Large time asymptotics for partially dissipative hyperbolic systems

Transcription

1 Larg tim asymtotics for artially dissiativ hyrbolic systms K. Bauchard, E. Zuazua Abstract This work is concrnd with (n-comonnt) hyrbolic systms of balanc laws in m sac dimnsions. First w considr linar systms with constant cofficints and analyz th ossibl bhavior of solutions as t. Using Fourir transform w xhibit th rol that control thortical tools, such as th classical Kalman rank condition, lay. W build Lyaunov functionals allowing to stablish xlicit dcay rats dnding on th frquncy variabl. In this way w xtnd th rvious analysis by Shizuta and Kawashima undr th so-calld algbraic condition (SK). In articular w show th xistnc of systms xhibiting a mor comlx bhavior than th on that th (SK) condition allows. W also discuss th link of this analysis with rvious litratur in th contxt of damd wav quations, hyolliticity and hyocorcivity. To conclud w analyz th xistnc of global solutions around constant quilibria for nonlinar systms of balanc laws. Our analysis of th linar cas allows roving xistnc rsults in situations that th rviously xisting thory dos not covr. Contnts 1 Introduction and main rsults Problm formulation Main rsults Rank conditions, (SK), hyocorcivity and dcay rats (SK) and Kalman rank conditions Lyaunov functionals and xlicit dcay rats A nw roof for Shizuta and Kawashima s dcomosition CMLA, ENS Cachan, CNRS, Univrsud, 61 avnu du Présidnt Wilson, F Cachan, Franc, mail: Karin.Bauchard@cmla.ns-cachan.fr IMDEA-Matmáticas & Dartamnto d Matmáticas, Facultad d Cincias, Univrsidad Autónoma, Cantoblanco, Madrid, Sain mail: nriqu.zuazua@uam.s Th work of th scond author has bn suortd by th Grant MTM , th DOMINO Projct CIT in th PROFIT rogram and th Ingnio Mathmatica (i-math) rojct of th Program CONSOLIDER INGENIO 2010 of th MEC (Sain) and th SIMUMAT rojct of th CAM (Sain). This work startd whil th first author was visiting th Univrsidad Autónoma d Madrid as a ostdoc within th SIMUMAT rojct and it finishd whil th scond author was visiting th Isaac Nwton Instiut in Cambridg within th rogram Highly Oscillatory Problms". Th authors acknowldg both Institutions for thir hositality and suort. 1

2 3 L 2 -stability and non dissiatd solutions (SK) is rarly satisfid Th st of dgnracy A NSC for strong L 2 stability A NSC for th xistnc of non dissiatd solutions Comlt classification in th cas m = Th cas D(A 1,..., A m, B) = R m : travling wavs? Dcomosition of th solutions whn D is th union of vctor subsacs Gnral statmnt Dcomosition whn n 1 = Discussion on xlicit xamls Comlt classification in th cas n = Summary array of th classification, on roblms and conjcturs Summary array of th classification On roblms and conjcturs Global xistnc around a constant quilibrium for th nonlinar systm Problm formulation Siz of th nighborhood for global xistnc undr (SK) Global xistnc without (SK) Introduction and main rsults 1.1 Problm formulation This work is concrnd with th following n-comonnt hyrbolic systm of balanc laws in m sac dimnsions: w m t + F j (w) x j = Q(w). (1) Hr m, n N, w : R R m R n, w = w(t, x) is th unknown and Q, F j : R n R n ar smooth functions. Such nonlinar systms tyically govrn non quilibrium rocsss in hysics, for mdia with hyrbolic rsons, as, for xaml, in gas dynamics. Thy also aris in th numrical simulation of consrvation laws by rlaxation schms (s [1], [3], [8], [24] and rfrncs citd thrin). In many alications, th sourc trm Q(w) has, or can b transformd by a linar transformation into th form ( ) 0 Q(w) = (2) q(w) with 0 R n1, q(w) R n2, whr n 1, n 2 N, n 1 + n 2 = n. It is wll known that (1) has local (in tim) smooth solutions (s [15], [19]), but ths solutions may dvlo singularitis (i.. shock wavs) in finit tim, 2

3 vn whn th initial data ar smooth and small (s [7], [19]). Howvr, in many hysical xamls, thanks to th intrlay btwn th sourc trm and th flux, thr xist global smooth solutions for a suitabl st of initial conditions. Total dissiation, which consists in rquiring th sourc daming trm to ntr in ach of th quations of th systm distributd all ovr th sac, is a wll known assumtion for global xistnc for suitabl classs of initial data (s [14]). But this condition is too strong and it is not satisfid by systms (1)-(2) with n 1 0 (in which th dissiation is not rsnt in all th comonnts of th systm) and mor gnral systms with rlaxation (s [6], [16]). This is th cas for th isntroic Eulr systm with daming u t v x = 0, v t + f(u) = v, (3) x with f (u) < 0 which has bn considrd in [17] and [13]. In this cas, th daming trm, vn if it ntrs only in th scond quation, may rvnt shock formation. W also rfr to [18] whr similar issus ar addrssd for systms with linar rincial art, which corrsond, in articular, to damd wav quations involving nonlinar convctiv trms. Th xistnc of global smooth solutions for systm (1)-(2) is on of th two main toics of this articl. In fact, th study of th nonlinar systms (1) w shall dvlo rlis on a linarization rincil, around constant quilibria, and this rquirs analyzing artially dissiativ linar hyrbolic systms of th form w m t + w A j = Bw (4) x j whr A 1,..., A m, B ar n n ral matrics, A j := (a (j) k,l ) 1 k,l n bing symmtric, for j = 1,..., m, and B such that B = ( D ), D R n2 n2, X t DX > 0, X R n2 {0}. (5) Not that D is not assumd to b symmtric. Th analysis of linar systms of th form (4) is rlvant, as mntiond abov, to undrstand th bhavior of nonlinar systms and, as w shall s, thy may xhibit a vry rich bhavior. Its analysis is th first goal of this articl. Th solutions of (4) with initial conditions w 0 L 2 (R m, R n ) ar xlicit. Indd, alying th Fourir transform in th x variabl, systm (4) can b rwrittn as m ŵ t + i A j ξ j ŵ = Bŵ, (6) or whr ŵ (t, ξ) = E(ξ)ŵ(t, ξ) (7) t E(ξ) := B ia(ξ), A(ξ) := m ξ j A j. (8) Solving this first ordr ordinary diffrntial quation, w gt ŵ(t, ξ) = x[e(ξ)t]ŵ 0 (ξ). (9) 3

4 Not that, whn n 2 n, th matrix E(ξ) is not corciv. Indd, with th notation ( ) X1 X = C n, X X 1 C n1, X 2 C n2, n 1 + n 2 = n, 2 w hav X t E(ξ)X = X t 2DX 2. Thus, this quadratic form dos not rovid any information on th X 1 comonnt. Howvr, it is by now wll known, in th contxt of linar finit-dimnsional systms, that this fact is not an obstacl for th solutions of (6) to dcay as t. Indd, th intraction of th dissiativ orator B with th timdynamics gnratd by (6) may vntually dissiat all comonnts of solutions. This can b viwd, for instanc, through th Kalman rank condition for th control of finit-dimnsional systms (s [4] and [25]) that w shall discuss in dtail in Sction 2. This fact is also wll known in svral othr contxts, and in articular, for dissiativ wav quations [10] which ar articular instancs of (4), and artially diffusiv artial diffrntial quations whr th notions of hyolliticity [11] and hyocorcivity [22] hav bn introducd to masur th global ffct of artial diffusion in th rgularity and th tim dcay of solutions, rsctivly. As w shall s, undr rathr gnral assumtions on th matrics A 1,..., A m and B, and, mor rcisly, undr th so calld Kalman rank condition for th air (A(ξ), B), it can b rovd that C > 0, λ(ξ) > 0 : x[e(ξ)t] C λ(ξ)t. (10) This dcay rorty, togthr with xlicit stimats on th ositivity and dndnc of λ(ξ) with rsct to ξ, allows dscribing accuratly th asymtotic bhavior of solutions of (4) as t, and driving a dcomosition of solutions in which various trms dcaying with diffrnt rats can b distinguishd. Obviously, th ovrall ictur dnds in a critical way on th rortis of th function ξ λ(ξ). Th analysis of this function is a comlx issu to which w will dvot a significant art of this articl. Thr is in fact an xtnsiv litratur on th subjct. For instanc, in [20], th authors study systms (4) undr th so-calld Shizuta-Kawashima condition: (SK) ξ R m, Kr(B) {ignvctors of A(ξ)} = {0}, (11) and rov that C, c > 0 s. t. ξ R m, x[e(ξ)t] C c min{1, ξ 2 }t. (12) Thanks to (9) and (12), th authors dduc that any solution w of (4) associatd to an initial condition w 0 L 1 L 2 (R m, R n ) can b dcomosd as w = w 1 + w 2 (13) whr w 1 (t) L2 (R m,r n ) C λt w 0 L2 (R m,r n ), t (0, + ), w 2 (t) L (R m,r n ) Ct m 2 w 0 L1 (R m,r n ), t (0, + ), (14) 4

