(1.1) (n 2 u 2 ) t + div(n 2 u 2 u 2 ) + q(n 2 ) = n 2 Ψ n 2 u 2,

Transcription

1 LAGE TIME BEHAVIO OF SOLUTIONS TO N-DIMENSIONAL BIPOLA HYDODYNAMIC MODEL FO SEMICONDUCTOS FEIMIN HUANG, MING MEI, AND YONG WANG Deicate to Professor Akitaka Matsumura on his 6th birthay Abstract. In this paper, we stuy the n-imensional (n ) bipolar hyroynamic moel for semiconuctors in the form of Euler-Poisson equations. In -D case, when the ifference between the initial electron mass an the initial hole mass is non-zero (switch-on case), the stability of nonlinear iffusion wave has been open for a long time. In orer to overcome this ifficulty, we ingeniously construct some new correction functions to elete the gaps between the original solutions an the iffusion waves in L -space, so that we can eal with the one imensional case for general perturbations, an prove the L -stability of iffusion waves in -D case. The optimal convergence rates are also obtaine. Furthermore, base on the -D results, we establish some crucial energy estimates an apply a new but key inequality to prove the stability of planar iffusion waves in n-d case. This is the first result for the multi-imensional bipolar hyroynamic moel of semiconuctors. Key wors. bipolar hyroynamic moel, semiconuctor, nonlinear amping, (planar) nonlinear iffusion waves, asymptotic behavior, convergence rates. AMS subject classifications. 35L5, 35L6, 35L65, 765. Introuction. In this paper, we stuy the n-d isentropic Euler-Poisson equations for the bipolar hyroynamic moel of semiconuctor evice n t + iv(n u ) =, (n u ) t + iv(n u u ) + p(n ) = n Ψ n u, n t + iv(n u ) =, (.) (n u ) t + iv(n u u ) + q(n ) = n Ψ n u, Ψ = n n, for x = (x, x,, x n ) n, n, t >. Here n = n (x, t), n = n (x, t), u = (u,..., u n )(x, t), u = (u,..., u n )(x, t), an Ψ(x, t) represent the electron ensity, the hole ensity, the electron velocity, the hole velocity, an the electrostatic potential, respectively. E := Ψ(x, t) is calle electric fiel. The nonlinear functions p(s) an q(s) enote the pressures of the electrons an the holes, respectively, which are smooth, strictly increasing an nonnegative. Since both p(s) an q(s) posses the same characters, for simplicity, here an after here we assume them to be ientical, namely, p(s) = q(s), p (s) = q (s) > for s >. (.) Institute of Applie Mathematics, Acaemy of Mathematics an Systems Science, Chinese Acaemy of Sciences, Beijing, 9, China (fhuang@amt.ac.cn), supporte in part by NSFC Grant No. 85 for istinguishe youth scholar, NSFC-NSAF Grant No an 973 project of China uner Grant No. 6CB859. Department of Mathematics, Champlain College, Saint-Lambert, Quebec J4P 3P, Canaa, an Department of Mathematics an Statistics, McGill University, Montreal, Quebec H3A K6, Canaa (ming.mei@mcgill.ca), supporte in part by Natural Sciences an Engineering esearch Council of Canaa uner the NSEC grant GPIN Corresponing author. Institute of Applie Mathematics, Acaemy of Mathematics an Systems Science, Chinese Acaemy of Sciences, Beijing, 9, China (yongwang@amss.ac.cn).

2 F.-M. HUANG, M. MEI, AND Y. WANG If the pressures p(s) an q(s) are ifferent, this will be a new story, an we will leave it for future (see emark below for etails). Hyroynamic moels of this type are generally use in the escription of the charge flui particles. Examples are electrons an holes in semiconuctor evices or positively an negatively charge ions in a plasma. These moels, which can be erive from kinetic moels, take an important place in the fiels of applie an computational mathematics. A stanar approach for this erivation is the moment metho. Accoring to the ifferent analysis for the phase space ensities, introuce to prescribe the epenence on the velocity, we recover ifferent limit moels an, in particular, the rift-iffusion equations an the hyroynamic (Euler-Poisson) systems. The hyroynamic moels are usually consiere to escribe high fiel phenomena of submicronic evices. For etails on the semiconuctor applications, see [7, 7]. For the applications in plasma physics, see [7, 38]. For unipolar isentropic an nonisentropic hyroynamic equations of semiconuctors on the whole space or the spatial boune omain, the main effort was mae on the mathematical moellings [7, 7] an on the rigorous mathematical analysis, such as well-poseness of steay-state solutions [, 4, 5], an their stability [8,, 6, 9, 3, 33], the global existence of classical an/or the entropy weak solutions [,, 6, 37, 4], the large time behavior of solutions [, 9, ], an the zero relaxation limit problems [7, ], etc. For bipolar hyroynamic semiconuctor equations, however, the stuy is quite limite an far from being well, especially for the high-imensional case. In -D case, Natalini [3], an Hsiao an Zhang [, 3] establishe the global entropic weak solutions in the framework of compensate compactness on the whole real line an spatial boune omain, respectively. Hattori an Zhu [9] prove the stability of steay-state solutions for a recombine bipolar hyroynamic moel. Gasser, Hsiao an H. Li [6], an Huang an Y. Li [5] investigate the large time behavior of both small smooth an weak solutions, respectively. Furthermore, Y. Li [] stuie the relaxation limit of a bipolar isentropic hyroynamic moels for semiconuctors with small momentum relaxation time. But the stuy in n-d case for the bipolar hyroynamic system of semiconuctors is never ealt with, so far as we know. Physically, the frictional amping usually causes the ynamical system to possess the nonlinear iffusive phenomena. Such interesting phenomena for -D compressible Euler equations with amping was investigate firstly by Hsiao an Liu []. Since then, this problem has attracte consierable attention, for example, see [5, 3, 34, 35, 36, 39, 4] an the references therein, see also the new progress mae by Mei [3] for the selection of the best asymptotic profiles with much faster convergence rates, recently. Here, our main interest is to investigate such iffusive phenomena for the Euler-Poisson equations (.). For one imensional (.), we enote J = n u an J = n u as the current ensities for the electrons an the holes, respectively, an E = Ψ x as the electric fiel for the sake of simplification. Thus, the -D isentropic Euler-Poisson equations of (.) is written as follows n t + J x =, J t + ( J n + p(n )) x = n E J, n t + J x =, J t + ( J n + p(n )) x = n E J, E x = n n, (.3)

3 with the initial ata BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 3 (n, n, J, J ) t= = (n, n, J, J )(x), (.4) where (n, n, J, J )(x) (n ±, n ±, J ±, J ± ) as x ±, an (n ±, n ±, J ±, J ± ) are the state constants, an J i± = n i± u i±. The nonlinear iffusive phenomena both in smooth an weak senses were also observe for the bipolar hyroynamic moel (.3) by Gasser, Hsiao an H. Li [6] an Huang an Y. Li [5], respectively. Namely, accoring to the Darcy s law, it is expecte that the solutions of (.3) (n, J, n, J, E)(x, t) converge in L -sense to the so-calle nonlinear iffusion waves ( n, J, n, J, )(x, t), where ( n, J) = ( n, J)((x + x )/ + t) (x is a shift constant) are the self-similar solutions to the following equations n t = p( n) xx, porous meium equation J := nū = p( n) x, Darcy s Law (.5) ( n, J) (n ±, ), as x ±. Such -D self-similar solutions ( n, ū)(x / + t) = ( n, ū)(x e/ + t) with e = (,,, ) n are also calle the planar iffusion waves to the n-imensional equations (.). In [6, 5], they nee the ifference of the initial mass of the electrons an the initial mass of the holes to be zero [n (x ) n (x )]x = an J i+ = J i, i =,. This implies, from the last equation of (.3), that the ifference of the electric fiels is zero E(+, t) E(, t) =. However, such a conition is too stiff, because it looks like a switch-off situation for the evice (no voltage). The most interesting but challenging case is E(+, t) E(, t), or equivalently [n (x ) n (x )]x. The large-time behavior of the solutions for this case has been open for a long time. In this case, the correction functions use in [6] for eleting the gaps between the original solutions an the iffusion waves at far fiels, (originally, such an iea was first introuce by Hsiao an Liu in []), can not be applie anymore because of the effect of the electric fiel, an still leave the perturbation equations with big gaps which are not in L (). In orer to overcome such a ifficulty, by a eep observation, we first make an heuristic analysis on the electric fiel an the current ensities, i.e, E an J i for i =, at far fiels, an then ingeniously construct some new correction functions to elete the gaps yiele by the original solutions an the corresponing iffusion waves. Then, we can prove the stability of nonlinear iffusion waves for the 5 5 bipolar hyroynamic moel of semiconuctors (.) in -D case. Precisely, for the iffusion waves ( n, ū)((x + x )/ + t) with some shift constant x, when the initial perturbations aroun the waves are small enough, we can prove the stability of the shifte waves with the optimal rates in the form x k t(n l k+l i n)(t) L () = O()( + t), k, l, i =,, (.6)

