Indirect stabilization of weakly coupled systems with hybrid boundary conditions

Manuscript submitted to W ebsite: http://AIMsciences.org AIMS’ Journals V olume X , Number 0X , XX 2 00X pp. X–XX INDIRECT ST ABILIZA TION O F WEAKL Y COUPLED SYSTEMS WITH HYBRID BOUNDAR Y CONDITIONS F a tiha Alabau-Boussouira Presen t posi tion D´ el´ egation CNRS at MAPMO, UMR 6628. Current p osition Universit ´ e Paul V erlaine-Metz, Ile du Saulcy , 57045 Metz Cedex 1, F rance Piermarco Cannarsa and Rober to Guglielmi Dipartiment o di Matematica - Universit` a di Roma “T or V ergata” Via della Ricerca Scient ifica 1 - 00133 R oma, IT AL Y (Commun icated by the as s o ciate editor name) Abstract. W e inv estigate stabili ty pr operties of indirectly damp ed s ystems of ev olution equations in Hilb ert spaces, under new compatibility assumptions. W e prov e p olynomial deca y f or the energy of solutions and optimize our results b y int erpol ation tec hniques, obtaining a f ull range of p o we r-like decay rates. In par ticular , we give explicit estimates w i th resp ect to the initial data. W e discuss several applications to hyperb olic systems with hybrid b oundary con- ditions, including the coupling of t w o wa ve equations sub ject to Diri c hlet and Robin ty pe boundary conditions, respectively . 1. Intr o duction. Ther e is no doubt that the interest of the sc ient ific communit y in the stabilization and con trol of sys tems of partial differential equations has r emark- ably increa sed, in re c ent y ears. This is probably due to the fact that such systems arise in several applied mathematical mo dels, such as those used for studying the vibrations o f flexible structures and netw o rks (see [ 19 ] and references therein), or fluids a nd fluid-structure interactions (se e, for ins tance, [ 8 ], [ 9 ], [ 16 ], [ 2 2 ], [ 28 ], [ 3 2 ]). When dealing with s ystems inv o lving quantities describe d by several components, pretending to co nt rol or observe all the s ta te v a riables might b e irrea listic. In applications to ma thematical mo dels for t he vibra tions of flexible structures (see [ 3 ] and [ 7 ]), e le c tromagnetism (see, for instance , [ 21 ]), o r fluid control (see [ 18 ] and the refere nc e s therein), it may happ en that o nly par t of suc h comp onents can be observed. This is why it b eco mes essential to study whether controlling only a reduced num b er of state v aria bles suffices to ensure the stability of the full s ystem. It turns out that c e r tain systems po s sess an in ternal structure tha t compens a tes for the aforementioned lack of control v aria bles. Such a phenomenon is refer red 2000 Mathematics Subje ct Classific ation. 93D15, 35L53, 47D06, 46B70. Key wor ds and phr ases. indir ect st abilization, energy estimates, i nterpolation spaces, ev ol ution equations, hyperb olic s ystems. The authors wish to thank Institut Henri Poincar ´ e (Paris, F rance) for prov iding a v ery stimulating environmen t dur i ng the ”Control of Partial and Differen tial Equations and Applications” program in the F al l 2010. Pa rt of this work was completed dur - ing the C.I.M.E. course ‘Control of partial differen tial equations’ (Cetraro, July 19–23, 2010). This researc h has b een p erformed in the framework of the GDRE CONEDP , see http://w ww.cerem ade.daup hine.fr/ ~ glass/GD RE/index .php . 1 2 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI to as indir e ct stabilization or indir e ct c ontr ol (se e [ 29 ]). An example o f indir ect stabilization o ccurs with the hyperb olic system      ∂ 2 t u − ∆ u + ∂ t u + αv = 0 in Ω × R ∂ 2 t v − ∆ v + αu = 0 in Ω × R u = 0 = v on ∂ Ω × R , (1) where Ω is a b ounded op en do main o f R N , and the ‘frictional’ term ∂ t u acts as a stabilizer. Indeed, a genera l result prov ed in [ 4 ] ensur es that, fo r sufficiently smo o th initial co nditions and | α | > 0 sma ll eno ugh, the energ y of the solution ( u, v ) of ( 1 ) decays to zero at a p olynomial rate as t → ∞ . The ab ov e indire c t stabilization prop er ty holds true for more genera l s y stems of partial differen tial equations, under the compatibility assumption ( 10 ) below, se e [ 4 ]. F o r applications to pro blems in mechanical engineering , howev er , it is extremely impo rtant to c onsider also bo undary conditions that fail to satisfy the assumption of [ 4 ]. This is the ca se of Neumann or Robin b oundary co nditio ns , which describ e different physical situations suc h as hinged or clamp ed devices. F or instance, let us change the b oundary conditions in ( 1 ) as follows:      ∂ 2 t u − ∆ u + ∂ t u + αv = 0 in Ω × R ∂ 2 t v − ∆ v + αu = 0 in Ω × R u + ∂ u ∂ ν = 0 = v on ∂ Ω × R . (2) Then, as is shown in P rop osition 2 b e low, the compatibility assumption ( 10 ) is not satisfied. Nev ertheles s, in this pape r we will pr ov e p oly no mial stability for system ( 2 ), using a new hypothesis which is