5 and C, λ ar ositiv constants dnding only on A 1,..., A m, B. Th two comonnts w 1 and w 2 corrsond, rsctivly, to th high and low frquncy comonnts. Th high frquncy comonnt dcays xonntially whil th low frquncy on dcays olynomially with th dcay rat of th hat krnl. W rfr to [18] for similar rsults for damd wav quations with nonlinar convction. This rsult has motivatd many othrs. For instanc, in [2], th authors rovd mor rcis dcay rats for th high frquncy comonnt w 1 undr (SK), and xtndd th analysis for non linar systms. Estimat (12) is quivalnt to saying that λ(ξ) c min{1, ξ 2 } in (10). In articular, (SK) imlis that λ(ξ) may only dgnrat quadratically at ξ = 0. Roughly saking, (SK) is a natural sufficint condition to guarant that th daming trm affcts all th comonnts of th systm and th L 2 -dcay of th solutions of (4) as t +. But it is not shar. Indd, as w shall s, thr ar many situations in which λ(ξ) dgnrats on othr oints than ξ = 0, but still th L 2 -dcay of solutions holds, togthr with dcomositions in th sirit of (13), but involving xtra trms, dcaying mor slowly than th m-dimnsional hat krnl. To do that w dvlo a carful analysis of th finit-dimnsional bhavior of systm (6) in trms of th multi-dimnsional aramtr ξ. This analysis is insird by control thortical tools. Indd, our aroach starts from th obsrvation that (SK) holds if th air of matrics (A(ξ), B) satisfis th Kalman rank condition: rk[b, A(ξ)B,..., A(ξ) n 1 B] = n. Th ky ingrdint to obtain a comlt dcomosition of solutions is th obtntion of a carful masur of th dcay rat λ(ξ) of solutions of (6) as a function of ξ. To do this w construct xlicit Lyaunov functionals, taking advantag of th intraction of th matrix B with th dynamics gnratd by A(ξ). This kind of Lyaunov function is similar to thos introducd by C. Villani (s [22]) for th analysis of th dcay of artially diffusiv systms and is also linkd, as w mntiond abov, to th xtnsiv litratur on damd wav quations (s [10] and [18], for instanc). Th aroach w dvlo hr, in addition of bing mor systmatic, has also th addd advantag of bing simlr to b carrid out from a tchnical viwoint sinc w avoid som of th long dvlomnts in [20] for roving th sufficincy of (SK) to achiv th dcomosition abov. Lt us now rturn to th nonlinar systms of balanc laws (1). Th xistnc of global smooth solutions in a nighborhood of a constant quilibrium W R n and Q(W ) = 0 was rovd in [23], undr a suitabl dissiation assumtion, whn th linarizd systm around W satisfis (SK). Lt us also mntion [9] for th sam rsult in on sac dimnsion (m = 1) with many xamls of alication. W also rfr to [5], whr th authors considr th multidimnsional isothrmal Eulr quations with a strong rlaxation and study th asymtotic bhavior of th solutions whn th rlaxation tim tnds to zro. Thy us th argumnts of [23] in ordr to rov th xistnc of global smooth solutions in a nighborhood of a constant quilibrium, whos siz is uniform with rsct to th rlaxation tim. Th tchniqus w dvlo in this articl, in th fram of [23], allow us to obtain xlicit stimats of th siz of th nighborhood of W whr th 5

6 xistnc of global smooth solutions holds. As a consqunc of this, using an argumnt insird in Coron s rturn mthod ([4]), w ar abl to rov a global xistnc rsult around a constant quilibrium that dos not fulfill th (SK) condition. 1.2 Main rsults Lt us now dscrib th contnt of th articl in mor dtail. Sctions 2, 3, 4 and 5 ar dvotd to th analysis of artially dissiativ linar hyrbolic systms and sction 6 to th nonlinar systms of balanc laws. Mor rcisly, sction 2, is concrnd with th Kalman rank condition, th (SK) condition, th notion of hyocorcivity and th dcay rats. Subsction 2.1 is dvotd to a rliminary discussion in which w show that (SK) is quivalnt to th classical Kalman rank condition in control thory for th airs (A(ξ), B) and all ξ 0. Thn w dvlo th roof of dcay (10) using Lyaunov functionals (s subsction 2.2). In articular, this yilds a mor systmatic aroach and a simlr way to gt th dcomosition (13)-(14) in [20] undr th condition (SK) (s subsction 2.3). In sction 3, using th tools dvlod in sction 2, w invstigat th asymtotic bhavior for (4) without th (SK) condition. Mor rcisly, w study th L 2 -stability and th non dissiatd solutions of (4). For doing that w introduc th st of dgnracy D(A 1,..., A m, B) := {ξ R m : Kr(B) {ignvctors of A(ξ)} {0}}, (15) i.. th st of valus of ξ for which (SK) fails. Undr th (SK) condition this st is th trivial on {0}. But, of cours, this is fals in gnral and, as w shall s, th structur of this st dtrmins th asymtotic bhavior of solutions. In subsction 3.2, using th fact that th Kalman condition fails on th st of dgnracy whn (SK) dos not hold, w rov that, ithr D(A 1,..., A m, B) is a strict algbraic submanifold of R m, or D(A 1,..., A m, B) = R m. In subsction 3.3, w rov that, in th first cas, (4) is strongly stabl in L 2, i.. all L 2 solutions tnd to zro in L 2 as t. In subsction 3.4, w rov that, in th scond cas, thr xist non dissiatd solutions of constant L 2 -norm. In subsction 3.5, w study th 1D cas m = 1 whatvr th siz n of th systm is. W show that, in this cas, th (SK) condition is shar in th sns that it charactrizs comltly th bhavior of th solutions : (SK) is a ncssary and sufficint condition for th strong stability in L 2 (R, R n ), (SK) is a ncssary and sufficint condition for th dcomosition (14), 6