4 4 F.-M. HUANG, M. MEI, AND Y. WANG k x l t(u i ū)(t) L () = O()( + t) 3 k+l, k, l, i =,. (.7) An particularly, we show the exponential ecay for the electronic ensity n towar the hole ensity n, an the electronic velocity u towar the hole velocity u, as well as the electric fiel E to in the form of k x l t(n n, u u, E)(t) L () = O()e ν t, k, l, for some ν >. (.8) This is our first contribution of the present paper. Obviously, the results presente in [6] is a special case of ours. From the structure of the correction functions, we fin also that it possesses much more interesting phenomena for non-zero mass case an is consistent with physical phenomena. For multi-imensional (.), there is no relevant literatures ealing with the stability of planar iffusion waves ue to some particular ifficulties. Firstly, the ifficulty comes from the complicate structure of the equations themselves. Seconly, the main ifficulty in the stuy is that the strategy of anti-erivative use in -D case is no longer effective for the n-d case. For the one-imensional problem, the anti-erivative strategy was successfully use to remove the a-priori assumption (c.f. [, 3] et al). However, for multi-imensional case, a irect generalization of the oneimensional iea leas to the implicity an complexity of the efine shift function which epens on the solutions, so that it oes not give a clear picture of the large time behavior of the solutions. To overcome this ifficulty, instea of taking antierivative of the perturbation to the ensity function, we apply a new an technical inequality, which was contribute by Huang, J. Li an Matsumura [4], to remove the a-priori assumption, then we can establish some key energy estimates to prove the stability of planar iffusion waves for the system of multi-imensional bipolar hyroynamic moel for semiconuctors. Namely, for the -D solutions of (.5) ( n, ū)(x, t), the so-calle planar iffusion waves to the n-d solutions (n, u, n, u, Ψ)(x, t) with x = (x, x,, x n ) n (for simplicity, we just consier n = 3) of (.) converge to the planar iffusion waves in the form (n n, n n)(t) L ( 3 ) = O()( + t) 3 4, (.9) (n n )(t) L ( 3 ) = O()( + t) 9 4, (.) ((u i, u i, u i3 ) (ū,, ))(t) L ( 3 ) = O()( + t) 5 4, i =,, (.) Ψ(t) L ( 3 ) = O()( + t) 7 4. (.) This is our secon contribution in the present paper. Note also that, such a stability result of planar iffusion waves for the multi-imensional isentropic Euler-Poisson equations is the first work as we know, so far. The rest of this paper is arrange as follows. In section, we give some wellknown results on the iffusion waves an one key inequality which will be use later for the stability proof in n-d case. In section 3, we prove the stability of iffusion waves in -D case. First of all, we trickily construct the correction functions to elete the gaps between the -D solutions an the -D iffusion waves at the far fiel, then we reformulate the original system of equations to a new one, an further prove the stability of iffusion waves by the energy metho. In section 4, the main purpose is to stuy the n-d case. Since the technique use in -D case is no long working for the n-d case, we employe a key inequality to establish some crucial energy estimates, then prove the stability of planar iffusion waves.

5 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 5 Notation. Through out this paper, the iffusion waves are always enote by ( n, J)(x / + t), an the gap functions (or say, correction functions) are enote by (ˆn, Ĵ, ˆn, Ĵ, Ê). C, Ci, c et al always enote some specific positive constants, an C enotes the generic positive constant. L ( n ) is the space of square integrable real value function efine on n with the norm, an H k ( n ) (H k without any ambiguity) enotes the usual Sobolev space with the norm k, especially =. Nonnegative multi-inex is α = (α,..., α n ) with orer α = α + + α n. Given a multi-inex α, we efine x α f(x) := x α... x αn n f(x). If k is a nonnegative integer, we efine k f(x) =: { α f(x) α = k}, an k f = ( α =k α f ) /.. Diffusion waves an some preliminaries. In this section, we are going to introuce some well-known results, i.e., the properties of the nonlinear iffusion waves an an important inequality which plays a funamental role later in the higher imensional case to prove the stability of planar iffusion waves. For the bipolar hyroynamic moel of semiconuctors (.), its corresponing -D porous meia equation is { n t = p( n) xx, J := nū = p( n) x, with or equivalently, { n t + J x =, J = p( n) x, (.) lim ( n, ū)(x, t) = (n ±, ). (.) x ± Let ( n, ū)( x +t ) be the self-similar solution of (.) satisfying the bounary conition (.). It has been prove in [3] (see also, for example, [], etc.) that the so-calle nonlinear iffusion wave ( n, ū)( x +t ) exists an behaves as follows. Such -D self-similar solution ( n, ū)( x +t ) is also calle the nonlinear planar iffusion wave for the n-d system of (.). It is note that the solution n( x +t ) is increasing if n < n + an ecreasing if n > n +, an satisfies Lemma. ([3]). For the self-similar solution of (.)-(.), let ζ = hols x +t, it n(ζ) n ζ> + n(ζ) n + ζ< C n + n e µζ, (.3) k x l t n C n + n ( + t) k+l e µx +t, k + l, k, l, (.4) k x l tū C n + n ( + t) k+l+ e µx +t, k, l, (.5) where µ > is a constant. Lemma. ([3]). For the self-similar solution of (.)-(.), it hols x n( ) n x + + t + x n( ) n + x + t C n + n ( + t), (.6) k x l t n x C n + n ( + t) l k, k + l. (.7)

6 6 F.-M. HUANG, M. MEI, AND Y. WANG Next we introuce a useful lemma given in [4], which will plays a key role for us to buil up the energy estimates in n-d case. Let µ > an Then G(x, t) = e µx x (+t) an g(x, t) = G(ξ, t)ξ. (.8) + t g t (x, t) = π µ + t G x (x, t), an g(, t) L () = µ. Lemma.3 ([4]). For < T, x, suppose that h(x, t) satisfies h L (, T ; L ()), h x L (, T ; L ()), h t L (, T ; H ()), then the following estimate hols for any t (, T ] t ( ) ( + τ) exp µx h (x, τ)x τ + τ t C(µ) { h(, ) L + h x (x, τ)x τ + t h t, hg H H τ }. (.9) 3. -D case: stability of iffusion waves. In this section, we stuy the -D bipolar hyroynamic moel of semiconuctors (.3) with the initial value conitions (.4) an the bounary conition at far fiel E(, t) = E. (3.) Here, for the sake of simplification, throughout this section, we still enote the -D spatial variable x as x without confusion. Note that, the bounary (3.) at far file x = (or replace it by E(+, t) = E + on x = + ) is proper an necessary. Because, as we show below on the state functions n i (±, t), J i (±, t) an E(±, t), without such a bounary conition, these state functions will be uneretermine, which then will cause the system (.3) an (.4) to be ill-pose. On the other han, from (.3) 5, we immeiately have an E t= = x E + := E (+ ) = (n (y) n (y))y + E =: E (x), + (n (y) n (y))y + E. The main target in this section is to prove that the solution (n, J, n, J, E)(x, t) of (.3) an (.4) with the conition (3.) converges to the nonlinear iffusion waves ( n, J, n, J, )( x+x +t ) for some shift constant x, even if [n (x) n (x)]x (or J i+ J i for i =, ). We will also erive the optimal convergence rates for the -D solutions to the corresponing iffusion waves, which are much better than what shown in [6]. More interesting, we fin that n n, an J J an E converge to zero time-exponentially. To construct the correction functions to elete the gaps between the original solutions an the iffusion waves is very tricky an plays a crucial role in the proof.