sp ecia lly designed to handle bo undary conditions as ab ov e—that w e call hybrid . More gener ally , in a rea l Hilbert space H , with scala r pro duct h· , ·i and nor m | · | , we shall s tudy the s ystem o f evolution equations ( u ′′ ( t ) + A 1 u ( t ) + B u ′ ( t ) + αv ( t ) = 0 v ′′ ( t ) + A 2 v ( t ) + αu ( t ) = 0 (3) where (H1) A i : D ( A i ) ⊂ H → H ( i = 1 , 2) ar e densely defined clos e d linear op erator s such that A i = A ∗ i , h A i u, u i ≥ ω i | u | 2 ∀ u ∈ D ( A i ) for so me ω 1 , ω 2 > 0, (H2) B is a b ounded linear op erator o n H such that B = B ∗ , h B u, u i ≥ β | u | 2 ∀ u ∈ H for so me β > 0, (H3) α is a r eal num b er such that 0 < | α | < √ ω 1 ω 2 . System ( 3 ), with the initial co nditions  u (0) = u 0 , u ′ (0) = u 1 , v (0) = v 0 , v ′ (0) = v 1 , (4) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 3 can b e formulated as a Cauch y pro blem for a cer ta in first order evolution equation in the pr o duct s pace H := D ( A 1 / 2 1 ) × H × D ( A 1 / 2 2 ) × H . More precisely , let us define the ener g ies as so ciated to op erator s A 1 , A 2 by E i ( u, p ) = 1 2  | A 1 / 2 i u | 2 + | p | 2  ∀ ( u, p ) ∈ D ( A 1 / 2 i ) × H ( i = 1 , 2 ) , (5) and the total energ y of the system as E ( U ) := E 1 ( u, p ) + E 2 ( v , q ) + α h u, v i (6) for every U = ( u , p , v , q ) ∈ H . The n, assumption (H1) y ie lds , for i = 1 , 2, | u | 2 ≤ 2 ω i E i ( u, p ) ∀ u ∈ D ( A 1 / 2 i ) , ∀ p ∈ H . (7) Moreov er, in view o f ( H 3), for a ll U = ( u, p , v , q ) ∈ H E ( U ) ≥ ν ( α ) h E 1 ( u, p ) + E 2 ( v , q ) i , (8) where ν ( α ) = 1 − | α | ( ω 1 ω 2 ) − 1 / 2 > 0. Let us introduce the bilinea r form on H ( U | b U ) = h A 1 / 2 1 u, A 1 / 2 1 b u i + h p, b p i + h A 1 / 2 2 v , A 1 / 2 2 b v i + h q , b q i + α h u, b v i + α h v , b u i . Since ( U | U ) = 2 E ( U ) ∀ U ∈ H , thanks to ( 8 ) the ab ove form is a sca lar pro duct on H , and H is a Hilb ert space with such a pro duct. Let now A : D ( A ) ⊂ H → H b e the o p erator defined b y ( D ( A ) = D ( A 1 ) × D ( A 1 / 2 1 ) × D ( A 2 ) × D ( A 1 / 2 2 ) A U = ( p , − A 1 u − B p − αv , q , − A 2 v − αu ) ∀ U ∈ D ( A ) . Then, problem ( 3 ) takes the eq uiv alent fo rm ( U ′ ( t ) = A U ( t ) U (0) = U 0 := ( u 0 , u 1 , v 0 , v 1 ) . (9) As will b e proved in L e mma 4.2 , A is a max imal dis s ipative op erator . Then, from classical results (s e e, for ins tance, [ 27 ]), it fo llows that A g enerates a C 0 -semigro up, e t A , o n H . Also , e t A U 0 = ( u ( t ) , p ( t ) , v ( t ) , q ( t )) , where ( u, v ) is the solution of pr oblem ( 3 )-( 4 ), and ( p, q ) = ( u ′ , v ′ ). In o rder to in tro duce our asy mpto tic ana lysis of system ( 3 )-( 4 )—or, equiv a lent ly , ( 9 )—let us o bserve that, a s is explained in [ 4 ], no exp onential s ta bilit y ca n b e exp ected. Therefore, w eaker decay rates at infinity , such as p olynomial ones, a re to be sought for . Polynomial decay results for ( 3 ) were obtained in [ 4 ] as s uming that, for so me integer j ≥ 2, | A 1 u | ≤ c | A j / 2 2 u | ∀ u ∈ D ( A j / 2 2 ) . (10) Similar decay estima tes for the ca s e o f bo undary damping (that is, when op erator B is unbounded) were derived in [ 2 ]. Also, we r efer the re ader to [ 13 ], [ 14 ] and [ 31 ] for indirect sta bilization with lo calize d damping, and to [ 6 ] for the study o f a one-dimensional wav e system coupled thr ough velocities. 4 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI The a symptotic b ehavior of wa ve-like equations and, in particular , the der iv ation of optimal decay ra tes for the energ y when the geometry o f the domain a nd damping region allow rays to b e tr app ed, hav e b een intensiv ely studied for several decades. F or such questions a nd res ults, we refer the rea der to Leb eau [ 23 ] and Bur q [ 17 ] (and the references therein). In [ 23 ], Leb eau consider e d a lo ca lly damp ed wa v e equation and proved o ptimal lo garithmic decay ra tes for the energy , provided that damping is active on a nonempty op en s et. The pr o of relies on o ptimal res olven t estimates for the co rresp onding infinitesimal generator of the asso ciated semigroup. L a ter on these results were completed by Burq in [ 17 ] in exter ior domains, in pa rticular for cases in which r ays may b e trapp ed b y the obstacle. Independently , indirec t stabiliza tio n for symmetric hyper bo lic systems was first considered by the first autho r in [ 1 ], and further developed in [ 2 , 4 , 5 ], using ene r gy t yp e metho ds, to gether with some new ideas s uch as the new integral inequality given in Theorem 2.5 (see [ 1 , 4 ]). In this approa ch, the pur p o se is ra ther to fo c us on the prop erties of the da ta —that is, the op erators A 1 , A 2 , B and the c o upling op erator —that allow to tra nsfer the damping action of the feedback to the un- damp ed equa tion. Subsequently , indirect stabilization of coupled