7 without (SK), any solution associatd to an L 2 initial condition can b dcomosd as w = w 1 + w 2 + w trw whr w 1, w 2 satisfy (14) and w trw is th sum of a finit numbr of travling wavs. This shows, in articular, that in 1D w may not hav a third comonnt dcaying at infinity with a slowr dcay rat. Th lattr may only aris in th multi-dimnsional cas. Subsction 3.6 is dvotd to analyz th most dgnrat cas in which D(A 1,..., A m, B) = R m. W show that this is a ncssary condition for th xistnc of travling wavs with L 2 rofils. Howvr, contrary to th cas n = 1, this condition fails to b sufficint whn m 2. Indd, th solutions of constant nrgy may, in gnral, hav a mor comlx structur that bing sums of travling wavs. In sction 4, w invstigat th analogu of th dcomosition (13)-(14) for th solutions of (4), whn (SK) is not fulfilld and whn th masur of th st of dgnracy D(A 1,..., A m, B) vanishs. W answr this qustion in som articular cass whr th st of dgnracy is th union of a finit numbr of vctor subsacs. In subsction 4.1, w rov a gnral statmnt giving a dcomosition for th solutions of (4) whn th st of dgnracy D(A 1,..., A m, B) is th union of a finit numbr of vctor subsacs and undr an assumtion of th following ty N (ω) Cdist(ω, D(A 1,..., A m, B)) α, ω S m 1, (16) for som constants C > 0 and α 2, whr n 1 N (ω) := min{ ɛ m k BA(ω) k x 2 ; x S n 1 }, (17) k=0 for a suitabl small nough ɛ and suitabl xonnts m k (s Proosition 1). Not that th valu of N (ω) rovids a quantitativ vrsion of th (SK) rorty, or th Kalman rank rorty, in th sns that, whn it fails on an isolatd oint ω, it holds in th nighboring ons (that do not blong to D(A 1,..., A m, B)), with an xlicit lowr bound on N as a owr of th distanc function. This dcomosition is of th form w = w 1 + w 2 + w 3 + w 4 whr w 1 and w 2 satisfy (14) and w 3 (rs. w 4 ) contains som (but not all) high (rs. low) frquncis and dcays lik t 1/α (rs. t r/α whr r N ). In this way w s that, in dimnsion m 2, thr is a whol class of hnomna that do not aris undr th condition (SK). Not that w alrady got a comlt classification of th ossibl dcomositions for th solutions of (4) whn m = 1 in subsction 3.5. In subsction 4.2, w study th articular cas n 1 = 1 (i.. Kr(B) = San( 1 )) in which th condition on th st of dgnracy bing th union of vctor subsacs is automatically satisfid. In that cas, (16) holds with a smallr xonnt α = 2. This lads to a dcomosition for th solutions of (4), in this articular cas n 1 = 1. In subsction 4.3, considring xlicit xamls, w show that thr ar situations in which α = 4 is th smallst xonnt on can hav in (16), and th aramtr r may tak any ositiv valu. 7

8 In subsction 4.4, w dduc from th rvious analysis a comlt classification in th cas n = 2, whatvr th sac dimnsion m is. In sction 5, w rcaitulat th various rsults obtaind in th rvious sctions, rsnting thm in a tabl. This classification is still incomlt in th sns that not all ossibl valus of m and n ar covrd by our analysis. Indd, w do not rovid a dcomosition in th gnral cas whr th st of dgnracy is th union of a finit numbr of vctor subsacs bcaus w only study articular xamls, whn th st of dgnracy D(A 1,..., A m, B) is an algbraic submanifold that is not a union of vctor subsacs, or whn D(A 1,..., A m, B) is th whol sac. W also mak rcis som on qustions and som conjcturs about th cass that our artial classification dos not covr. In Sction 6, w study th xistnc of global (in tim) smooth solutions for th non linar systm (1), locally around a constant quilibrium W R n. In subsction 6.1 w mak rcis th contxt of our work. Th novlty of our study with rsct to [23] is that w do not imos th (SK) condition on th linarizd systm around W. To do that w rocd in two sts. W first assum that (SK) holds and taking advantag of th xlicit Lyaunov function introducd in subsction 2.2, w mak th rsult in [23] mor rcis by giving an xlicit stimat on th siz for th nighborhood of W on which global xistnc holds. Thn, in subsction 6.3, using ths xlicit stimats and undr som suitabl assumtions on th nonlinaritis, w rov th xistnc of global solutions for (1), around a constant quilibrium W that dos not satisfy (SK). This is don assuming th xistnc of a family of constant quilibria fulfilling (SK) convrging to W. This aroach is insird by Coron s rturn mthod for th controllability of nonlinar systms (s [4] for an introduction and xamls of alications), that taks advantag of th nonlinarity of th systm. W conclud subsction 6.3 with an xaml of alication of th rvious thorm. At this oint, it is convnint to not that th fact that (SK) was not ncssary for th xistnc of global smooth solutions around a constant quilibrium was alrady known. Indd, Zng rovd in [24] th xistnc of global 1D (m = 1) smooth solutions for an quation of gas dynamics, around a constant stat, without th (SK) condition, slitting th systm in two arts, on of thm bing linarly dgnrat. But, to our knowldg, thr wr no gnral rsult of global xistnc for artially dissiativ hyrbolic systms without (SK) (for arbitrary m, n). Our rsult is th first on in that dirction. On may xct that a furthr analysis of th linarizd systms along th lins w do in sctions 2, 3, 4 togthr with this way of aroaching th nonlinar on, will allow to gnraliz furthr th rsult w ar rsnting hr. Notations: In this articl, R(.), I(.) dnot th ral and imaginary arts of comlx numbrs, ( 1,..., n ) is th canonical basis of R n and.,. dnots th hrmitian roduct in C n, X, Y = X t Y. Th notations givn in this introduction (m, n, n 1, n 2, A j = (a (j) k,l ) 1 k,l n, B, D, D(A 1,..., A m, B)) will b usd all along th ar. 8

9 2 Rank conditions, (SK), hyocorcivity and dcay rats In subsction 2.1, w show th quivalnc btwn th (SK) condition for hyrbolic systms and th Kalman rank condition in control thory. In subsction 2.2, taking advantag of this quivalnc, w rov an xlicit dcay rat for λ(ξ) in (10), by mans of a Lyaunov functional. Finally, in subsction 2.3, w rcovr Shizuta-Kawashima s dcomosition (13)-(14) for th solutions of (4) undr th (SK) condition. 2.1 (SK) and Kalman rank conditions Th following lmma is on of th ky ingrdints of this sction. Lmma 1 Lt n N, A, B b n n matrics with ral cofficints, such that B has th form (5). Th following statmnts ar quivalnt (1) A and B satisfy (SK) : {ignvctors of A} Kr(B) = {0}, (2) y C n {0}, t B x(at)y dos not vanish on R, (3) y C n {0}, thr xists k {0, 1,..., n 1} such that BA k y 0, (4) for vry a 0,..., a n 1 > 0, th xrssion dfins a norm on C n N(y) := ( n 1 ) 1/2 a k BA k y 2 k=0 (5) (A, B) satisfis th Kalman rank condition : th (n 2 ) n Kalman matrix B K := BA... (18) BA n 1 has rank n. Ths quivalncs ar classical (s for xaml [4, Chatr 1.3], [21, Thorm 2.2.1]). W giv a roof for th sak of comltnss. Proof of Lmma 1: Lt us rov (1) (2)". W assum that (2) is fals. Thn, thr xists y C n {0} such that B x(at)y 0. (19) Lt ν dt(a XI n ) := (X λ j ) rj 9

10 b th charactristic olynomial of A. W hav Th quality (19) givs C n = ν y = ν y j. ν r j 1 λjt k=0 Kr[(A λ j ) rj ] t k k! B(A λ j) k y j 0. Thanks to th linar indndnc of th family w dduc that {t k λjt ; 0 k r j 1, 1 j ν} B(A λ j ) k y j = 0, k {0,..., r j 1}, j {1,..., ν}. Lt j {1,..., ν} b such that y j 0 and k {0,..., r j 1} b th largst intgr such that (A λ j ) k y j 0. Thn, (A λ j ) k y j is an ignvctor of A in th null sac of B. Thus, (1) is fals. Lt us rov now that (2) (3)". W assum that (3) is fals. Thn, thr xists y C n {0} such that BA k y = 0 for k = 0,..., n 1 and thanks to th Cayly-Hamilton thorm, BA k y = 0 for vry k N. Thus B x(at)y 0 and (2) is fals. Th imlication (3) (4)" is clar. Lt us rov (4) (5)". W assum (4). If y Kr(K), thn BA k y = 0 for k = 0,..., n 1. Thus N(y) = 0 and so y = 0. This rovs that K is injctiv and thus rk(k) = n. Finally, lt us rov (5) (1)". W assum (1) is fals. Lt v S n 1 b an ignvctor of A in th null sac of B. Thn Ky = 0 and, thus, K is not injctiv. Thrfor rk(k) < n, i.. (5) is fals. 2.2 Lyaunov functionals and xlicit dcay rats According to th rsults of th rvious sction thr is a vry xlicit connction btwn th (SK) condition, which ariss naturally in th analysis of th dcay of artially dissiativ hyrbolic systms, and th Kalman rank condition in finit dimnsional control thory. In this subsction w dvlo on of th ky lmnts of this articl which consists in driving xlicit dcay rats for th finit-dimnsional systm (6) in trms of ξ. To do this w construct xlicit Lyaunov functionals, which ar insird on thos rank conditions and mor rcisly in th statmnt (4) of Lmma 1. W st ξ = ρω, with ρ > 0 and ω S m 1, and, consquntly, rwrit (6) in th form: ẋ = (B + iρa(ω))x, x(0) = x 0 C n. (20) 10