7 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS eformulation of -D system. First of all, as in [3], let us look into the behaviors of the solutions to (.3) an (.4) at the far fiels x = ±. Then we may unerstan how big the gaps are between the solutions an the iffusion waves at the far fiels. Let n ± i (t) := n i(±, t), J ± i (t) := J i(±, t), i =,. (3.) E ± (t) := E(±, t), From (.3) an (.3) 3, since x J i x=± = for i =,, it can be easily seen that n ± i (t) = n i(±, t) n ±. (3.3) Differentiating (.3) 5 with respect to t an applying (.3) an (.3) 3, we have E xt = (n n ) t = (J J ) x, an integrating it with respect to x over (, ), we obtain t E+ (t) t E (t) = [J + (t) J + (t)] + [J (t) J (t)]. (3.4) Taking x = ± to (.3) an (.3) 4, we also formally have t J ± (t) = n ±E ± (t) J ± (t), (3.5) t J ± (t) = n ±E ± (t) J ± (t), (3.6) It can be easily seen that, five equations (3.4)-(3.6) cannot uniquely etermine the six unknown state functions J ± i (t) (i =, ) an E± (t). So, we nee to a one bounary conition at far file like (3.). Otherwise, J ± i (t) (i =, ) an E± (t) will be uneretermine, an this will cause the system (.3) to be ill-pose. Therefore, without loss of generality, we may assume as before Then, from (.3) an (.3) 4, we can easily get E(, t) = E. (3.7) J i (t) = J i e t, i =,. (3.8) From (.3), (.3) 4 an (3.3), we get t J + (t) = n +E + (t) J + (t), t J + (t) = n +E + (t) J + (t), J + i () = J i+, i =,, (3.9) an from (3.4), (3.7) an (3.8), we have t E+ (t) = [J + (t) J + (t)] + (J J )e t. (3.)

8 8 F.-M. HUANG, M. MEI, AND Y. WANG Combining (3.9) an (3.), we establish the following system of ODEs for J + (t), J + (t) an E+ (t): t J + (t) = n +E + (t) J + (t), t J + (t) = n +E + (t) J + (t), t E+ (t) = [J + (t) J + (t)] + (J J )e t, J + i () = J i+, i =,, E + () = E +. (3.) Aing (3.) an (3.), we get { t [J + (t) + J + (t)] = [J + (t) + J + (t)], [J + (t) + J + (t)] t= = J + + J +, (3.) which can be solve as J + (t) + J + (t) = (J + + J + )e t. (3.3) On the other han, subtracting (3.) from (3.), we have t [J + (t) J + (t)] = n +E + (t) [J + (t) J + (t)], t E+ (t) = [J + (t) J + (t)] + (J J )e t, [J + (t) J + (t)] t= = J + J +, E + (t) t= = E +, (3.4) which, by substituting the secon equation of (3.4), i.e., J + (t) J + (t) = t E+ (t) + (J J )e t, (3.5) into the first equation of (3.4), can be reuce to t E + (t) + t E+ (t) + n + E + (t) =, E + () = E +, t E+ t= = (J + J ) (J + J ). Notice that, the eigenvalues of the secon orer ODE (3.6) are (3.6) λ = 8n + an λ = + 8n +. (3.7) Thus, accoring to the signs of 8n +, we can irectly but teiously solve the equations (3.3), (3.5) an (3.6) to have the solutions J + (t), J + (t) an E+ (t) as follows. Case : When 8n + =, then J + (t) = (J + + J + + J J )e t + { e t (J + J ) (J + J ) + [(J + J ) (J + J ) + } E +]t, (3.8)

9 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 9 J + (t) = (J + + J + J + J )e t { e t (J + J ) (J + J ) + [(J + J ) (J + J ) + } E +]t, (3.9) { E + (t) = e t E + + [(J + J ) (J + J ) + } E +]t. (3.) Case : When 8n + <, then J + (t) = (J + + J + + J J )e t + { 4 e t 8n+ [(J + J ) (J + J )] cos( t) ( ((J+ J ) (J + J )) + E ) 8n + E + 8n+ 8n+ } sin( t), (3.) J + (t) = (J + + J + J + J )e t { 4 e t 8n+ [(J + J ) (J + J )] cos( t) ( [(J+ J ) (J + J )] + E ) 8n + E + 8n+ 8n+ } sin t, (3.) { E + (t) = e t 8n+ E + cos t + [(J + J ) (J + J )] + E + 8n+ 8n+ sin( } t). (3.3) Case 3: When 8n + >, then J + (t) = (J + + J + + J J )e t { λ Ae λt + λ Be λt}, (3.4) J + (t) = (J + + J + J + J )e t + { λ Ae λt + λ Be λt}, (3.5) E + (t) = Ae λt + Be λt, (3.6) where λ an λ are the eigenvalues given in (3.7), an A an B are A = (J+ J ) (J+ J ) λe+ 8n+ B = (J+ J ) (J+ J ) λe+. 8n+ (3.7)

10 F.-M. HUANG, M. MEI, AND Y. WANG From (3.3), (3.7), (3.8) an (3.8) (3.6), we have n i (±, t) = n ±, i =, J i (+, t) = O()e νt, i =, J i (, t) = J i e t, i =, E(+, t) = O()e νt, E(, t) = E =, for some constant < ν <, which combine with the iffusion waves ( n, J)(±, t) = (n ±, ) to yiel n i (±, t) n(±, t) =, i =, J i (+, t) J(+, t) = O()e νt, i =, J i (, t) J(, t) = J i e t, i =, E(+, t) = O()e νt, E(, t) = E =. Obviously, there are some gaps between J i (±, t) an J(±, t), an E(+, t) an Ē, which lea J i (x, t) J(x, t) an E(x, t) L (). To elete these gaps, we nee to introuce some gap functions (calle also the correction functions). However, the usual manner for constructing the correction functions in [] for -D Euler equations with linear amping an in [3] with nonlinear amping/accumulating cannot be applie. So, we nee to look for something else. Observing the structure of the solutions at far fiels (see (3.)), we ingeniously construct the correction functions (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) satisfying the following linear equations ˆn t + Ĵx = Ĵ t = nê Ĵ ˆn t + Ĵx = Ĵ t = nê Ĵ Ê x = ˆn ˆn with Ĵ i (x, t) J ± i (t) as x ± Ê(x, t) as x Ê(x, t) E + (t) as x +. (3.8) Here n = n(x), Ĵ i (x, ) an Ê(x, ) will be trickily constructe in (3.9) an (3.3) below such that n(x) n ±, Ĵ i (x, ) J i±, an Ê(x, ) E±, as x ±. (3.9) In orer to get (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) to (3.8), we consier the following linear system with some tricky selection on n = n(x), Ĵ i (x, ) an Ê(x, ) Ĵ t = nê Ĵ Ĵ t = nê Ĵ Ê t = (Ĵ Ĵ) + (J J )e t Ĵ i (x, ) = J i + (J i+ J i ) x Ê(x, ) = E + m (y)y, x m (y)y, i =, (3.3)