s ystems was investigated in [ 10 ] and [ 2 4 ]. In [ 10 ], reso lven t es tima tes were obtained and sp ectral analy s is was used to prove p oly nomial decay for ( 3 ), cov er ing some o f the exa mples trea ted in [ 4 ]. In [ 24 ], where a Riesz ba s is approa ch is follow ed, p olynomial decay r ates for the energy were der ived for a simplified case of co upled system, where op erators A 1 and A 2 are suppo sed to be equal (to A ) and the damping op erator is a nonp ositive fractional p ow er of A . More recently , ins pired b y [ 23 ] and [ 1 7 ], and, throug h [ 10 ], by [ 1 , 2 , 4 ], the optimality of sp ectra l-analysis -derived decay r a tes was shown in [ 11 ] and [ 15 ], taking int o acco un t the a symptotic b ehaviour of the res olven t on the imagina ry axis. In the c ontext of indirect stabilization for coupled systems, we would like to stress the fact that ch ecking the a ssumptions o n the data— A 1 , A 2 , B and the c o upling op erator —that are needed to ensure decay , may b e a difficult task. In particular, resolven t estimates may b e hard to obtain when A 1 and A 2 do not commute, or damping and coupling o p e rators do not commute with A 1 and A 2 . F or r esults in this directio n we re fer the reader to [ 1 , 2 ]. The case of lo c alized or b oundary damping, together with lo calized coupling, is ana lyzed in [ 5 ], where A 1 = A 2 = A , but B a nd the coupling o p er ator do not c o mmut e with A . Moreov er, since co upling is lo calized, the co rresp onding op era tor is no lo nger co ercive. This fact g enerates additional difficulties. In this pap er, we will repla c e ( 10 ) by D ( A 2 ) ⊂ D ( A 1 / 2 1 ) and | A 1 / 2 1 u | ≤ c | A 2 u | ∀ u ∈ D ( A 2 ) , (11) which is sa tisfied by a large cla ss o f s y stems including ( 2 ) as a sp ecial case (see Section 5 b elow). Under s uch a conditio n we will show that any solution U of ( 9 ) satisfies the in tegral inequality Z T 0 E ( U ( t )) dt ≤ c 1 4 X k =0 E ( U ( k ) (0)) ∀ T > 0 , U 0 ∈ D ( A 4 ) . (12) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 5 Moreov er, since the ener gy of solutions is decreasing in time, ( 12 ) implies, in turn, the p olyno mial decay of or der n of E , that is, E ( U ( t )) ≤ c n t n 4 n X k =0 E ( U ( k ) (0)) ∀ t > 0 (13) for all n ≥ 1 and U 0 ∈ D ( A 4 n ) (see C o rollar y 2 below). No tice that ( 13 ) yields, in particular, the strong sta bilit y o f e t A . The compa tibility c ondition ( 11 ) is equiv a le n t to the b oundedness of A 1 / 2 1 A − 1 2 . Let us point out that this hypothesis is sufficient but no t necessar y . Such a fact can b e observed tak ing, for example, A 2 = A τ 1 with τ ∈ (0 , 1 / 2 ). In this case, condition ( 11 ) is violated, but it is e a sy to check that condition ( 10 ) holds for the smallest integer j such that j > 2 /τ . O n the other hand, condition ( 11 ) is satis fie d for all τ ≥ 1 / 2. This example shows that the present r esults a nd those of [ 4 ] are in some sense complementary —and, for A 2 = A τ 1 ,( τ ≥ 0) exa ctly complementary . One should also note that, for genera l op erator s A 1 and A 2 , the tw o co mpatibility conditions ( 10 ) and ( 11 ) do not cov e r all p ossible cases. Passing from p olynomial to a general power-lik e decay estimate is quite natural, once ( 13 ) has b een establis he d. Indeed, in Section 4 , using interp olation t he ory , we obtain the fractional decay ra te E ( U ( t )) ≤ c n t n/ 4 n X k =0 E ( U ( k ) (0)) ∀ t > 0 (14) for a ll n ≥ 1 and U 0 ∈ D ( A n ) (see Co rollary 4 b elow). Moreover, taking initia l da ta in  H , D ( A n )  θ , 2 for any 0 < θ < 1, we deduce the co nt inu ous decay rate k U ( t ) k 2 H ≤ c n,θ t nθ / 4 k U 0 k 2 ( H ,D ( A n )) θ, 2 ∀ t > 0 . (15) Notice that a somewhat compar able result is obtained in [ 10 , Prop ositio n 3.1] using a different technique. In particular , fo r n = 1, ( 14 ) implies that, fo r every U 0 ∈ D ( A ) , the so lution U of problem ( 9 ) s atisfies E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c t 1 / 4 k U 0 k 2 D ( A ) ∀ t > 0 , (16) and there exists c 1 > 0 such that k U 0 k 2 D ( A ) ≤ c 1  | A 1 u 0 | 2 + | A 1 / 2 1 u 1 | 2 + | A 2 v 0 | 2 + | A 1 / 2 2 v 1 | 2  . Thu s, interpola tion theory applied to systems satisfying ( 11 ) allows to prove contin uous energ y decay ra tes , tog ether with decay rates under explicit smo othness conditions on the initial data . F urthermo re, we would like to p oint out that it also yields stronger results in the fra mework studied in [ 4 ], that is, under c o ndition ( 10 ). W e descr ib e suc h applicatio ns in Section 6 , where we s how how to deduce p ower- like decay rates from the energy estimates of [ 4 ], thus recovering, in a mor e genera l set-up, rela ted asymptotic e stimates that can b e o btained by spec tr al ana ly sis. Let us now mention some op en ques tio ns. One interesting problem is to de- rive optimal decay