11 Proosition 1 W fix a family of non-ngativ ral numbrs (m k ) 0 k n such that 0 = m 0 < m 1 <... < m n, (21) m k m k 1 + m k+1 δ > 0, k = 1,..., n 1, (22) 2 for som δ > 0. Lt A 1,..., A m, B b n n ral matrics such that B has form (5). For ω S m 1, ɛ > 0 w dfin th symmtric non-ngativ matrix and its minimal ignvalu n 1 M ɛ (ω) := ɛ m k (A(ω) t ) k B t BA(ω) k (23) k=0 n 1 N,ɛ (ω) := min{ x, M ɛ (ω)x ; x S n 1 } = min{ ɛ m k BA(ω) k x 2 ; x S n 1 }. (24) Thn, thr xist ɛ = ɛ (A 1,..., A m, B) (0, 1) and c = c(a 1,..., A m, B) > 0 such that, for vry ɛ (0, ɛ ), x 0 C n, ρ (0, + ) and ω S m 1, th solution of (20) satisfis k=0 x(t) 3 x 0 cn,ɛ(ω) min{1,ρ2 }t, t (0, + ). (25) Rmark 1 This rsult rovids an xonntial dcay rat for x[e(ρω)t], which is xlicit in trms of ρ and ω, and mor rcisly on N,ɛ (ω). As w will s in sctions 2.3 and 4, th main intrst of this Proosition is that it rducs th roblm of th asymtotic bhavior of th solutions of (4) to th study of th ral valud ma ω S m 1 N,ɛ (ω) R +. Not that Proosition 1 holds without assuming th (SK) condition, which, in fact, only ntrs whn trying to obtain a uniform lowr bound on N,ɛ (ω) for ω S m 1, in sction 2.3. In articular, it could b that, for som valu ω of ω, N,ɛ (ω ) = 0. In that cas, to gt xlicit dcay rats for (4), on has to analyz th bhavior of N,ɛ (ω) for valus ω clos to ω. This fact will lay an imortant rol whn driving dcomositions of solutions in th absnc of th (SK) condition and analyzing thir dcay rats as t in sction 4. Rmark 2 Th constant 3 in (25) can b rlacd by any constant C > 1. Indd, an asy adatation of th roof blow rovids th following mor gnral statmnt: for vry C > 1, thr xist ɛ = ɛ (A 1,..., A m, B) > 0 and c = c( C, A 1,..., A m, B) > 0 such that th sam conclusion holds with 3 rlacd by C. On th othr hand, C bing fixd, it is natural to rais th issu of finding th valu of ɛ that maximizs th dcay rat. This is an on roblm. Rmark 3 Notic that Proosition 1 holds without assuming th symmtry of th matrics A 1,..., A m. This will b usd in sction 6. Proof of Proosition 1: First, lt us introduc som notations. Sinc B has th form (5), thr xists C 1 = C 1 (B) > 0 such that R Bx, x C 1 Bx 2, x C n. (26) 11

12 W considr th charactristic olynomial n 1 dt(xi n A(ω)) = X n + a k (ω)x k, k=0 a k (ω) bing th cofficint of its k th ordr trm. W also st M 1 := max{ BA(ω) k : 0 k n 1, ω S m 1 }, (27) M 2 := max{ a k (ω) : 0 k n 1, ω S m 1 }. (28) Lt ɛ = ɛ (A 1,..., A m, B) > 0 b small nough so that, for vry ω S m 1, n 1 ɛ m k (A(ω) t ) k B t BA(ω) k 1 < 1 2, (29) k=1 M 1 ɛ m1 + ɛ δ + nm 2 2ɛ δ < C 1 2, (30) M 2 1(n 1) ɛ m1 + ɛ δ C (1 + nm 2 2) < (31) For ɛ (0, ɛ ), ρ (0, + ), ω S m 1 and x C n w considr L ρ,ω,ɛ (x) := x 2 + min{ρ, 1 n 1 ρ } ɛ m k I BA(ω) k 1 x, BA(ω) k x. (32) Thanks to (29), on has k=1 1 2 x 2 L ρ,ω,ɛ (x) 3 2 x 2, x C n, ρ (0, + ), ω S m 1, ɛ (0, ɛ ). (33) Lt ɛ (0, ɛ ). In ordr to simlify th notations and sinc ɛ is fixd, in th squl, w will rathr writ N (ω) and L ρ,ω instad of N,ɛ (ω) and L ρ,ω,ɛ. Whn x solvs (20), on has d dt [L ρ,ω(x)] = 2R( (B + iρa ω )x, x ) min{ρ, 1 ρ } n 1 k=1 ɛm k I( BA k 1 ω min{ρ, 1 ρ } n 1 whr A ω := A(ω). W now distinguish two cass: 0 < ρ < 1 and ρ > 1. Th cas ρ (0, 1): Using (26), w gt d dt [L ρ,ω(x)] 2C 1 Bx 2 ρ 2 n 1 k=1 ɛm k BA k ωx 2 +ρ n 1 k=1 ɛm k Bx [ BA k 1 ω +ρ 2 n 1 k=1 ɛm k BA k 1 Morovr, by (27) and (21) w gt (B + iρa ω )x, BA k ωx ) k=1 ɛm k I(BA k 1 ω x, BA k ω(b + iρa ω )x ) (34) ω x BA k+1 ω BA k ωx + BA k ω BA k 1 ω x. ρ n 1 k=1 ɛm k Bx [ BA k 1 ω BA k ωx + BA k ω BA k 1 ω x ] M 1 ρ n 1 k=1 ɛm k Bx [ BA k ωx + BA k 1 ω x ] M 1 ρɛ m1 Bx 2 + n 1 k=1 2M 1ρɛ m k Bx BA k ωx M 1 ρɛ m1 Bx 2 + ( ) n 1 C 1 k=1 n 1 Bx 2 + ρ 2 M 2 1 (n 1) C 1 ɛ 2m k BA k ωx 2 (C 1 + M 1 ρɛ m1 ) Bx 2 + ρ 2 M 2 1 (n 1) n 1 C 1 k=1 ɛ2m k BA k ωx x ] (35) (36)