11 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS where m (x) an n(x) are also trickily selecte as m (x), m C (), supp m [ L, L ], n(x) = n + (n + n ) x+l m (y)y m (y)y = (3.3) with some constant L >. When x < L, we have Ê(x, ). So, it can be easily seen that (3.3) possesses the particular solutions Ĵ i (x, t) = J i e t, i =,, Ê(x, t) =, for < x < L. (3.3) When x L, we have n(x) n +. Similarly to (3.), by a straightforwar but complicate calculation, we can solve (3.3) as the following (3.33)-(3.35), or (3.38)- (3.4), or (3.43)-(3.45) for L x <. However, we can verify that these solutions imply also the solutions given in (3.3) for x < L. Therefore, we summarize them as follows. Case : when 8n + =, then, for x, Ĵ (x, t) = J e t + { [(J + J ) + (J + J )]e t [ +e t (J + J ) (J + J ) + [(J + J ) (J + J ) + ]} x E +]t Ĵ (x, t) = J e t + { [(J + J ) + (J + J )]e t [ e t (J + J ) (J + J ) + [(J + J ) (J + J ) + ]} x E +]t Ê(x, t) = e t { [ E + + Thus, let us trickily construct J + J (J + J ) + E + ] t m (y)y, (3.33) m (y)y, (3.34) } x m (y)y. (3.35) ˆn (x, t) = { m (x) ((J + J ) + (J + J ))e t ( +e t E + + ((J + J ) (J + J ) + )} E +)t, (3.36) ˆn (x, t) = { m (x) ((J + J ) + (J + J ))e t ( e t E + + ((J + J ) (J + J ) + )} E +)t, (3.37) then we can verify that (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) satisfy (3.8) for (x, t) +. Case : when 8n + <, then the solutions of (3.3) are, for x, Ĵ (x, t) = J e t + (J + J ) + (J + J ) x e t m (y)y

12 F.-M. HUANG, M. MEI, AND Y. WANG So, let + { 4 e t 8n+ ((J + J ) (J + J )) cos( t) ( ((J+ J ) (J + J )) + E + + 8n+ + ) 8n+ } x 8n + E + sin( t) m (y)y, (3.38) Ĵ (x, t) = J e t + (J + J ) + (J + J ) x e t m (y)y { 4 e t 8n+ ((J + J ) (J + J )) cos( t) ( ((J+ J ) (J + J )) + E + + 8n+ 8n+ + 8n + E + ) sin( { Ê + (x, t) = e t 8n+ E + cos( 8n+ sin( t) } x } x t) m (y)y, (3.39) t) + (J + J + J J + ) + E + 8n+ ˆn (x, t) = (J + J ) + (J + J ) m (x)e t + { m (x)e t 8n+ 8n + E + cos( t) 6n + m (y)y. (3.4) 8n+ +[ (E + + ((J + J ) (J + J ))) + ((J + J ) (J + J )) + E ] + 8n+ } sin( t), (3.4) 8n+ ˆn (x, t) = (J + J ) + (J + J ) m (x)e t { m (x)e t 8n+ 8n + E + cos( t) 6n + 8n+ +[ (E + + ((J + J ) (J + J ))) + ((J + J ) (J + J )) + E ] + 8n+ } sin( t), (3.4) 8n+ then the solutions (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) satisfy (3.8) for (x, t) +. Case 3: when 8n + >, then Ĵ (x, t) = J e t + (J + J ) + (J + J ) x e t m (y)y (λ Ae λt + λ Be λt) x m (y)y, (3.43) Ĵ (x, t) = J e t + (J + J ) + (J + J ) x e t m (y)y

13 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 3 + (λ Ae λt + λ Be λt) x m (y)y, (3.44) Ê(x, t) = (λ Ae λt + λ Be λt) x m (y)y, (3.45) Similarly, we construct ˆn (x, t) = ((J + J ) + (J + J ))m (x)e t + ( m (x) λ Ae λt + λ Be λt), (3.46) ˆn (x, t) = ((J + J ) + (J + J ))m (x)e t ( m (x) λ Ae λt + λ Be λt), (3.47) then we can check that (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) are the solutions of (3.8) for (x, t) +. Clearly, the correction functions (ˆn, ˆn, Ĵ, Ĵ, Ê)(x, t) expresse in each case have an exponential ecay with respect to t, an ˆn an ˆn have the same compact support with m (x). Namely, we prove Lemma 3.. There hol an (ˆn, ˆn, Ĵ, Ĵ, Ê)(t) L () Cδe νt (3.48) supp ˆn = supp ˆn = supp m [ L, L ] for δ := n + n + J + + J + J + + J + E + an < ν <. Now we are going to make a perturbation of (.3) to the iffusion waves (.). From (.3), (3.8) an (.), we have (n ˆn n) t + (J Ĵ J) x =, (J Ĵ J) t + ( J n + p(n ) p( n)) x = n E nê (J Ĵ J) + p( n) xt, (n ˆn n) t + (J Ĵ J) x =, (J Ĵ J) t + ( J n + p(n ) p( n)) x = n E + nê (J Ĵ J) + p( n) xt (E Ê) x = (n ˆn n) (n ˆn n), (3.49) where ( n, J) = ( n, J)(x + x, t) are the shifte iffusion waves with some shift x, which will be specifie later. Let us integrate the first an thir equation of (3.49) over (, + ) with respect to x, an note that J i (±, t) = Ĵi(±, t) for i =, an J(±, t) = as shown before, we have [n i (x, t) ˆn i (x, t) n(x + x, t)]x t = [J i (+, t) Ĵi(+, t) J(+, t)]

14 4 F.-M. HUANG, M. MEI, AND Y. WANG +[J i (, t) Ĵi(, t) J(, t)] =, i =,, then integrate the above equation with respect to t to have [n i (x, t) ˆn i (x, t) n(x + x, t)]x = [n i (x, ) ˆn i (x, ) n(x + x, )]x =: I i (x ), i =,. (3.5) Now we are going to etermine x such that I i (x ) =. Since I i(x ) = ( ) [n i (x, ) ˆn i (x, ) n(x + x, )]x = (n + n ), i =,, x which gives, with I i (x ) =, that x := [n i (x, ) ˆn i (x, ) n(x, )]x, i =,. (3.5) n + n This implies that, we nee [n (x, ) ˆn (x, ) n(x, )]x = [n (x, ) ˆn (x, ) n(x, )]x, namely, [n (x) n (x)]x = [ˆn (x, ) ˆn (x, )]x. (3.5) However, such a conition is always true, an automatically guarantee by the system (.3). In fact, from (3.36), (3.37), (3.4), (3.4), (3.46) an (3.47), for each case we always have ˆn (x, ) = m (x)[(j + J ) + (J + J ) + E + ], ˆn (x, ) = m (x)[(j + J ) + (J + J ) E + ], which, with the fact m (x)x =, gives [ˆn (x, ) ˆn (x, )]x = E +. Substituting this to (3.5), we nee [n (x) n (x)]x = E +. (3.53) However, by integrating (.3) 5 with respect to x over (, ) an taking t =, as well as noting E =, we immeiately obtain (3.53). Hence, (3.5) automatically hols.

15 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 5 Thus, (3.5) with I i (x ) = for such selecte x in (3.5) implies that the integration of the perturbe equations (3.49) over (, x] coul be set in L space. Therefore, by efining x φ i (x, t) := [n i (ξ, t) ˆn i (ξ, t) n(ξ + x, t)]ξ ψ i (x, t) := J i (x, t) Ĵi(x, t) J(x (3.54) + x, t), i =, H(x, t) := E(x, t) Ê(x, t), namely, φ ix = n i ˆn i n an φ it = ψ i = J i Ĵi J, we euce (3.49) into φ t + ψ ( =, ) ψ t + ( φt+ĵ+ J) φ x+ˆn + n + p(φ x + ˆn + n) p( n) x = (φ x + ˆn + n)h + (φ x + ˆn + n n)ê ψ + p( n) xt, φ t + ψ =, (3.55) ( ) ψ t + ( φt+ĵ+ J) φ x+ˆn + n + p(φ x + ˆn + n) p( n) x = (φ x + ˆn + n)h (φ x + ˆn + n n)ê ψ + p( n) xt, H = φ φ, with the initial ata φ i (x) := φ i (x, ) = x [n i (ξ) ˆn i (ξ, ) n(ξ + x, )]ξ ψ i (x) := ψ i (x, ) = J i (x) Ĵi(x, ) J(x + x, t), i =,, H (x) := φ (x) φ (x). From (3.55), we obtain ( ) φ tt + φ t p(φ x + ˆn + n) p( n) + (φ x + ˆn + n)h x = ( f + g x p( n) xt, ) φ tt + φ t p(φ x + ˆn + n) p( n) (φ x + ˆn + n)h x = f + g x p( n) xt, where with the initial ata (3.56) (3.57) f i = (φ ix + ˆn i + n n)ê, g i = ( φ it + Ĵi + J), i =,, (3.58) φ ix + ˆn i + n φ i (x, ) = φ i (x), φ it (x, ) = ψ i (x), i =,. (3.59) Furthermore, subtracting (3.57) from (3.57), we get the IVP for the following ampe Klein-Goron type equation where H tt + H t + nh = h := h x h h 3 + h 4x, (3.6) h = p(φ x + ˆn + n) p(φ x + ˆn + n), h = (φ x + φ x + ˆn + ˆn )H, h 3 = [φ x + φ x + ˆn + ˆn + ( n n)]ê, ( φt+ĵ+ J) ( φt+ĵ+ J) h 4 = φ x+ˆn + n φ. x+ˆn + n (3.6)