rates for the energy of an indirectly damp ed coupled system in geometric situa tio ns for which tra pped rays may exis t for the uncoupled damped equation. Mo re prec is ely , it would be very interesting to g eneralize Leb e a u’s res o l- ven t analy sis in [ 23 ] to suc h coupled systems o btaining o ptima l energy es timates. In a somewhat different spirit, another op en question would b e to determine if it is p oss ible to combine the results of [ 23 ] and [ 17 ] with the techniques develope d 6 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI in [ 2 , 4 , 5 ] in order to der ive s ha rp upp er decay rates for the ener gy . In a ll the examples w e discuss in the present work—as well as in [ 1 , 2 , 4 , 5 ]—oper ators A 1 and A 2 happ en to hav e compact r esolven ts. I t would b e very interesting to see if explicit energ y decay r ates can be derived in different situations . F or instance, it would b e nice to ex tend Burq’s approa ch [ 17 ] in or der to obta in indirect damping of coupled systems in exterior domains, and prove decay of the lo c al total energ y of solutions. This pap er is organized as fo llows. Section 2 reca lls preliminary notions, ma inly related to interpolatio n theor y which is so relev ant for mos t o f this pap er. Section 3 is devoted to our p olynomia l decay result and its pro of. In Section 4 , we co mplete the analys is with estimates in interpola tion spaces. In Section 5 , we describ e s e veral applications to s ystems of partial differen tial oper ators. Finally , in Sec tio n 6 , we show how to improv e the results of [ 4 ] by in terp olation. 2. Preli m inaries. In this sec tion, we introduce the main to ols r equired to deal with int erp olation theor y betw een Ba na ch spaces. F or a general exp ositio n of this theory the re ader is referred to [ 30 ] and [ 26 ]. Interesting intro ductio ns ar e also given in [ 1 2 ] from the p oint of view of control theory , and [ 25 ] for the sp e cific case of analytic semigroups. In this sectio n ( X , | · | X ) stands for a r eal Banach spa ce. Let ( Y , | · | Y ) b e another Ba na ch spa ce. W e say that Y is contin uously embedded in to X , and we write Y ֒ → X , if Y ⊂ X and | x | X ≤ c | x | Y ∀ x ∈ Y for so me constant c > 0. W e denote b y L ( Y ; X ) the B anach spac e of all bo unded linea r op era to rs T : Y → X equipp ed with the standar d op er a tor no r m. If Y = X , we r efer to such a space as L ( X ). F or any given subspa ce D o f X , w e denote by T | D the re s triction of T to D . Definition 2.1. Le t ( D , | · | D ) b e a clo sed subspace of X . A s ubspace ( Y , | · | Y ) of X is said to b e an interpo lation space betw een D and X if (a) D ֒ → Y ֒ → X , and (b) for every T ∈ L ( X ) such that T | D ∈ L ( D ), we hav e that T | Y ∈ L ( Y ). Let X , D b e Banach spac e s, with D contin uous ly em bedded into X . F or a ny α ∈ [0 , 1], we denote by J α ( X, D ) the family of all s ubspaces Y of X containing D such tha t | x | Y ≤ c | x | α D | x | 1 − α X ∀ x ∈ D for so me constant c > 0. Let us introduce, for ea ch x ∈ X a nd t > 0, the quantit y K ( t, x, X , D ) := inf x = a + b, a ∈ X, b ∈ D ( | a | X + t | b | D ) . (17) Let 0 < θ < 1 b e fixed. W e define ( X, D ) θ , 2 :=  x ∈ X : Z + ∞ 0 | t − θ − 1 2 K ( t, x, X , D ) | 2 dt < + ∞  (18) and | x | 2 θ , 2 := Z + ∞ 0 | t − θ − 1 2 K ( t, x, X , D ) | 2 dt . INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 7 The spac e ( X, D ) θ , 2 , endow ed with the norm | · | θ , 2 , is a Banach space. The reader is referred to [ 26 ] for the pr o of of the following r esults. Theorem 2.2. L et X 1 , X 2 , D 1 , D 2 b e Banach sp ac es such that D i is c ont inuously emb e dde d in X i , for i = 1 , 2 . If T ∈ L ( X 1 ; X 2 ) ∩ L ( D 1 ; D 2 ) , then we have that T ∈ L (( X 1 , D 1 ) θ , 2 ; ( X 2 , D 2 ) θ , 2 ) for every θ ∈ (0 , 1) . Mor e over, k T k L (( X 1 ,D 1 ) θ, 2 ;( X 2 ,D 2 ) θ, 2 ) ≤ k T k 1 − θ L ( X 1 ; X 2 ) k T k θ L ( D 1 ; D 2 ) . Consequently , the space ( X , D ) θ , 2 belo ngs to J θ ( X, D ) for every θ ∈ (0 , 1 ). Let α ∈ [0 , 1 ] a nd denote by K α ( X, D ) the family of all subspaces ( Y , | · | Y ) of X containing D such that sup t> 0 , x ∈ Y K ( t, x, X , D ) t α | x | Y < + ∞ . Theorem 2. 3 (Reitera tion Theorem) . L et 0 < θ 0 < θ 1 < 1 . Fix θ ∈ ]0 , 1[ and set ω = (1 − θ ) θ 0 + θθ 1 . 1) If E i ∈ K θ i ( X, D ) , i = 0 , 1 , then ( E 0 , E 1 ) θ , 2 ⊂ ( X , D ) ω , 2 . 