13 Using (22), w gt ρ 2 n 1 k=1 ɛm k BA k 1 ω x BA k+1 ω x ρ 2 ɛ δ n 1 k=1 ɛ mk 1+m k+1 2 BA k 1 ω x BA k+1 ω x 1 2 ρ2 ɛ δ n 1 k=1 ɛm k 1 BA k 1 ω x 2 + ɛ m k+1 BA k+1 ρ 2 ɛ δ n k=0 ɛm k BA k ωx 2. ω x 2 (37) Thanks to Cayly-Hamilton thorm, Cauchy-Schwartz inquality, (28) and (21), th last trm of th right hand sid in th rvious inquality satisfis ρ 2 ɛ δ+mn BA n ωx 2 = ρ 2 ɛ δ+mn n 1 k=0 a k(ω)ba k ωx 2 nm 2 2ρ 2 ɛ δ+mn n 1 k=0 BAk ωx 2 nm 2 2ρ 2 ɛ δ n 1 k=0 ɛm k BA k ωx 2. (38) Finally, using (36), (37) and (38) in (35) and thanks to (30), (31), w gt d dt [L ρ,ω(x)] C 1 2 Bx 2 ρ2 n 1 ɛ m k BA k 2 ωx 2. Thrfor, using (24) and (33) w gt k=1 whr d dt [L ρ,ω(x(t))] 2 cn (ω)ρ 2 L ρ,ω (x(t)) (39) c := 1 6 min{c 1, 1}. Finally, thanks to (33), w gt x(t) 3 x 0 cn (ω)ρ2t. Th cas ρ (1, + ) : Now, lt us justify th dcay stimat in Proosition 1 for ρ (1, + ). Lt ρ (1, + ) and ω S m 1. Whn x solvs (20), with th sam argumnts as in th rvious cas, w gt d dt [L ρ,ω(x(t))] 2C 1 Bx 2 n 1 k=1 ɛm k BA k ωx 2 and + 1 n 1 ρ k=1 ɛm k Bx [ BA k 1 ω + n 1 k=1 ɛm k BA k 1 ω x BA k+1 BA k ωx + BA k ω BA k 1 ω x ] ω x d dt [L ρ,ω(x(t))] C 1 2 Bx 2 1 n 1 ɛ m k BA k 2 ωx 2. k=1 (40) Not that th stimats w gt ar vry clos to thos of th cas ρ (0, 1) xct for th fact that, du to th wight 1/ρ on th scond trm in (32), all trms in which ρ aars hav to b dividd by ρ. Finally, with th sam comutations as in th first cas, w gt x(t) 3 x 0 cn (ω)t. 13

14 Rmark 4 Th xlicit Lyaunov function (32) is insird by thos introducd by Villani in [22] to driv dcay stimats for artially diffusiv systms. In [22] th orators undr considration ar of th form L = A A + B whr B is antisymmtric. In our cas th orator has rathr th form L = B + iρa(ω) whr iρa(ω) is antisymmtric but B dos not ncssarily hav any symmtry rorty. Th Lyaunov functional w us is also similar to thos usd in th analysis of th dcay rortis of dissiativ wav quations, as, for instanc, v tt v + v t = 0. Thr th systms undr considration ar scond ordr (in tim), and in th Fourir stting thy tak th form of th following dissiatd harmonic oscillator ˆx + ξ 2ˆx + ˆx = 0. Th nrgy of th systm is thn givn by (t) = 1 2 [ ˆx 2 + ˆx 2], whil th Lyaunov functional to b usd to driv th dcay is of th form L(t) = 1 2 [ ˆx 2 + ˆx 2] + εˆxˆx. This corrsonds rcisly to functionals of th form (32) in th articular cas in which n = A nw roof for Shizuta and Kawashima s dcomosition As a consqunc of Proosition 1, th dcomosition (14) is straightforward. Thorm 1 W assum that B has form (5), A 1,..., A m ar symmtric and (SK) is satisfid. Thn, thr xist C = C(A 1,..., A m, B), λ = λ(a 1,..., A m, B) > 0 such that, for vry w 0 L 1 L 2 (R m, R n ), th solution w(t, x) of (4) can b dcomosd as in (13) whr (14) holds. Proof of Thorm 1 : Lt ɛ > 0 b as in Proosition 1 and ɛ (0, ɛ ). Thanks to th imlication (1) (4)" in Lmma 1, and in viw of th (SK) assumtion, w hav N,ɛ (ω) > 0 for vry ω S m 1. Morovr, th function ω N,ɛ (ω) is continuous on th comact st S m 1 and, thrfor, thr xists N > 0 such that N,ɛ (ω) N, for vry ω S m 1. W dfin w 1 and w 2 by ŵ 1 (t, ξ) := ŵ 1 (t, ξ)1 ξ >1 and ŵ 2 (t, ξ) := ŵ 1 (t, ξ)1 ξ <1. Tanks to Proosition 20, on has, ŵ 1 (t, ξ) 3 ŵ 0 (ξ) cn t, ŵ 2 (t, ξ) 3 ŵ 0 (ξ) cn ξ 2t, for all ξ R m and t (0, + ). Thus, on has w 1 (t) L2 (R m,r n ) 3 w 0 L2 (R m,r n ) cn t, w 2 (t) L (R m,r n ) ŵ 2 (t) L 1 (R m,r n ) 3 ŵ 0 (ξ) cn ξ 2t dξ R m Ct m/2 w 0 L 1 (R m,r n ). 14

15 Rmark 5 Lt us comar th tools dvlod hr with thos in [20]. In [20], th authors us algbraic tools to justify th quivalnc btwn (SK) and th xistnc of a comnsating function (notion dfind blow) for (4). Thn, thy us this comnsating function to rov th dcomosition of Proosition 1 with an nrgy aroach. Dfinition 1 A C ma ω S m 1 K(ω) C n n is a comnsating function for (4) if K( ω) = K(ω), ω S m 1, K(ω) is a skw-symmtric matrix, for vry ω S m 1, B +B (K(ω)A(ω) + (K(ω)A(ω))t ) is ositiv dfinit for vry ω S m 1. Th roof of Proosition 1 contains th argumnts to justify that, whn ɛ δ (1+ nm 2 2) < 1/2, th xrssion n 1 K(ω) := ɛ m k [(A(ω) t ) k B t BA(ω) k 1 (A(ω) t ) k 1 B t BA(ω) k ] k=1 dfins a comnsating function and that B + B (K(ω)A(ω) + (K(ω)A(ω)) ) 1 2 min{c 1, 1}N,ɛ (ω). Notic that, onc a comnsating function K(ω) is known, th xrssion L ɛ (x) := x 2 + ɛ min{1, ρ}i K(ω)x, x for ɛ > 0 small nough, rovids a Lyaunov function for th roof of Proosition 1. Our roof is howvr much mor dirct and yilds th dsird dcay rat by mans of an xlicit Lyaunov function, insird by Lmma 1. 3 L 2 -stability and non dissiatd solutions This sction is dvotd to th study of th L 2 -stability and th non dissiatd solutions of (4) whn (SK) dos not hold i.. D(A 1,..., A m, B) {0}. In subsction 3.1, w show that thr ar many situations in which (SK) dos not hold, which is a motivation for th analysis dvlod in th following subsctions. In subsction 3.2 w rov that, ithr D(A 1,..., A m, B) is a strict algbraic submanifold of R m, or, D(A 1,..., A m, B) = R m. In subsction 3.3, w rov that, in th first cas, (4) is strongly stabl in L 2, i.. all L 2 solutions tnd to zro in L 2 as t. In subsction 3.4, w rov that, in th scond on, thr xist non dissiatd solutions of constant L 2 -norm. In subsction 3.5, w dduc a comlt classification of th dcomosition of th solutions in 1D (m = 1). In subsction 3.6, w rov that D(A 1,..., A m, B) = R m is a ncssary condition for th xistnc of travling wavs with L 2 rofils but this condition fails to b sufficint whn m 2. 15