16 6 F.-M. HUANG, M. MEI, AND Y. WANG 3.. Convergence theorem in -D case. In this subsection, we are going to state the convergence result of the bipolar hyroynamic moel for semiconuctors in the one imensional case. Theorem 3.. Let (φ, φ, ψ, ψ )(x) H 3 () H 3 () H () H (), δ := n + n + J + + J + J + + J + E + an Φ := (φ, φ ) 3 + (ψ, ψ ). Then, there is a δ > such that if δ + Φ δ, the solutions (n, n, J, J, E) of IVP (.3)), (.4) an (3.) are unique an globally exist, an satisfy ( + t) k+ x(n k n, n n)(t) k= + ( + t) k+ x(j k Ĵ J, J Ĵ J)(t) C(δ + Φ ), (3.6) k= (n n )(t) + (J Ĵ J + Ĵ)(t) + (E Ê)(t) C(δ + Φ )e νt, (3.63) for some constant ν >. Furthermore, if (φ, φ ) L (), then the optimal L p () ( p + ) ecay rates hol x(n k n, n n)(t) L p () C(δ + Φ ) ( + t) ( k+ p ), (3.64) x(j k Ĵ J, J Ĵ J)(t) Lp () C(δ + Φ ) ( + t) ( k+ p ),(3.65) for k if p =, an k if p (, + ]. Corollary 3.3. Let (φ, φ ) L (). Then, for i =,, k x l t(n i n)(t) L () C(δ + Φ ) ( + t) k+l, k, l, (3.66) k x l t(u i ū)(t) L () C(δ + Φ ) ( + t) 3 k+l, k, l, (3.67) k x l t(n n )(t) L () C(δ + Φ ) e ν t, k, l, (3.68) k x l t(u u )(t) L () C(δ + Φ ) e ν t, k, l, (3.69) k x l te(t) L () C(δ + Φ ) e ν t, k, l, (3.7) where u i = J i /n i. emark.. Although the iffusion waves ( n, ū)(x+x, t) are the asymptotic profiles of the original solutions (n i, u i )(x, t), i =,, with algebraic ecay, the much better asymptotic profiles of the original solutions (n, u )(x, t) (or (n, u )(x, t)), in fact, are just their partner solutions (n, u )(x, t) (or (n, u )(x, t)), because the corresponing ecay as showe in (3.68) an (3.69) are exponential.. If the pressures p(s) an q(s) in (.) are ifferent, the question will become more complicate an challenging, because the clarification of the asymptotic profiles (iffusion waves) in this case an the construction of the corresponing correction functions will be much harer, an totally ifferent from the simple issue we stuy here. So, this will be something left for us in future. 3. The new technique for constructing the correction functions can be also applie to solve the stability of iffusion waves for other hyroynamic moels of semiconuctors, for example, the unipolar hyroynamic moel of semiconuctors, the nonisentropic full Euler-Poisson system with fractional amping external-force, an so on.

17 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS A priori estimates in -D case. It is known that Theorem 3. can be prove by the classical energy metho with the continuation argument base on the local existence an the a priori estimates, c.f. [8, 9]. Since the local existence of the solutions of (3.57), (3.59) an (3.6) can be prove in the stanar iteration metho together with the energy estimates, so the main effort in this subsection is to establish the a priori estimates for the solutions, which is usually technical an crucial in the proof of stability. Let T (, + ], we efine the solution space for (3.57), (3.59) an (3.6) as follows { X(T ) = (φ, φ t, φ, φ t, H)(x, t) j t φ i C(, T ; H 3 j ()), i =,, j =,, } j t H C(, T ; H j ()), j =,, t T with the norm N(T ) = sup t T { 3 ( + t) k x(φ k, φ )(t) + k= + i+j= e νt i t j xh(t) }. ( + t) k+ x(φ k t, φ t )(t) Let N(T ) ε, where ε is sufficiently small an will be etermine later. Notice that, by Sobolev inequality k xf L () C k xf / k+ x f /, we have, i =,, ( + t) 4 + k k x φ i (t) L () + k= ( + t) k k x φ it (t) L () Cε. k= Clearly, there exists a positive constant c such that < c n i = φ ix + ˆn i + n c, i =,. (3.7) Now we first establish the following basic energy estimate. Lemma 3.4. It hols that (H, H x, H t, H xx, H xt, H tt )(t) C(δ + Φ )e νt (3.7) provie ε + δ. Proof. Step : Multiplying (3.6) by H + H t an integrate it over (, + ), we obtain ( Ht + [ ) ( t + n]h + H t H x + Ht + ( n n t )H ) x = (H + H t )(h x h h 3 + h 4x )x. (3.73) Applying Taylor s formula to (3.6), namely, h x = p (n )H xx + p (n )(ˆn x ˆn x ) + O()(φ xx + ˆn x + n x )(H x + ˆn ˆn ),

18 8 F.-M. HUANG, M. MEI, AND Y. WANG then the first term of the right-han sie term of (3.73) can be estimate as follows h x (H + H t )x := I + I + I 3 with so we have I = p (n )H xx (H + H t )x p (n )H t xx p (n )Hxx + C(ε + δ) (H, H x, H t )(t), I = p (n )(ˆn x ˆn x )(H + H t )x κ (H, H t )(t) + C(κ)δ e νt, for some small constant κ >, I 3 = O()(φ xx + ˆn x + n x )(H x + ˆn ˆn )(H + H t )x C(ε + δ) (H, H x, H t )(t) + Cδ e νt, h x (H + H t )x t p (n )Hxx p (n )Hxx +C(ε + δ + κ) (H, H x, H t )(t) + Cδ e νt, (3.74) where we use (3.48) an κ is small an it will be etermine later. Similarly, noticing (3.48) an (3.6), which imply φ ix CN(t) Cε an ˆn i Cδe νt, we can prove h (H + H t )x C(ε + δ) (H, H t )(t), (3.75) h 3 (H + H t )x C(ε + δ) (H, H t )(t) + Cδ e νt, (3.76) where for (3.76), we also use the facts ( n n) x Cδ ( + t), which, as showe in (.6) (Lemma.), can be prove from the construction of n(x) n ± as x ± an the property of the iffusion wave n( x+x +t ). Noticing h 4x = J n H xx J H xt + O()(ˆn x + ˆn x + n Ĵx + Ĵx) +O()(φ xx + φ xt + n x + J x + ˆn x + Ĵx) (H x + H t + ˆn + ˆn + Ĵ + Ĵ), (3.77) then, by integrating it by parts an using Cauchy inequality, we have h 4x (H + H t )x ( J ) t n Hxx +C(ε + δ + κ) (H, H x, H t )(t) + Cδ e νt. (3.78) Substituting (3.74)-(3.78) into (3.73), an noticing the smallness of ε, δ, κ, we obtain ( Ht + ( t + n)h + H t H + (p (n ) J ) n )Hxx