2) If E i ∈ J θ i ( X, D ) , i = 0 , 1 , then ( X , D ) ω , 2 ⊂ ( E 0 , E 1 ) θ , 2 . Conse quently, if E i ∈ J θ i ( X, D ) ∩ K θ i ( X, D ) , i = 0 , 1 , then ( E 0 , E 1 ) θ , 2 = ( X , D ) ω , 2 , with e quivalenc e b et we en the r esp e ctive norms. Remark 1. Since ( X , D ) θ , 2 is contained in J θ ( X, D ) ∩ K θ ( X, D ), for every 0 < θ 0 , θ 1 < 1 we hav e (( X, D ) θ 0 , 2 , ( X, D ) θ 1 , 2 ) θ , 2 = ( X , D ) (1 − θ ) θ 0 + θ θ 1 , 2 . (19) Since X ∈ J 0 ( X, D ) ∩ K 0 ( X, D ) and D ∈ J 1 ( X, D ) ∩ K 1 ( X, D ), we also have ( X, ( X , D ) θ 1 , 2 ) θ , 2 = ( X , D ) θ θ 1 , 2 and (20) (( X, D ) θ 0 , 2 , D ) θ , 2 = ( X , D ) (1 − θ ) θ 0 + θ , 2 . (21) 2.1. In terp olation spaces and fractional p ow ers of op e rators. L e t ( H, h · , · i ) be a real Hilber t space, with norm | · | . Le t A : D ( A ) ⊂ H → H b e a densely defined closed linea r op erator o n H such that h Ax, x i ≥ δ | x | 2 , ∀ x ∈ D ( A ) (22) for some δ > 0. As usua l, we denote by A θ the fractional p ow er of A for a ny θ ∈ R (see, for instance, [ 12 , Cha pter 1 - Section 5]), and by A ∗ the a djoint of A . W e recall that A is self-adjoint if D ( A ) = D ( A ∗ ) a nd h Ax, y i = h x, Ay i for every x, y ∈ D ( A ). F or the pr o of of the following result we r efer to [ 26 , Theorem 4.36]. Theorem 2.4 . L et A b e a self-adjoint op er ator satisfying ( 22 ) . Then, for every θ ∈ (0 , 1) , α, β ∈ R su ch that β > α ≥ 0 , ( D ( A α ) , D ( A β )) θ , 2 = D ( A (1 − θ ) α + θ β ) . (23) In p articular, ( H, D ( A β )) θ , 2 = D ( A β θ ) . (24) W e say that A is a n m-accre tive op erator if ( h Ax, x i ≥ 0 ∀ x ∈ D ( A ) ( accre tiv ity ) ( λI + A ) D ( A ) = H for some λ > 0 ( maximal ity ) 8 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Notice that, if the ab ove maxima lit y condition is sa tisfies for some λ > 0 , then the same condition holds for every λ > 0. Moreover, we say that A is m-dissipative if − A is m-accretive. W e refer the rea der to [ 26 , Section 4.3] for the pro of of the next result. Prop ositi o n 1. L et ( A, D ( A )) b e an m- ac cr etive op er ator on a H ilb ert sp ac e H , with A − 1 b ounde d in H . Then for every α, β ∈ R , β > α ≥ 0 , θ ∈ (0 , 1) , A satisfies ( 23 ) and ( 24 ) . In p articular, D ( A θ ) = ( H , D ( A )) θ , 2 ∀ 0 < θ < 1 . (25) Corollary 1. If A is t he infinitesimal gener ator of a C 0 -semigr oup of c ontr actions on H , with A − 1 b ounde d in H , then D ( A m ) = ( H , D ( A k )) θ , 2 for every k ∈ N , θ ∈ (0 , 1) such that m = θk is an inte ger. 2.2. An abstract deca y result. W e recall an abstract result obtained in [ 1 ] in a slightly differen t form, and in [ 4 , Theor em 2.1] in the current v ersion. Let A : D ( A ) ⊂ H → H b e the infinitesimal genera to r of a C 0 -semigro up of b ounded linear op erato rs on H . Theorem 2.5. L et L : H → [0 , + ∞ ) b e a c ont inuous function such that, for some inte ger K ≥ 0 and some c onstant c ≥ 0 , Z T 0 L ( e tA x ) dt ≤ c K X k =0 L ( A k x ) ∀ T ≥ 0 , ∀ x ∈ D ( A K ) . (26) Then, for any inte ger n ≥ 1 , any x ∈ D ( A nK ) and any 0 ≤ s ≤ T Z T s L ( e tA x ) ( t − s ) n − 1 ( n − 1)! dt ≤ c n (1 + K ) n − 1 nK X k =0 L ( e sA A k x ) . (27) If, in addition, L ( e tA x ) ≤ L ( e sA x ) for any x ∈ H and any 0 ≤ s ≤ t , then L ( e tA x ) ≤ c n (1 + K ) n − 1 n ! t n nK X k =0 L ( A k x ) ∀ t > 0 (28) for any inte ger n ≥ 1 and any x ∈ D ( A nK ) . 3. Main result. W e a re now ready to state a nd prove the p olynomial decay of solutions to w eakly coupled systems. In addition to the s tanding assumptions ( H 1 ) , ( H 2) , ( H 3), we will as sume that D ( A 2 ) ⊂ D ( A 1 / 2 1 ) and | A 1 / 2 1 u | ≤ c | A 2 u | ∀ u ∈ D ( A 2 ) (29) for some constant c > 0. Condition ( 29 ) can be formulated in the following e quiv a- lent ways. Lemma 3.1. Under assumption ( H 1) the fol lowing pr op erties ar e e quivalent. (a) Assumption ( 29 ) holds. (b) A 1 / 2 1 A − 1 2 ∈ L ( H ) . (c) F or some c onstant c > 0 |h A 1 u, v i| ≤ c | A 2 v |h A 1 u, u i 1 / 2 ∀ u ∈ D ( A 1 ) , ∀ v ∈ D ( A 2 ) . (30) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 9 Pr o of. The implicatio ns (a) ⇔ (b) ⇒ (c) b eing straightforward, let us pr o ceed to show that (c) ⇒ (a). Consider the Hilb ert space V 1 = D ( A 1 / 2 1 ) with the s calar pro duct h u, v i V 1 = h A 1 / 2 1 u, A 1 / 2 1 v i and recall tha t D ( A 1 ) is a dense subspa c e of V 1 . Let v ∈ D ( A 2 ) and define the linear functional φ v : D ( A 1 ) → R by φ v ( u ) = h A 1 u, v i ∀ u ∈ D ( A 1 ) . Owing to (c), φ v can b e extended to a b ounded linea r functiona l on V 1 (still denoted by φ v ) sa tisfying k φ v k ≤ c | A 2 v | . Ther efore, b y the Riesz Theo rem, ther e is a unique vector w ∈ V 1 such that φ v ( u ) = h A 1 / 2 1 u, A 1 / 2 1 w i ∀ u ∈ V 1 . Hence, h A 1 u, ( v − w ) i = 0 for a ll u ∈ D ( A 1 ), and so v = w ∈ V 1 since A 1 is inv ertible. Mor eov er , | A 1 / 2 1 v | = | w | V 1 ≤ c | A 2 v | . The main result of this section is the following. Theorem 3 . 