16 3.1 (SK) is rarly satisfid Th goal of this sction is to mhasiz that thr ar many situations in which (SK) dos no hold. This fact motivats th analysis dvlod in th following subsctions. Th first cas is whn m > n and B has form (5) with n 2 < n. Indd, in that cas th st {ξ R m ; m a (j) k,1 ξ j = 0 for k = 2,..., n} is a non mty vctor subsac of R m. It is containd in D(A 1,..., A m, B). Indd, for vry ξ in this st, w hav A(ξ) 1 = ( m a(j) 1,1 ξ j) 1, thus 1 is an ignvctor of A(ξ) that blongs to Kr(B). As a consqunc, th rorty (SK) may only b satisfid whn m n. Howvr, for any air (m, n) with m n, and any valu of n 1 thr ar xamls for which (SK) dos not hold. Indd, lt m, n N with m n, n 1 {1,..., n 1} and n n ral matrics A 1,..., A m, B with A 1,..., A m symmtric and B of th form (5). W assum that a (1) 2,1 0, a(j) 2,1 = 0 for j = 2,..., m, and a(j) k,1 = 0 for j = 1,..., m and k = 3,..., n. Thn A(ξ) 1 = ( m a(j) 1,1 ξ j) 1 + a (1) 2,1 ξ 1 2. Thus, for vry ξ in th hyrlan {ξ R m ; ξ 1 = 0}, 1 is a ignvctor of A(ξ) that blongs to Kr(B). Thrfor, D(A 1,..., A m, B) contains this hyrlan and (SK) is not fulfilld. 3.2 Th st of dgnracy Proosition 2 Lt A 1,..., A m, B b n n ral matrics such that A 1,..., A m ar symmtric and B has form (5). Th st of dgnracy D(A 1,..., A m, B) is an algbraic submanifold of R m. In othr words, thr xists a finit family of olynomials (P j ) j J R[X] such that D(A 1,..., A m, B) = {ξ R m ; P j (ξ) = 0, j J}. (41) Thus, ithr D(A 1,..., A m, B) = R m, or D(A 1,..., A m, B) has zro masur. Morovr, D(A 1,..., A m, B) is stabl by homothtis. Rmark 6 Not that it can han that D(A 1,..., A m, B) = R m, th dissiation matrix B bing non-trivial. Indd, th fact that D(A 1,..., A m, B) = R m mans only that, for ach ξ, thr is a dirction in which th dissiation mchanism is not ffctiv but this is comatibl, as w shall s, with B bing non-trivial (s Sction 3.6). Proof of Proosition 2 : Thanks to Lmma 1, on may charactriz th st of dgnracy D(A 1,..., A m, B) as th st of thos ξ R m such that any n n subdtrminant of th Kalman matrix K dfind in (18) (with A rlacd by A(ξ)) vanishs. Th olynomials P j in th statmnt of Proosition 2 ar rcisly thos corrsonding to ths subdtrminants. 16

17 Thn, ithr all th olynomials vanish idntically, and thn D(A 1,..., A m, B) = R m, or, if som of thm do not, th masur of D(A 1,..., A m, B) vanishs. With th dfinition (15) of th st of dgnracy, it is clar that, whn ξ D(A 1,..., A m, B) and α R, thn, αξ D(A 1,..., A m, B). According to th rvious roositions, thr ar thrfor only two ossibl cass, dnding on th siz of th st of dgnracy. Of cours, smallr D(A 1,..., A m, B) is, bttr dcay rortis w xct. Th cas in which (SK) holds is a vry articular instanc whr th masur of D(A 1,..., A m, B) vanishs, sinc, in that cas, actually, D(A 1,..., A m, B) is rducd to th trivial st {0}. On of th main goals of this sction is to dscrib what hans whn th masur of th st of dgnracy vanishs, but (SK) dos not hold. To do that it is first convnint to considr som xamls. Th first conclusion is that, in som cass, D(A 1,..., A m, B) is a vctor subsac. But, this is not ncssarily always th cas. This maks th analysis of th dcay rat of (7) as a function of ξ mor dlicat. W introduc th notation α k,l = α k,l (ξ) := m a (j) k,l ξ j, for ξ R m, 1 k, l n, whr A j = (a (j) k,l ) 1 k,l n. Thn A(ξ) = (α k,l ) 1 k,l n. Obviously, α k,l = α l,k, 1 k, l n. In th following xamls, it is asir to us th dfinition (15) of D(A 1,..., A m, B) in ordr to comut th olynomials than using th minors of th Kalman matrix. Examl 1 : Lt us considr th cas n 1 = 1. Th dfinition givs D(A 1,..., A m, B) = {ξ R m ; α k,1 = 0, k {2,..., n}}. Thus, D(A 1,..., A m, B) is a vctor subsac. Examl 2 : Lt us considr th cas n 1 = 2. Th dfinition allows charactrizing D(A 1,..., A m, B) as th st of thos ξ R m for which thr xists α, β, γ R such that ( α1,1 α 1,2 α 1,2 α 2,2 ) ( α β ) ( α = γ β ) and αα k,1 + βα k,2 = 0, k 3. Comuting xlicitly α, β as functions of α 1,1, α 1,2, α 2,2, distinguishing th cass (α, β) = (1, 0), (α, β) = (0, 1) and (α 0 and β 0), w gt whr D = D 1 D 2 D 3+ D 3, D j (A 1,...A m, B) := {ξ R m ; α k,j = 0, k j} for j {1, 2}, and D 3± (A 1,...A m, B) ar th sts of thos ξ R m such that, for vry k 3, ( ) 2α 1,2 α k,1 = α 1,1 α 2,2 ± (α 1,1 α 2,2 ) 2 + 4α1,2 2 α k,2. 17

18 Now, w distinguish two cass. Examl 2.a : Lt us considr matrics A 1,..., A m such that a (j) 1,1 a(j) 2,2 = δa (j) 1,2 for j = 1,..., m, for som δ R. Thn α 1,1 α 2,2 = δα 1,2 and D 3+ D 3 (A 1,...A m, B) = {ξ R m ; 2α k,1 = [δ ± δ 2 + 4]α k,2, 3 k n}. Thus D(A 1,..., A m, B) is th union of a finit numbr of vctor subsacs of R m. Examl 2.b : Now, lt us tak n = 3, m = 5 and A 1,..., A 5 dfind by A 1 := , A 2 := , A 3 := , A 4 := , A 5 := whr th notation mans that th corrsonding cofficint may hav any ral valu. Thn, α 1,1 = ξ 1, α 2,2 = ξ 2, α 1,2 = ξ 3, α 3,1 = ξ 4, α 3,2 = ξ 5. Thus D 3± (A 1,...A 5, B) = {ξ R 5 ; 2ξ 3 ξ 4 + (ξ 2 ξ 1 )ξ 5 = ±ξ 5 (ξ 1 ξ 2 ) 2 + 4ξ 2 3 }. Obviously, D 3± ar not vctor subsacs and, consquntly, th st of dgnracy can not b writtn as th union of a finit numbr of vctor subsacs of R m nithr. 3.3 A NSC for strong L 2 stability Proosition 3 Lt A 1,..., A m, B b n n ral matrics with B of th form (5), A 1,..., A m bing symmtric. Th following statmnts ar quivalnt, (1) th masur of D(A 1,..., A m, B) vanishs, (2) any solution of (4) with L 2 (R m, R n ) initial condition convrgs to zro strongly in L 2 (R m, R n ) as t. Proof: First, w rov that (1) (2)". W assum that th masur of D(A 1,..., A m, B) vanishs. Lt w 0 L 2 (R m, R n ) and w b th solution of (4) with initial condition w 0. Alying Proosition 1, w gt, for ɛ > 0 small nough, ŵ(t, ξ) 3 ŵ 0 (ξ) c min{ ξ 2,1}N,ɛ(ω)t whr ω := ξ/ ξ. Thanks to th imlication (1) (4)" in Lmma 1 and sinc th masur of D(A 1,..., A m, B) vanishs, thn, for almost vry ω S m 1, N,ɛ (ω) > 0. Thus ŵ(t, ξ) 0 whn t + for almost vry ξ R m. Morovr, ŵ(t,.) ŵ 0 (.) L 2 (R m, R n ). Thus, th dominatd convrgnc thorm imlis that ŵ(t) convrgs to zro in L 2 (R m, R n ) whn t +. Now, w rov (2) (1)". Lt us assum (2) holds. Lt ϕ S(R m, R) b such that ϕ > 0 on R m. Lt w (1) 0 L 2 (R m, R n ) b dfind by ŵ (1) 0 (ξ) := ϕ(ξ) 1 and w (1) b th solution of (4) with initial condition w (1) 0. On has ŵ (1) (t, ξ) = x[e(ξ)t] 1 ϕ(ξ). 18