19 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 9 + Cδ e νt. ( ) Ht + ( n n t )H + p (n )Hx x Applying Gronwall s lemma to above ifferential inequality, we obtain (H, H x, H t )(t) C(δ + Φ )e νt, (3.79) where ν is some positive constant. Step : Differentiating (3.6) with respect to x, we obtain H xtt + H xt + nh x + n x H = h x = h xx h x h 3x + h 4xx. (3.8) Multiplying (3.8) by H x + H xt an integrating it over (, + ), we obtain ( [H xt + ( ] ) t + n)h x + H xt H x x [ ] + Hxt + ( n n t )Hx x = n x H(H x + H xt )x + (H x + H xt )(h xx h x h 3x + h 4xx )x. (3.8) By using (3.48) an (3.79), the terms in the right-han sie of (3.8) can be similarly estimate as follows n x H(H x + H xt )x Cδ H xt (t) + C(δ + Φ )e νt, (3.8) an x + H xt )h xx x (H ( ) p (n )H t xxx p (n )Hxxx +C(ε + δ + κ) (H xx, H xt )(t) + Cδ e min(ν,ν)t,(3.83) an an an (H x + H xt )h x x C(ε + δ) H xt (t) + C(δ + Φ )e min(ν,ν)t, (3.84) (H x + H xt )h 3x x κ H xt (t) + Cδ e min(ν,ν)t, (3.85) x + H xt )h 4xx x (H ( J ) t n Hxxx + C(ε + δ + κ) (H xx, H xt )(t) +C(δ + Φ )e min(ν,ν)t. (3.86) Substituting (3.8)-(3.86) into (3.8), an noticing the smallness of ε, δ, κ, we obtain ( Hxt + ( t + n)h x + H xt H x + (p (n ) J ) n )Hxxx

20 F.-M. HUANG, M. MEI, AND Y. WANG + ( ) Hxt + ( n n t )Hx + p (n )Hxx x C(δ + Φ )e min(ν,ν)t. (3.87) Again, applying Gronwall s iequality to above ifferential inequality, we obtain (H x, H xx, H xt )(t) C(δ + Φ )e νt (3.88) for some constant ν >. Furthermore, by applying (3.79) an (3.87) to the equation (3.6), we can prove H tt (t) C(δ + Φ )e ν3t (3.89) for some constant ν 3 >. Finally, let ν = min(ν, ν, ν, ν 3 ). Thus, (3.79), (3.88) an (3.89) imply (3.7). The proof of this lemma is complete. Since now, we will state more higher orer energy estimates for the solutions φ i (i =, ) to the wave equations (3.57) in ifferent lemmas with sketchy proofs. Lemma 3.5. It hols that, for i =,, (φ i, φ ix, φ it )(t) + t (φ ix, φ iτ )(τ) τ C(δ + Φ ) (3.9) provie ε + δ. Proof. By taking [(3.57) i (φ i + λφ it )]x for some large number λ >, an applying Lemma., Lemma. an Lemma 3.4, with a teious calculation we the complete the proof of Lemma 3.5. Lemma 3.6. It hols that, for i =,, ( + t) (φ ix, φ it )(t) + t ( + τ) φ iτ (τ) τ C(δ + Φ ) (3.9) provie ε + δ. Proof. By taking t (+τ)(3.57) i φ itxτ, i =, an integrating the resultant equation with respect to t over [, t], an applying Lemma 3.5, we obtain ( + t) (φ ix, φ it )(t) + t Hence, we complete the proof of Lemma 3.6. Lemma 3.7. It hols that, for i =,, ( + t) (φ ixx, φ ixt )(t) + t ( + τ) φ iτ (τ) τ C(δ + Φ ). (3.9) [( + τ) φ ixτ (τ) + ( + τ) φ ixx (τ) ]τ C(δ + Φ ) (3.93) provie ε + δ. Proof. By taking t (+τ) [ x (3.57) i φ ix + x (3.57) i φ ixτ ]xτ, an applying Lemma 3.6, as showe before, we can similarly prove (3.93). Lemma 3.8. It hols that, for i =,, ( + t) 3 (φ ixxx, φ ixxt )(t) + t [( + τ) 3 φ ixxτ (τ) + ( + τ) φ ixxx (τ) ]τ

21 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS C(δ + Φ ) (3.94) provie ε + δ. Proof. By taking t ( + τ)3 [ xxτ (3.57) φ xxτ + xxτ (3.57) φ xxτ ]xτ, an applying Lemma 3.7, we can prove (3.94). Lemma 3.9. It hols that, for i =,, ( + t) (φ it, φ ixt, φ itt )(t) + t ( + τ) (φ ixτ, φ iττ )(τ) τ C(δ + Φ ) (3.95) provie ε + δ. Proof. By taking t ( + τ) [ τ (3.57) (φ τ + φ ττ ) + τ (3.57) (φ τ + φ ττ )]xτ, an applying Lemma 3.8, we can prove (3.95). Lemma 3.. It hols that ( + t) 3 (φ ixt, φ itt )(t) + t ( + τ) 3 φ iττ (τ) τ C(δ + Φ ) (3.96) for i =,, provie ε + δ. Proof. By taking t ( + τ)3 [ τ (3.57) φ ττ + τ (3.57) φ ττ ]xτ, an applying Lemma 3.9, we can prove (3.96). Lemma 3.. It hols that, for i =,, ( + t) 3 (φ ixt, φ ixxt, φ ixtt )(t) + t ( + τ) 3 (φ ixxτ, φ ixττ )(τ) τ C(δ + Φ ) (3.97) provie ε + δ. Proof. By taking t ( + τ)3 [ τ (3.57) (φ xτ + φ xττ ) + τ (3.57) (φ xτ + φ xττ )]xτ, an applying Lemma 3., we can prove (3.97). Lemma 3.. It hols that, for i =,, ( + t) 4 (φ ixxt, φ ixtt )(t) + t ( + τ) 4 φ ixττ (τ) τ C(δ + Φ ) (3.98) provie ε + δ. Proof. By taking t ( + τ)4 [ τ (3.57) φ xττ + τ (3.57) φ xττ ]xτ, an applying Lemma 3., we can prove (3.98). Now we are going to prove the optimal ecays (3.64) an (3.65), if the initial perturbation is further in L (). Let us rewrite the equations (3.57) as follows where φ it (a i (x, t)φ ix ) x = F i φ itt, i =,, (3.99) a i (x, t) : = p ( n i (x, t)) C >, i =,, F : = f + g x p( n ) xt (φ x + ˆn + n )H (p(φ x + ˆn + n ) p( n ) p ( n )φ x ) x, F : = f + g x p( n ) xt + (φ x + ˆn + n )H (p(φ x + ˆn + n ) p( n ) p ( n )φ x ) x.

22 F.-M. HUANG, M. MEI, AND Y. WANG As shown in [36], we can similarly construct the minimizing Green functions as follows ( G i (x, t; y, s) = 4πa i (x, t)(t s) an rewrite (3.99) in the integral form where φ i (x, t) = t + + ) / exp ( G i (x, t; y, )φ i (y)y t (x y) ), i =,, 4a i (y, s)(t s) G i (x, t; y, s)[f i (y, s) φ iss (y, s)]ys Gi (x, t; y, s)φ i (y, s)]ys, i =,, (3.) Gi (x, t; y, s) := s G i (x, t; y, s) + y {a i (y, s) y G i (x, t; y, s)}, i =,. Differentiating (3.) with respect to x an t, we have, for l an k + l 3, l t k xφ i (x, t) = l t k x + l t k x + l t k x = : I l,k i G i (x, t; y, )φ i (y)y t t + Il,k i G i (x, t; y, s)[f i (y, s) φ iss (y, s)]ys Gi (x, t; y, s)φ i (y, s)]ys + Il,k i3, i =,. (3.) Base on the ecay rates we obtaine in Lemmas , an on the estimates of ecay rates for the approximating Green functions as shown in [36], by a similar but teious calculation to [36], we can similarly prove I l,k i + Il,k i + Il,k i = O()( + t) 4 l k, for l, l + k 3, i =,. (3.) Here, the etails are omitte. Hence, applying (3.) to (3.), we obtain the optimal ecay rates as follows. Lemma 3.3. Furthermore, if (φ, φ ) L (), then k xφ i (t) = O()( + t) 4 k, k =,,, 3, i =,, (3.3) k x t φ i (t) = O()( + t) 5 4 k, k =,, i =,. (3.4) Proof of Theorem 3.. Base on the local existence an the a priori estimates given in Lemma 3.4 Lemma 3., by using the continuity argument (c.f. [8]), we can prove the global existence of the unique solutions of the IVP (.3)-(.4) with the esire ecay rates (3.6) an (3.63). Furthermore, when the initial perturbation is in L (), from Lemma 3.3, we immeiately obtain the optimal ecay rates (3.64) an (3.65).