2. A s s ume ( H 1) , ( H 2) , ( H 3) and ( 29 ) . If U 0 ∈ D ( A 4 ) , then the solu- tion U of pr oblem ( 9 ) satisfies Z T 0 E ( U ( t )) dt ≤ c 1 4 X k =0 E ( U ( k ) (0)) ∀ T > 0 (31) for some c onstant c 1 > 0 . The pro of of T heo rem 3.2 will b e g iven in several steps. First, let us recall that, as showed in [ 4 , Lemma 3.3], system ( 9 ) is dissipative. Indeed, under the only assumptions ( H 1) and ( H 2), the energy of the s o lution U = ( u , u ′ , v , v ′ ) of problem ( 9 ) with U 0 ∈ D ( A ) sa tisfies d dt E ( U ( t )) = −| B 1 / 2 u ′ ( t ) | 2 ∀ t ≥ 0 . (32) In particular , t 7→ E ( U ( t )) is nonincreasing o n [0 , ∞ ). Corollary 2. A ssume ( H 1) , ( H 2) , ( H 3) and ( 29 ) . (a) If U 0 ∈ D ( A 4 n ) for some inte ger n ≥ 1 , then the solution U of pr oblem ( 9 ) satisfies E ( U ( t )) ≤ c n t n 4 n X k =0 E ( U ( k ) (0)) ∀ t > 0 (33) for some c onstant c n > 0 . (b) F or every U 0 ∈ H we have E ( U ( t )) → 0 as t → + ∞ . Pr o of. Statement ( a ) follows by combining the dissipa tion rela tion ( 32 ), Theo- rem 3.2 , and Theorem 2.5 . T o prov e part ( b ), we fix U 0 ∈ H a nd co nsider a sequence ( U n 0 ) n ∈ N such that U n 0 ∈ D ( A 4 n ) for e very n ≥ 1 and U n 0 → U 0 in H for n → + ∞ . W e set U n ( t ) = e t A U n 0 and U ( t ) = e t A U 0 for t ≥ 0. Then, by linearity and the contraction prop erty of ( e t A ) t ≥ 0 , we hav e || U n ( t ) − U ( t ) || ≤ || U n 0 − U 0 || , ∀ t ≥ 0 , n ∈ N . 10 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Therefore, r ecalling the definition of E , we deduce that E ( U n ( . )) conv erges to E ( U ( . )) as n → + ∞ , uniformly o n [0 , ∞ ). Since, for any fixed n ∈ N , E ( U n ( t )) conv erges to 0 a s t → ∞ , we easily obtain the conclusion. W e now p ro ceed with the pro of of Theor em 3.2 . Hereafter, C will denote a g eneric po sitive cons ta nt , independent of α . T o beg in with, let us recall that, thanks to [ 4 , Lemma 3.4], the so lution of ( 9 ) with U 0 ∈ D ( A ) verifies Z T 0 E ( U ( t )) dt ≤ Z T 0 | v ′ ( t ) | 2 dt + C E ( U (0)) (34) for some consta nt C ≥ 0 and every T ≥ 0. Hence, the main technical p oint o f the pro of is to bo und the r ight-hand side of ( 34 ) by the total ener gy of U (and a finite nu m ber of its deriv a tives) a t 0. Lemma 3. 3. L et U = ( u, u ′ , v , v ′ ) b e the solution of pr oblem ( 9 ) with U 0 ∈ D ( A ) . Then Z T 0 | A − 1 / 2 1 v | 2 dt ≤ C Z T 0 | A − 1 / 2 2 u | 2 dt + C α 2 h E ( U (0)) + E ( U ′ (0)) i . (35) Pr o of. Rewr ite ( 9 ) a s system ( 3 ) to obtain Z T 0 h u ′′ + A 1 u + B u ′ + αv , A − 1 1 v i dt − Z T 0 h v ′′ + A 2 v + αu , A − 1 2 u i dt = 0 . Hence, by straightforward computations, α Z T 0 | A − 1 / 2 1 v | 2 dt ≤ α Z T 0 | A − 1 / 2 2 u | 2 dt − Z T 0 h B u ′ , A − 1 1 v i dt + Z T 0  h v ′′ , A − 1 2 u i − h u ′′ , A − 1 1 v i  dt . Int egration by parts tr a nsforms the last inequality into α Z T 0 | A − 1 / 2 1 v | 2 dt ≤ α Z T 0 | A − 1 / 2 2 u | 2 dt − Z T 0 h A − 1 / 2 1 B u ′ , A − 1 / 2 1 v i dt + Z T 0  h A − 1 / 2 1 v , A 1 / 2 1 A − 1 2 u ′′ i − h A − 1 / 2 1 u ′′ , A − 1 / 2 1 v i  dt + h h v ′ , A − 1 2 u i − h v , A − 1 2 u ′ i i T 0 . (36) W e now pro ceed to bo und the right-hand side of ( 36 ). W e hav e      Z T 0 h A − 1 / 2 1 B u ′ , A − 1 / 2 1 v i dt      ≤ α 4 Z T 0 | A − 1 / 2 1 v | 2 dt + C α Z T 0 | B 1 / 2 u ′ | 2 dt . Similarly , thanks to assumption ( 29 ) and the fact that B is po sitive definite,      Z T 0 h A − 1 / 2 1 v , A 1 / 2 1 A − 1 2 u ′′ i dt      ≤ α 4 Z T 0 | A − 1 / 2 1 v | 2 dt + C α Z T 0 | B 1 / 2 u ′′ | 2 dt . Also,      Z T 0 h A − 1 / 2 1 u ′′ , A − 1 / 2 1 v i dt      ≤ α 4 Z T 0 | A − 1 / 2 1 v | 2 dt + C α Z T 0 | B 1 / 2 u ′′ | 2 dt . INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 11 Finally , observe that the last term in ( 36 ) can b e b ounded as follows     h h v ′ , A − 1 2 u i − h v, A − 1 2 u ′ i i T 0     ≤ C E ( U (0)) . Combining the ab ov e estimates with ( 36 ), we o btain Z T 0 | A − 1 / 2 1 v | 2 dt ≤ C Z T 0 | A − 1 / 2 2 u | 2 dt + C α E ( U (0)) + C α 2 Z T 0  | B 1 / 2 u ′ | 2 + | B 1 / 2 u ′′ | 2  dt . The conclusion fo llows from the ab ov e inequality and the diss ipation ident it y ( 32 ) applied to u a nd u ′ . Lemma 3. 4. L et U = ( u, u ′ , v , v ′ ) b e the solution of pr oblem ( 9 ) with U 0 ∈ D ( A ) . Then Z T 0 | v | 2 dt ≤ C α 2 Z T 0 | u | 2 dt + C α 2 3 X k =1 E ( U ( k ) (0)) . (37) Pr o of. Since h v ′′ + A 2 v + αu, A − 1 2 v i = 0, in tegrating over [0 , T ] we have Z T 0 | v | 2 dt = − α Z T 0 h v , A − 1 2 u i dt − Z T 0 h v ′′ , A − 1 2 v i dt . (38) The last term in the ab ov e iden tit y can b e b ounded using assumption ( 29 ) a nd Lemma 3.1 a s follows      Z T 0 h v ′′ , A − 1 2 v i dt      =      Z T 0 h A − 1 / 2 1 v ′′ , A 1 / 2 1 A − 1 2 v i dt      ≤ 1 4 Z T 0 | v | 2 dt + C Z T 0 | A − 1 / 2 1 v ′′ | 2 dt . (39) Now, a pplying ( 35 ) to v ′′ and ( 32 ) to u ′ , we obtain Z T 0 | A − 1 / 2 1 v ′′ | 2 dt ≤ C Z T 0 | A − 1 / 2 1 u ′′ | 2 dt + C α 2  E ( U ′′ (0)) + E ( U ′′′ (0))  ≤ C E ( U ′ (0)) + C α 2  E ( U ′′ (0)) + E ( U ′′′ (0))  . (40) On the other hand,      α Z T 0 h v , A − 1 2 u i dt      ≤ 1 4 Z T 0 | v | 2 dt + C α 2 Z T 0 | u | 2 dt . (41) The conclusio n follows combining ( 38 ),. . . ,( 41 ). Let us now complete the pro of o f Theo r em 3.2 . Pr o of of The or em 3.2 . T o prove ( 31 ) it suffices to apply ( 37 ) to v ′ and use the resulting e s timate to b ound the r ight-hand side of ( 34 ). Since B is p ositive definite, the co nclus ion follows b y the dissipation identit y ( 32 ). 