19 Sinc w (1) (t) 0 strongly in L 2 (R m, R n ), thr xists an incrasing squnc ) N R + such that (t (1) x[e(ξ)t (1) ] 1 0 whn [ + ] for almost vry ξ R m. Lt w (2) 0 L 2 (R m, R n ) b dfind by ŵ (2) 0 (ξ) := ϕ(ξ) 2 and w (2) b th solution of (4) with initial condition w (2) 0. Th squnc (w(2) (t (1) )) N tnds to zro strongly in L 2 (R m, R n ). Thus, thr xists a subsqunc (t (2) ) N of (t (1) ) N such that x[e(ξ)t (2) ] 2 0 whn [ + ] for almost vry ξ R m. Itrating this rocss for k = 1,..., n, on gts an incrasing squnc (t ) N R + and a subst N of R m of zro masur such that x[e(ξ)t ] 0 in R n n, ξ R m N. (42) Lt ξ R m N. Th convrgnc (42) imlis that th ral art of any ignvalu of E(ξ) has a ngativ ral art. Thus, A(ξ) has no ignvctor in Kr(B) sinc, othrwis, on would gt an ignvalu λ of E(ξ) with R(λ) = 0. In conclusion D(A 1,..., A m, B) is containd in N and, consquntly, its masur vanishs. Rmark 7 As a consqunc of this rsult, and in viw of Proosition 2 which classifis th ossibl structurs for th st of dgnracy, w dduc that, whnvr D(A 1,..., A m, B) is a strict subst of R m, all solutions tnd to zro in L 2 strongly as t. In th following sction w show that, in th othr cas, i.. whn D(A 1,..., A m, B) coincids with R m, thr ar non-trivial solutions of constant L 2 -norm. 3.4 A NSC for th xistnc of non dissiatd solutions Proosition 4 Lt A 1,..., A m, B b n n ral matrics with B of th form (5) and A 1,..., A m symmtric. Th following statmnts ar quivalnt (1) D(A 1,..., A m, B) = R m, (2) thr xists a non dissiatd solution of (4), i.. with a constant L 2 - norm. Th following Lmma will b imortant in th roof of Proosition 4. Lmma 2 Lt A 1,..., A m, B b n n ral matrics with B of th form (5) and A 1,..., A m symmtric. Lt w 0 L 2 (R m, R n ) and w b th solution of (4) with initial condition w 0. Th following statmnts ar quivalnt (1) w is not dissiatd i.. w(t) L 2 w 0 L 2, (2) for almost vry ξ R m, ŵ 0 (ξ) blongs to V (ξ) := {v C n, BA(ξ) k v = 0, k N}. 19

20 Proof of Lmma 2 : Lt us rov (1) (2). W assum w is not dissiatd. W hav d w(t, x) 2 dx = 2 Bw(t, x), w(t, x) dx 0, dt R m R m thus Bw 0, which imlis Bŵ 0. Thus th xrssion (9) can b writtn Th quality Bŵ 0 imlis ŵ(t, ξ) = x[ ia(ξ)t]ŵ 0 (ξ). B x[ ia(ξ)t]ŵ 0 (ξ), for vry t R +, for almost vry ξ R m. Diffrntiating k tims this quality with rsct to tim at t = 0, w gt (2). Now, lt us rov (2) (1). W assum that, for almost vry ξ R m, ŵ 0 (ξ) V (ξ). By dfinition, V (ξ) is a vctor subsac of Kr(B) stabl by A(ξ), thus (9) givs ŵ(t, ξ) = k=0 t k k! [ B ia(ξ)]k ŵ 0 (ξ) = k=0 t k k! [ ia(ξ)]k ŵ 0 (ξ) = x[ ia(ξ)t]ŵ 0 (ξ). Thrfor w(t) L 2 = ŵ(t) L 2 w 0 L 2, i.. (1) holds. Proof of Proosition 4 : Lt us rov (2) (1). W assum thr xists a non dissiatd solution w, associatd to an initial condition w 0 L 2 (R m, R n ) with w 0 0. Thanks to Lmma 2, w know that, for almost vry ξ R m, ŵ 0 (ξ) V (ξ). So, for almost vry ξ Su(w 0 ) (th suort of w 0 ), V (ξ) is a non mty vctor subsac of Kr(B) stabl by A(ξ), morovr A(ξ) V (ξ) is symmtric, thus it has an ignvctor. W hav rovd that Su(w 0 ) D(A 1,..., A m, B), thus th masur of D(A 1,..., A m, B) dos not vanish and Proosition 2 imlis that (1) holds. Now, lt us rov (1) (2). W assum D(A 1,..., A m, B) = R m. Thn for vry ξ R m, V (ξ) {0} bcaus V (ξ) contains at last an ignvctor of A(ξ) that blongs to Kr(B). In ordr to rov that (2) holds, w build an L 2 (R m, R n ) initial condition w 0 such that ŵ 0 (ξ) V (ξ), for vry ξ R m, which givs th conclusion thanks to Lmma 2. Th Cayly-Hamilton thorm justifis that V (ξ) = {v C n ; BA(ξ) k v = 0 for k = 0,..., n 1} = Kr(M 1 (ξ)), whr M 1 (ξ) = n 1 k=0 A(ξ)k B t BA(ξ) k dnds olynomially in ξ. For ξ R m, lt P (ξ) b th orthogonal rojction R n Kr(M 1 (ξ)). Lt ξ R m and z S n 1 Kr(M 1 (ξ )). Th ma ξ R m P (ξ)z is continuous at ξ bcaus M 1 (ξ) is an analytic rturbation of M 1 (ξ ) (s, for xaml [12, Chatr 2, Sction 1.4]). Thus, thr xists a nighborhood Ω 1 of ξ in R m with a ositiv finit masur such that P (ξ)z > 1/2, for vry ξ Ω 1. W dfin ŵ 0 (ξ) := P (ξ)z 1 Ω1 ( Ω 1)(ξ). Thn w 0 L 2 (R m, R n ) bcaus ŵ 0 (ξ) 1 Ω1 ( Ω 1)(ξ) and ŵ 0 ( ξ) = ŵ 0 (ξ). Morovr, w 0 0 bcaus ŵ 0 (ξ) 1/2, ξ Ω 1 ( Ω 1 ). 20

21 Rmark 8 According to th rsults in this sction and th rvious on, w dduc that thr ar only two ossibilitis for th st of dgnracy which lad to diffrnt asymtotic rortis as t : Whnvr D(A 1,..., A m, B) is a strict subst of R m, its masur vanishs, and all solutions tnd to zro in L 2 strongly as t. Whn D(A 1,..., A m, B) coincids with R m, thr ar non-trivial solutions of constant L 2 -norm. 3.5 Comlt classification in th cas m = 1 In this sction w considr th 1D roblm w t + Aw x = Bw. (43) whr A, B R n n, w R n and x R. Not that this siml cas can b tratd dirctly and without th analysis dvlod in th rvious subsctions. Howvr, it is clarifying to trat it in th framwork of th gnral thory w hav dvlod. Whn m = 1, A(ξ) = ξa, sinc ξ is now scalar. Thrfor, th (SK) condition dos not involv th Fourir variabl ξ and can b writtn simly as (SK) : Kr(B) {ignvctors of A} = {0}. Obviously, this is quivalnt to rquiring that th air (A, B) satisfis th Kalman rank condition In 1D (m = 1), th (SK) condition charactrizs comltly th asymtotic bhavior of th solutions of (43). Proosition 5 W assum that A is symmtric and B has th form (5). Th following holds: Th following statmnts ar quivalnt (1) A and B satisfy (SK) : {ignvctors of A} Kr(B) = {0}, (2) thr xist C, λ > 0 such that, for vry w 0 L 1 L 2 (R, R n ), th solution of (43) can b dcomosd as w = w 1 + w 2 whr (14) holds with m = 1. Morovr, if (SK) is not satisfid, any solution of (43) with initial condition in L 2 (R, R n ) may b dcomosd as w = w 1 + w 2 + w trw whr w 1, w 2 satisfy th rvious stimats and w trw is a ur transort trm constitutd by a finit sum of travling wavs, w trw (t, x) = r w trw,j(x λ j t), whr r n, λ j R and w trw,j L 2 (R, R n ) for j = 0,..., r. Proof of Proosition 5 : First, th imlication (1) (2)" was rovd in Thorm 1. Now, w rov th scond statmnt, that also givs (2) (1)". Assum that (SK) is fals. Lt V b th subsac of Kr(B) which is stabl by A (AV V ) with maximal dimnsion constitutd by th sum of all subsacs 21