23 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 3 4. n-d case: stability of planar iffusion waves. In this section, we are going to stuy the multi-imensional isentropic Euler-Poisson equations for the bipolar hyroynamic moel of semiconuctors (.) with the initial ata { n i (x, ) = n i (x) n ±, as x ± i =,, (4.) u i (x, ) = u i (x) (u i±,, ), as x ± where n i± an u i± for i =, are constants. We will prove that the solutions of the n-d equations (.) an (4.) converge to the -D nonlinear iffusion waves of (.), the so-calle planar iffusion waves to the n-d equations (.). It must be pointe out that, the strategy of the antierivatives for the problem setting up use in -D case (see (3.54) before) is no long effective in the multi-imensional case, because the shift x efine in (3.5) in -D case will be an implicit function in n-d case, an it epens on the solution (n, u, n, u, Ψ)(x, t) of the system (.) an, rather than the initial ata. Instea of this, we are going to apply the key Lemma.3 to establish some crucial energy estimates an then to prove the stability of planar iffusion waves. For simplicity, we just consier the 3-D case, an enote x = (x, x, x 3 ) 3. Let (ñ, ñ, J, J, Ẽ)(x, t) be the solutions of -D isentropic Euler-Poisson equations (.3) with small perturbations, i.e., Φ given in Theorem 3., an efine Ũ i (x, t) = (ũ i (x, t),, ), Ū(x, t) = (ū(x, t),, ), for i =,. (4.) Ẽ(x, t) = (Ẽ(x, t),, ), Base on the above preparation, we are going to make a perturbation of (.) to the one imensional solutions (ñ, ñ, J, J, Ẽ)(x, t) of (.3) that we just remin above. We efine z = n ñ, z = n ñ, w = u Ũ, w = u Ũ. Combining (.) with (.3), we obtain z t + iv(z w + ñ w + ũ z ) =, w t + w + θ z ( Ψ Ẽ) = L N, z t + iv(z w + ñ w + ũ z ) =, (4.3) w t + w + θ z + ( Ψ Ẽ) = L N, iv( Ψ Ẽ) = z z, where { Li = w i ũ i + ũ i w i, N i = w i w i + θ i z i + θ i ñ i, θ i = p (ñ i) ñ i, θ i = p (ñ i+z i) ñ i+z i p (ñ i) ñ i, i =,. (4.4) Noticing that curl( Ψ Ẽ) =, so there exists E such that E = Ψ Ẽ. We can reuce (4.3) into z t + iv(z w + ñ w + ũ z ) =, w t + w + θ z E = L N, z t + iv(z w + ñ w + ũ z ) =, (4.5) w t + w + θ z + E = L N, E = z z,

24 4 F.-M. HUANG, M. MEI, AND Y. WANG provie with the initial ata { z i := z i (x, ) = n i (x) ñ i (x ) = n i (x) n(x ) H 4 ( 3 ), w i := w i (x, ) = u i (x) Ũi(x, ) = u i (x) Ū(x, ) H4 ( 3 ), (4.6) an the bounary conition E, as x +. (4.7) For later use, we efine { E (x) := z (x) z (x), E (x), as x +, an { Ē(x) := z t (x, ) z t (x, ), Ē (x), as x +, (4.8) an η := (n i ñ i, u i Ũi)() 4 + ( Ψ Ẽ)() 4 + ( Ψ t Ẽt)() 3, i= where Ψ(x, ) = E (x) + Ẽ(x, ) an Ψ t(x, ) = Ē(x) + Ẽt(x, ). 4.. Convergence theorem in 3-D case. We now state the stability results for the planar iffusion waves in the multi-imensional case as follows. Theorem 4.. Let δ = n + n + u + + u + u + + u + Φ. Then, if η + δ, there exists a unique global smooth solution (n, n, u, u, Ψ) for the 3-D bipolar hyroynamic moel for semiconuctors system (.), (4.) an (4.7) an satisfies n i ñ i, u i Ũi, Ψ Ẽ C([, ), H4 ( 3 )) C ([, ), H 3 ( 3 )), for i =,, an 4 ( + t) k+ k ( Ψ Ẽ)(t) k= + i= k= 4 ( + t) k ( k (n i ñ i ), k (u i Ũi))(t) L ( 3 ) C(δ + η ). (4.9) Theorem 4.. Uner the conitions in Theorem 4., then (n ñ, n ñ )(t) L ( 3 ) C(δ 4 + η)( + t) 3 4, (n n )(t) L ( 3 ) C(δ 4 + η)( + t) 9 4, (u Ũ, u Ũ)(t) L ( 3 ) C(δ 4 + η)( + t) 5 4, ( Ψ Ẽ)(t) L ( 3 ) C(δ 4 + η)( + t) 7 4. (4.) Base on Corollary 3.3 an Theorem 4., we immeiately obtain the convergence of the solution (n, n, u, u, E)(x, t) for the 3-D equations (.) an (4.) to the -D iffusion wave ( n, ū)(x / + t) for (.), namely, the stability of the planar iffusion waves.

25 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 5 Corollary 4.3. Uner the conitions in Theorem 4., then (n n, n n)(t) L ( 3 ) C(δ 4 + η)( + t) 3 4, (u Ū, u Ū)(t) L ( 3 ) C(δ 4 + η)( + t) 5 4, Ψ(t) L ( 3 ) C(δ 4 + η)( + t) 7 4. (4.) emark. Notice that, for one-imensional case, the electron fiel ecay exponentially fast, but here the ecay rate of electron fiel for 3-D case is algebraic only. 4.. A priori estimates in 3-D case. Let T (, + ], we efine the solution space of (4.5) as, for t T, { } X (T ) = (z, z, w, w, E)(x, t) z i, w i, E C(, T ; H 4 ()), i =,. The local existence of (4.5) can be establishe by a stanar contraction mapping argument, for example, see [8, 4]. The main uty in the rest of the present paper is to establish some crucial energy estimates. We are going to establish the a priori estimates of (z, z, w, w, E), which will be the main effort of this section. We efine N (T ) = sup t T 4 k= k= [ ( + t) k ( k z i, k w i )(t) + ( + t) k+ k+ E(t) ]. Let N (T ) ε, where ε is sufficiently small an will be etermine later. Then Gagliaro Nirenberg inequality guarantees, for i =,, k =,,, ( + t) k ( k z i, k w i )(t) L + ( + t) k k+ E(t) L Cε. (4.) emark 3. Before we establish the a priori estimates for the solutions, we nee to estimate E(x, t). Since E = z z an E, E L ( 3 ), we can formally solve it by using Green function an estimate it as E(x, t) = = x y (z z )(y, t)y + C = x y E(y, t)y + C ( ) 3 E(y, t)y + C C, for all x 3. (4.3) x y 3 3 Now we first establish the following useful estimate, which plays a funamental role in n-d case. Lemma 4.4. It hols that t B(τ)τ C(δ + η ) + C provie δ + ε, where B(t) := 3 t (w, w, z, z )(τ) τ (4.4) µx + t e +t [z (x, t) + z(x, t)]x.