12 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Remark 2 . ( i ) Similar results can be obtained for systems of equations coupled with different co efficients, such as ( u ′′ ( t ) + A 1 u ( t ) + B u ′ ( t ) + α 1 v ( t ) = 0 v ′′ ( t ) + A 2 v ( t ) + α 2 u ( t ) = 0 . (42) In this case, ( H 3) sho uld b e r eplaced with (H3’) α 1 , α 2 are tw o real num ber s such that 0 < α 1 α 2 < ω 1 ω 2 . Let us explain how to adapt o ur appr oach to the cas e of α 1 6 = α 2 , when α 1 , α 2 > 0. The total energy is defined by E ( U ) := α 2 E 1 ( u, p ) + α 1 E 2 ( v , q ) + α 1 α 2 h u, v i , where E 1 and E 2 are the energ ies of the tw o comp onents, defined in ( 5 ). Moreover, for each U ∈ H , E ( U ) ≥ ν ( α 1 , α 2 ) h α 2 E 1 ( u, p ) + α 1 E 2 ( v , q ) i , with ν ( α 1 , α 2 ) = 1 − ( α 1 α 2 ) 1 / 2 ( ω 1 ω 2 ) − 1 / 2 > 0. Finally , for ea ch U 0 ∈ D ( A ), the solution U ( t ) = ( u ( t ) , p ( t ) , v ( t ) , q ( t )) of the first o rder evolution equation as s o ciated with sys tem ( 42 ) sa tis fie s d dt E ( U ( t )) = − α 2 | B 1 / 2 u ′ ( t ) | 2 ∀ t ≥ 0 . ( 43) In particula r, t 7→ E ( U ( t )) is nonincreasing on [0 , ∞ ). F rom this point, reasoning as in the ab ove pr o of, the reader ca n easily derive the conclusion of T he o rem 3.2 . ( ii ) Another interesting situation o ccurs when α 1 = 0 , that is, when the first equation of sys tem ( 42 ) is damp ed, wherea s the second co mp one nt is undamp ed and weakly coupled with the first o ne. In this case there is no hop e to stabilize the full system b y o ne single feedback. Indee d, let A 1 = A 2 =: A and consider the sequence of p ositive eigenv alues ( ω k ) k ≥ 1 of A , satisfying ω k → + ∞ , with ass o ciated eigenspaces ( Z k ) k ≥ 1 . Moreover, let B = 2 β I , with 0 < β < √ ω 1 , and λ k = p ω k − β 2 . Then, the equa tion u ′′ ( t ) + Au ( t ) + 2 β u ′ ( t ) = 0 (44) with initial conditions u (0) = u 0 = X k ≥ 1 u 0 k , u ′ (0) = u 1 = X k ≥ 1 u 1 k , where u i k ∈ Z k for every k ≥ 1, ( i = 1 , 2 ), admits the so lution u ( t ) = e − β t X k ≥ 1  u 0 k cos( λ k t ) + u 1 k + β u 0 k λ k sin( λ k t )  . In par ticular, cho osing u 0 ∈ Z 1 and u 1 ∈ Z 1 , we have that u ( t ) lies in Z 1 for every t ≥ 0. On the other ha nd, the solution to v ′′ ( t ) + Av ( t ) + αu ( t ) = 0 (45) is coupled with ( 44 ) o nly in the co mp one nt in Z 1 , while it is conser v ative in Z ⊥ 1 . More precisely , writing v ( t ) = v 1 ( t ) + v 2 ( t ) ∈ Z 1 + Z ⊥ 1 , equa tion ( 45 ) implies that ( v ′′ 1 ( t ) + ω 1 v 1 ( t ) + αu ( t ) = 0 v ′′ 2 ( t ) + Av 2 ( t ) = 0 . (46) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 13 Therefore, taking v (0) = v 0 / ∈ Z 1 and v ′ (0) = v 1 / ∈ Z 1 , E ( v 2 ( t ) , v ′ 2 ( t )) = 1 2  | v ′ 2 ( t ) | 2 + h Av 2 ( t ) , v 2 ( t ) i  = E ( v (0) , v ′ (0)) > 0 for all t ≥ 0. So , system ( 42 ) is not sta bilizable. 4. Res ults with data in i n terp olation spaces. When the initial data b elong to an int erp olation s pace b etw een H and the domain of a p ow e r of A we can improve Corollar y 2 as follows. Theorem 4.1. Assume ( H 1) , ( H 2) , ( H 3) and ( 29 ) . If U 0 ∈ ( H , D ( A 4 n )) θ , 2 for some n ≥ 1 and 0 < θ < 1 , t hen the solution U of pr oblem ( 9 ) satisfies k U ( t ) k 2 H ≤ c n,θ t nθ k U 0 k 2 ( H ,D ( A 4 n )) θ, 2 ∀ t > 0 (47) for some c onstant c n,θ > 0 . Pr o of. The pr o of easily follows from the interpolatio n results rec alled in Section 2 applied to the o p erator Λ t : H → H defined by Λ t ( U 0 ) = e t A U 0 ∈ H for each U 0 ∈ H . Although ( H , D ( A 4 n )) θ , 2 is usually difficult to identify explicitly , we can single out imp ortant sp ecia l case s where such an identification is p os sible. W e need a preliminary res ult. Lemma 4.2. The op er ator A : D ( A ) → H is invertible, with A − 1 b ounde d. Mor e- over, A is m-dissip ative (thus, A gener ates a C 0 -semigr oup of c ontra ctions on H ) . Pr o of. F or a ny U = ( u, p, v , q ), b U = ( ˆ u, ˆ p, ˆ v , ˆ q ) ∈ H , the identit y A U = b U is equiv alent to p = ˆ u , − A 1 u − B p − αv = ˆ p , q = ˆ v , − A 2 v − αu = ˆ q . Hence, p = ˆ u ∈ D ( A 1 / 2 1 ), q = ˆ v ∈ D ( A 1 / 2 2 ). So, in order to compute A − 1 it suffices to solve the system ( A 1 u + αv = f A 2 v + αu = g , (48) for suitably chosen f , g ∈ H . Since I − α 2 A − 1 1 A − 1 2 is inv er tible thanks to ( H 3), it is easy to chec k that ( 48 ) admits the solution ( ¯ u =  I − α 2 A − 1 1 A − 1 2  − 1 A − 1 1 ( f − αA − 1 2 g ) ∈ D ( A 1 ) ¯ v = A − 1 2 ( g − α ¯ u ) ∈ D ( A 2 ) . Thu s, A is inv er tible, and A − 1 is b ounded. Moreover, A is dissipative, since ( A U | U ) ≤ −h B p, p i H ≤ − β | p | 2 H ≤ 0 ∀ U ∈ D ( A ) . In addition, it