22 of Kr(B) stabl by A. Lt r :=dim(v ). Th ndomorhism A V is symmtric thus, thr xists an orthonormal basis (v 1,..., v r ) of V mad of ignvctors of A : Av j = λ j v j, j = 1,..., r. Lt V b th orthogonal sulmntary of V in Kr(B) (i.. Kr(B) = V + V ) and (v r+1,..., v n1 ) b an orthonormal basis of V. Thn, (v 1,..., v n1 ) is an orthonormal basis of Kr(B). Sinc Kr(B) = San( 1,..., n1 ), thn V := (v 1,..., v n1, n1+1,..., n ) is an orthonormal basis of R n. Lt P b th basis chang matrix from th canonical basis ( 1,..., n ) to th basis V (th columns of P ar th comonnts of th vctors of V in th canonical basis ( 1,..., n )). Thn P 1 BP = B, Thus Av j, v i = v j, Av i = λ i v j, v i = 0, j {r + 1,..., n 1 }, i {1,..., r}, A j, v i = j, Av i = λ i j, v i = 0, j {n 1 + 1,..., n}, i {1,..., r}. Ã := P 1 AP = λ λ r whr th first diagonal block is of dimnsion r r, th scond on is of dimnsion (n 1 r) (n 1 r) and th third on is of dimnsion n 2 n 2. Thn w := P 1 w solvs w t + Ã w x = B w and w j (t, x) = w j (0, x λ j t) for j = 1,..., r. Ths comonnts ar ur transort trms. W dfin w 1... w trw := P w r Th (n r) latst comonnts of w solv an hyrbolic systm fulfilling th (SK) condition. Thus, it may b dcomosd as w 1 + w 2 with (14) for m = 1. Finally, w tak w 1 := P w 1 and w 2 := P w 2. This comlts th dcomosition in thr trms of th scond statmnt. Not that this argumnt shows that (2) (1)", as claimd. Indd, if (1) is not satisfid, i.. th (SK) is not fulfilld, th argumnt abov shows that, thn, thr is a travling wav comonnt that dos not dcay, thus showing that (2) dos not hold ithr. 3.6 Th cas D(A 1,..., A m, B) = R m : travling wavs? Obviously, th cas D(A 1,..., A m, B) = R m is th most dgnrat on. Proosition 4 shows that, in this cas, thr xist non dissiatd solutions. In 1D 22

23 (m = 1), any non dissiatd solution is a finit sum of travling wavs (s Proosition 5). Thus, it is natural to ask whthr this is also tru for m 2. As w shall s, this is not th cas. Proosition 6 W assum m 2. Th quality D(A 1,..., A m, B) = R m is ncssary for th xistnc of a non trivial travling wav solution of (4) with an L 2 (R m, R n )-rofil but it is not sufficint. Rmark 9 As a consqunc, whn m 2, th non dissiatd solutions xhibit mor comlx structurs that bing sums of travling wavs. Proof : First, w rov that D(A 1,..., A m, B) = R m is ncssary for th xistnc of travling wavs solutions. Indd, any non trivial travling wav solution is a non dissiatd solution. Thus Proosition 4 givs th rsult. Now, w rov that D(A 1,..., A m, B) = R m is not sufficint for th xistnc of travling wavs. Lt us rmark that, whn (4) has a travling wav solution, thn thr xists c R m and Ω R m with ositiv masur, such that, for almost vry ξ Ω, A(ξ) has an ignvalu λ(ξ) of th form λ(ξ) = m c j ξ j (44) associatd to an ignvctor v(ξ) that blongs to Kr(B). As w will s, on can asily build xamls of matrics A 1,..., A m, B for which th latr is imossibl, but D(A 1,..., A m, B) = R m. W considr th systm (4) with m = 2 and A 1 := a 1 a 1,2 a 1,2 a with a 1,2 0, c 1,2 0 and, A 2 := c 1 c 1,2 c 1,2 c 2 0 0, B := δ a a 1,2 δ c c 1,2, (45) 4δ a δ c a 1,2 c 1,2 (δ a c 1,2 ) 2 + (δ c a 1,2 ) 2, (46) whr δ a := a 2 a 1 and δ c := c 2 c 1. Th notation " mans that th corrsonding cofficint may hav any valu. W claim that, for this choic of th aramtrs a 1, a 1,2, a 2, c 1, c 1,2, c 2, on has D(A 1, A 2, B) = R 2 but th ignvalus of (ξ 1 A 1 + ξ 2 A 2 ) Kr(B) cannot hav th form (44) on a subst of R 2 with ositiv masur. For (ξ 1, ξ 2 ) R 2 {0}, th subsac Kr(B) is stabl by A(ξ) = ξ 1 A 1 + ξ 2 A 2, and th rstriction A(ξ) Kr(B) is symmtric. Thus it has an ignvctor. Thrfor D(A 1, A 2, B) = R 2. Th ignvalus of (ξ 1 A 1 + ξ 2 A 2 ) Kr(B) ar λ ± (ξ 1, ξ 2 ) := 1 2 ( (a 1 + a 2 )ξ 1 + (c 1 + c 2 )ξ 2 ± ) (ξ 1, ξ 2 ) 23

24 whr (ξ 1, ξ 2 ) = [δ a ξ 1 + δ c ξ 2 ] 2 + 4[a 1,2 ξ 1 + c 1,2 ξ 2 ] 2. Thanks to (45), w hav (ξ 1, ξ 2 ) > 0 and thr dos not xist any α, β R such that (ξ 1, ξ 2 ) = (αξ 1 + βξ 2 ) 2 on a subst of R m with ositiv masur bcaus of (46). travling wav solutions may not xist. Thus, non trivial Rmark 10 Th structur of th non dissiatd solutions in a gnral contxt nds to b furthr invstigatd. 4 Dcomosition of th solutions whn D is th union of vctor subsacs As rovd in rvious sctions, whnvr th masur of D(A 1,..., A m, B) vanishs, all solutions tnd to zro in L 2 as t. This sction is dvotd to analyz th asymtotic bhavior in som mor dtail. As mhasizd in Rmark 1, th asymtotic bhavior of th solutions of (4) rducs to th study of th ral valud function ω S m 1 N,ɛ (ω), dfind in Proosition 1. This study is th ky oint of this sction. In all this sction, w will considr situations in which th st of dgnracy D(A 1,..., A m, B) has zro masur and is th union of vctor subsacs. This assumtion is rstrictiv bcaus, in gnral, whn th masur of D(A 1,..., A m, B) vanishs, this st is an algbraic submanifold (s Proosition 2), which is not ncssarily a union of vctor subsacs (s xamls 1, 2.a, 2.b in subsction 3.2). Howvr, this assumtion holds in many articular xamls, studid in this sction. In subsction 4.1, w stat a dcomosition for th solutions of (4) whn D(A 1,..., A m, B) is a union of vctor subsacs and undr th additional assumtion N,ɛ (ω) cdist(ω, S m 1 D) α, ω S m 1 (47) for som α 2, c > 0 (s Proosition 7). In subsction 4.2, w rov that, in th articular cas n 1 = 1 (i.. Kr(B) = San( 1 )), D(A 1,..., A m, B) is a vctor subsac of R m and (47) holds with α = 2 (s Proosition 8). This lads to a dcomosition of th solutions of (4) with n 1 = 1 (s Thorm 2). In subsction 4.3, w considr diffrnt xlicit xamls, in which th dgnrat st D(A 1,..., A m, B) is a union of vctor subsacs and th smallst xonnt in (47) is α = 2 or α = 4. W dduc a dcomosition of th solutions of (4) for ths articular cass (s Thorm 3). With ths xamls, w s that, whn m 2, thr is a whol class of hnomna that do not aris undr (SK). Finally, in subsction 4.4, w writ a comlt classification of th diffrnt ossibl asymtotic bhaviors whn n = 2. 24