26 6 F.-M. HUANG, M. MEI, AND Y. WANG Proof. From Lemma.3, it hols that t 3 e µx +τ { + τ (z + z)xτ C(µ) (z, z )() + + i= t where z i is the solution of (4.5). Notice also that, i= = = t z it, z i g H H τ t i= t i= 3 t ( z, z )(τ) τ z it, z i g H H τ }, (4.5) 3 z i g iv(z i w i + ñ i w i + ũ i z i )xτ [(z i + ñ i )w i + ũ i z i ] (z i g )xτ =: Ĩ + Ĩ. (4.6) By using Cauchy inequality an noticing (.8), Corollary 3.3 an Lemma., we have Ĩ = = C t i= t i= i= (z i + ñ i )w i (z i g )xτ 3 t 3 (z i + ñ i )w i ( z i g + z i g g)xτ (w i, z i )(τ) τ t + e µx +τ 8C(µ) + τ (z + z)(x, τ)xτ, (4.7) 3 t Ĩ = ũ i z i (z i g )xτ i= 3 t ( ũ i ū + ū )z i ( z i g + z i g g)xτ 3 i= t i= 3 Cδ + C +Cδ t (Cδ ( + t) 3 e µx (+t) + Cδ )z i ( z i g + z i g g)xτ + t i= 3 t z i (τ) τ e µx +τ + τ (z + z )(x, τ)xτ. (4.8)

27 Thus, we have i= t BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 7 t z it, z i g H H τ Cδ + C (w i, z i )(τ) τ +[Cδ + 8C(µ) ] t 3 e µx +τ + τ (z + z)xτ. (4.9) Substituting (4.9) into (4.5) an using the smallness of δ, we complete the proof of Lemma 4.4. Lemma 4.5. It hols that (z i, w i )(t) + t C(δ + η ) + C w i (τ) τ t t E(τ) τ + C(δ + ε) ( w i, z i )(τ) τ (4.) for i =,, provie δ + ε. Proof. By taking (4.5) z, (4.5) w, (4.5) 3 z, an (4.5) 4 w, we have ñ θ ñ θ ( z O()ñ t z ñ )t + z ivw = z {iv(z w + ũ z ) + w ñ }, ñ ( w θ )t ( z ñ )t ( w θ )t O()ñ t w + w θ + z w E w θ = w θ {L + N }, O()ñ t z + z ivw = z ñ {iv(z w + ũ z ) + w ñ }, O()ñ t w + w θ + z w + E w θ = w θ {L + N }. (4.) Combining them an applying Cauchy s inequality an the smallness of δ an ε, we obtain ( z + z + w + w + C (w, w ) ñ ñ θ θ )t + iv(z w + z w ) C E + O()ñ t z + O()ñ t z w θ {L + N } w θ {L + N } z ñ {iv(z w + ũ z ) + w ñ } z ñ {iv(z w + ũ z ) + w ñ }. (4.) Integrating (4.) with respect to x over 3 an using the smallness of ε an δ, we have ( z + z + w + w ) x + C (w, w )(t) t ñ 3 ñ θ θ C(δ + ε) ( w, w, z, z )(t) +C E(t) + Cδ ( + t) + Cδ B(t), (4.3) where we have use ( z {iv(z w + ũ z ) + w ñ } + z ) {iv(z w + ũ z ) + w ñ } x ñ ñ 3

28 8 F.-M. HUANG, M. MEI, AND Y. WANG C(δ + ε) (w, w, w, w, z, z )(t) +Cδ ( + t) + Cδ B(t), (4.4) an ( w {L + N } + w ) {L + N } x θ 3 θ C(δ + ε) (w, w, w, w, z, z )(t) +Cδ ( + t) + Cδ B(t). (4.5) Integrating (4.3) over [, t], an using Lemma 4.4 an the smallness of δ an ε, we prove Lemma 4.5. Lemma 4.6. It hols that ( z i, w i )(t) + t w i (τ) τ t C(δ + η ) + C(δ + ε) ( z i, w i )(τ) τ + C t E(τ) τ (4.6) for i =,, provie δ + ε. Proof. Let α be multi-inex an α =. Integrating x α (4.5) x α w + x α (4.5) 4 x α w with respect to x over 3, we have t ( w, w )(t) + ( w, w )(t) + ( x α w x α w ) x α Ex α = 3 + [ x α (θ z ) x α w + x α (θ z ) x α w ]x α = 3 = [ x α (L + N ) x α w + x α (L + N ) x α w ]x. (4.7) 3 α = Using Cauchy inequality, we have α = 3 ( α x w α x w ) α x Ex 3 ( w, w )(t) C E(t). (4.8) Let H 3 (t) := F 3 (t) := 3 3 ( θ θ (z + ñ ) z + (z + ñ ) z ) x, (4.9) ( θ (z + ñ ) z θ + (z + ñ ) z ) x. (4.3) We are going to estimate the other integral terms in (4.7) as follows [ x α (θ z ) x α w + x α (θ z ) x α w ]x 3 α = = i= α = 3 β =,β α C β β θ i α β z i α w i x

29 BIPOLA HYDODYNAMIC MODEL OF SEMICONDUCTOS 9 Cδ ( w, w ) Cδ( + t) ( z, z ) θ i α z i α ivw i x. (4.3) 3 i= α = In orer to control the last integral of (4.3), we first note that, by noting (4.5) an (4.5) 3, α ivw i = { α z it + z i + ñ i So, utilizing (4.3), we have i= α = = α = + 3 α β =,β α α β =,β α C β β (ñ i + z i ) α β ivw i } + α [z i ivũ i + ũ i z i + w i ñ i + w i z i ], i =,. (4.3) 3 θ i α z i α ivw i x { θ i α z i α z it + z i + ñ i α β =,β α C β β ñ i α β ivw i x C β β z i α β ivw i + α [ z i ivũ i + ũ i z i + w i ñ i + w i z i ]}x C(δ + ε) ( w, w )(t) C(δ + ε)( + t) ( z, z, w, w )(t) Cδ ( + t) 4 Cδ ( + t) B(t) + t H 3(t). (4.33) Here, we use the following estimates to complete (4.33): i= α = 3 θ i z i + ñ i α z i α z it H 3t C(δ + ε)( + t) ( z, z )(t), (4.34) an an i= α = 3 θ i z i + ñ i α z i β =,β α C β β ñ i α β ivw i x C(δ + ε)( + t) ( z, z )(t) C(δ + ε) ( w, w ), (4.35) i= α = 3 θ i z i + ñ i α z i β =,β α C β β z i α β ivw i x C(δ + ε)( + t) ( z, z )(t) C(δ + ε) ( w, w ), (4.36) an i= α = 3 [ ] θ i α z i α z ivū + ū z + w ρ + w z x z i + ñ i

30 3 F.-M. HUANG, M. MEI, AND Y. WANG C(δ + ε) ( w, w )(t) C(δ + ε)( + t) ( z, z, w, w )(t) Cδ ( + t) 4 Cδ ( + t) B(t). (4.37) Applying (4.33) to (4.3), we obtain [ x α (θ z ) x α w + x α (θ z ) x α w ]x 3 α = C(δ + ε) ( w, w )(t) C(δ + ε)( + t) ( z, z, w, w )(t) Cδ ( + t) 4 Cδ ( + t) B(t) + t H 3(t). (4.38) Similarly, we can estimate the last term of (4.7) as [ x α (L + N ) x α w + x α (L + N ) x α w ]x 3 α = C(δ + ε) ( w, w )(t) + C(δ + ε)( + t) ( z, z, w, w )(t) +Cδ ( + t) 4 + Cδ ( + t) B(t) t F 3(t). (4.39) Let M 3 (t) = H 3 (t) + F 3 (t) + ( w, w )(t). Substituting (4.8)-(4.39) into (4.7), an noticing the smallness of ε, an δ, we obtain t M 3(t) ( w, w )(t) C E(t) + C(δ + ε)( + t) ( z, z, w, w )(t) +Cδ ( + t) 4 + Cδ ( + t) B(t). (4.4) Integrating (4.4) with respect to τ over [, t], an using Lemma 4.4 an the smallness of δ an ε, we then prove Lemma 4.6. By a teious computation to [(4.5) z 3 +(4.5) 4 z ]x an applying Lemma 4.4, we can prove Lemma 4.7. It hols that t z i (τ) τ C(δ + η ) + C (w i, z i )(t) + C for i =,, provie δ + ε. Lemma 4.8. It hols that ( E, E t, E)(t) + t ( E, E t, E)(τ) τ t w i (τ) τ (4.4) t C(δ + η ) + C(δ + ε) [ (w, w )(τ) + ( z, z )(τ) ]τ (4.4) provie δ + ε. Proof. By a teious computation to 3 [(ñ + z ) (4.5) 4 (ñ + z ) (4.5) ] Ex