) Lp≤CE0 for eachpwith 2≤p≤q.
The local existence of solutions to the problem (4.3), (4.4) can be proved by the standard method (cf.[12]).
Therefore, for the proof of Theorem 4.1, it suffices to show the a priori estimate of solutions to the prob- lem (4.3), (4.4). We use the following notations.
Ep(t) = sup
0≤τ≤t (w, z)(τ) Lp, Dp(t)p=
t
0 ∇(|(w, z)|p/2)(τ) 2L2+ (w−z)(τ) pLpdτ, where 2≤p <∞. Oura priori estimate is then given as follows.
Proposition 4.2 (a priori Lp estimate). Let n≥3andq1≤q <∞, whereq1 is the same as in Theorem4.1.
Let (¯u,¯v,ψ)¯ be the stationary solution satisfying (4.1)and let(w, z, φ)be a solution to the problem (4.3),(4.4) corresponding to the initial data (w0, z0) ∈ L2∩Lq. If E0+ ˜g in Theorem 4.1 is suitably small, then the solution (w, z, φ) satisfies the following a prioriLp estimate:
(Ep+Dp)(t)≤CE0 (4.7)
for p= 2andq1≤p≤q.
Proof. Letn≥3. The proof is based on theLp energy method employed in [8]. We divide the proof into four parts.
Step 1. Let 2≤p≤q. We derive the Lp energy inequality for the problem (4.3), (4.4). We multiply (4.3a) by|w|p−2w. A straightforward computation gives
1 p
|w|p
t+c0|∇(|w|p/2)|2+u∞|w|p−2w(w−z)− ∇ ·(|w|p−2w∇w)
=−∇ ·
|w|p−2wf1+|w|p∇φ
+ (p−1)
|w|p−2∇w·f1+|w|p−2w∇w· ∇φ
, (4.8)
SYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE DRIFT-DIFFUSION MODEL IN THE WHOLE SPACE
wherec0= 4(p−1)/p2. Here we have used (4.3c). We integrate (4.8) overRn to obtain 1
p d
dt w pLp+c0 ∇(|w|p/2) 2L2+u∞
Rn|w|p−2w(w−z) dx≤(p−1)(I+J), (4.9) where we put
I=
Rn|w|p−2|∇w| |f1|dx, J =
Rn|w|p−1|∇w| |∇φ|dx. (4.10) It follows from (4.5) thatI≤I1+I2 with
I1=
Rn|∇ψ¯| |w|p−1|∇w|dx, I2=
Rn|¯u−u∞| |w|p−2|∇w| |∇φ|dx. (4.11) We integrate (4.9) with respect tot. This yields
w(t) pLp+ t
0 ∇(|w|p/2)(τ) 2L2dτ+ t
0
Rn|w|p−2w(w−z)(τ) dxdτ ≤C w0 pLp+C t
0
(I+J)(τ) dτ, (4.12) where we have usedu∞>0. We have a similar energy inequality also forz.
Step 2. Letq1≤p≤q. We show the followingLpenergy inequality:
(Ep+Dp)(t)p≤CE0p+C ˜g (E2+D2+Dp)(t)p+C(Ep+D2+Dp)(t)p+1. (4.13) We use (4.12). First, applying the H¨older inequality with 1/n+ 1/2∗+ 1/2 = 1 (where 1/2∗= 1/2−1/n) and using (2.6), we estimate the term I1 as
I1= (2/p)
Rn|∇ψ¯| |w|p/2|∇(|w|p/2)|dx
≤C ∇ψ¯ Ln (|w|p/2) L2∗ ∇(|w|p/2) L2 ≤C ˜g ∇(|w|p/2) 2L2. (4.14)
Thus we obtain t
0
I1(τ) dτ ≤C ˜g t
0 ∇(|w|p/2)(τ) 2L2dτ ≤C ˜g Dp(t)p. Next we estimate the termI2. We determinerwith 1< r < nand 2< r < p such that
1 r = θ
2+1−θ
p , θ= 2
n· (4.15)
For this choice ofr, we see that 1/n+(1−2/p)/2∗+1/2+1/r∗= 1, where 1/2∗= 1/2−1/nand 1/r∗= 1/r−1/n.
Applying the H¨older inequality with this relation and using (2.6) and (2.3), we have I2= (2/p)
Rn|¯u−u∞| |w|(p/2)(1−2/p)|∇(|w|p/2)| |∇φ|dx
≤C ¯u−u∞ Ln (|w|p/2) 1−2/pL2∗ ∇(|w|p/2) L2 ∇φ Lr∗
≤C ˜g ∇(|w|p/2) 2(1−1/p)L2 w−z Lr
≤C ˜g ∇(|w|p/2) 2(1−1/p)L2 w−z θL2 w−z 1−θLp , (4.16)