is easy to c heck that ther e exists λ > 0 suc h that the range o f λI − A equals H . Thus, by the Lumer-Phillips Theorem (see, e.g., [ 27 , Theorem 4.3 ]), A generates a C 0 -semigro up of contractions on H . Applying Co rollar y 1 , we obtain the following result. Corollary 3. If θ k = m , for some 0 < θ < 1 and k , m ∈ N , then D ( A m ) = ( H , D ( A k )) θ , 2 . (49) 14 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Remark 3. In par ticula r, let us ta ke k = 4 n ( n ≥ 1) and θ j = j 4 n for j = 1 , . . . , 4 n − 1. Then, ( 49 ) yields ( H , D ( A 4 n )) θ j , 2 = D ( A j ) ( j = 1 , . . . , 4 n − 1) . (50) Thu s, a pply ing Theorem 4.1 to the a bove v alues o f θ j , one can show that, if U 0 ∈ D ( A j ), then the a sso ciated solution U ( t ) of pr oblem ( 9 ) s atisfies k U ( t ) k 2 H ≤ c n,j t j / 4 k U 0 k 2 D ( A j ) ∀ t > 0 for some constant c n,j > 0. Moreover, we claim that c n,j can b e chosen independent of n . Indeed, since j 6 = 4 n , one can take the smallest p os itive n j such that j < 4 n j , and us e ( 50 ) with θ j = j / (4 n j ) to co nclude that c n j ,j = c j . As already mentioned in the int ro duction, this result c a n be compare d with the one in [ 1 0 , P rop osition 3.1], which was obtained by a different metho d. Corollary 4. A ssume ( H 1) , ( H 2) , ( H 3) and ( 29 ) . i) If U 0 ∈ D ( A n ) for some n ≥ 1 , then t he solution of ( 9 ) satisfies E ( U ( t )) ≤ c n t n/ 4 n X k =0 E ( U ( k ) (0)) ∀ t > 0 (51) for some c onstant c n > 0 . ii) If U 0 ∈ ( H , D ( A n )) θ , 2 for some n ≥ 1 wher e 0 < θ < 1 , then the solution of ( 9 ) satisfies k U ( t ) k 2 H ≤ c n,θ t nθ / 4 k U 0 k 2 ( H ,D ( A n )) θ, 2 ∀ t > 0 (52) for some c onstant c n,θ > 0 . iii) If U 0 ∈ D (( −A ) θ ) for some 0 < θ < 1 , then t he solution of pr oblem ( 9 ) satisfies k U ( t ) k 2 H ≤ c θ t θ / 4 k U 0 k 2 D (( −A ) θ ) ∀ t > 0 (53) for some c onstant c θ > 0 . Pr o of. Points i ) and ii ) derive fro m Corolla ry 2 a nd following the pro o f of Theo- rem 4.1 , thanks to Remark 3 . In or der to prov e p oint iii ), first we deduce from Lemma 4.2 that −A is in vertible with b ounded inv er se. Mo reov er, it is m-acc retive on H , hence ( 25 ) yields ( H , D ( A )) θ , 2 = ( H , D ( −A )) θ , 2 = D (( −A ) θ ) for every 0 < θ < 1. The conclusion follows applying ii ) with n = 1. Under further assumptions, the norm in ( H , D ( A )) θ , 2 can b e g iven a mor e explicit form. F o r this purp ose, for each k ≥ 0 consider the space H k = D ( A ( k +1) / 2 1 ) × D ( A k/ 2 1 ) × D ( A ( k +1) / 2 2 ) × D ( A k/ 2 2 ) . W e recall the following result (see [ 4 , Lemma 3.1]). Lemma 4.3. Assum e ( H 1 ) , and ( H 2) . L et n ≥ 1 b e su ch that B D ( A ( k +1) / 2 1 ) ⊂ D ( A k/ 2 1 ) (5 4) D ( A ( k/ 2)+1 1 ) ⊂ D ( A k/ 2 2 ) (55) D ( A ( k/ 2)+1 2 ) ⊂ D ( A k/ 2 1 ) (56) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 15 for every inte ger k satisfying 0 < k ≤ n − 1 . (no assumption is made if n = 1 ). Then H k ⊂ D ( A k ) for every 0 ≤ k ≤ n . In [ 4 ], it is a ls o shown that H k = D ( A k ) for every 0 ≤ k ≤ n , provided ( 5 5 ) and ( 56 ) ar e replaced by the s tronger as sumptions D ( A ( k +1) / 2 1 ) ⊂ D ( A k/ 2 2 ) D ( A ( k +1) / 2 2 ) ⊂ D ( A k/ 2 1 ) for every 0 < k ≤ n − 1 . Let 0 < θ < 1 and k ≥ 1 b e fixed. As a dir ect cons equence of Theor em 2.2 , choosing appropr iate spaces and o pe rator T , one can show that, if H k is contained in D ( A k ), then ( H , H k ) θ , 2 is contained in ( H , D ( A k )) θ , 2 . Mo reov er, ( H , H k ) θ , 2 equals H k,θ := ( D ( A 1 / 2 1 ) , D ( A ( k +1) / 2 1 )) θ , 2 × ( H, D ( A k/ 2 1 )) θ , 2 × ( D ( A 1 / 2 2 ) , D ( A ( k +1) / 2 2 )) θ , 2 × ( H, D ( A k/ 2 2 )) θ , 2 . Notice that, since A i is self-a djoint a nd ( 22 ) holds for i = 1 , 2, applying Theorem 2.4 we hav e, for every 0 ≤ α < β ( i = 1 , 2), ( D ( A α i ) , D ( A β i )) θ , 2 = D ( A (1 − θ ) α + θ β i ) . Therefore, H k,θ equals D ( A 1 2 + k 2 θ 1 ) × D ( A k 2 θ 1 ) × D ( A 1 2 + k 2 θ 2 ) × D ( A k 2 θ 2 ) . Observing that, for initia l data in H n,θ , we can b ound (a bove and b elow) the norm of U 0 by the norms of its comp onents, w e have the following. Corollary 5. A ssume ( H 1) , ( H 2) , ( H 3) and ( 29 ) . 1) If H n ⊂ D ( A n ) for some n ≥ 2 , then for e ach U 0 ∈ H n the s olut ion U of pr oblem ( 9 ) satisfies k U ( t ) k 2 H ≤ c n t n/ 4 k U 0 k 2 H n ∀ t > 0 (57) for some c onstant c n > 0 , wher e k U 0 k 2 H n = | u 0 | 2 D ( A ( n +1) / 2 1 ) + | u 1 | 2 D ( A n/ 2 1 ) + | v 0 | 2 D ( A ( n +1) / 2 2 ) + | v 1 | 2 D ( A n/ 2 2 ) . 2) L et n ≥ 1 and 0 < θ < 1 b e fixe d. If H n ⊂ D ( A n ) , then for every U 0 ∈ H n,θ the solution U of ( 9 ) satisfies k U ( t ) k 2 H ≤ c n,θ t nθ / 4 k U 0 k 2 H n,θ ∀ t > 0 (58) for some c onstant c n,θ > 0 , with k U 0 k 2 H n,θ ≍ | u 0 | 2 D ( A (1+ nθ ) / 2 1 ) + | u 1 | 2 D ( A nθ/ 2 1 ) + | v 0 | 2 D ( A (1+ nθ ) / 2 2 ) + | v 1 | 2 D ( A nθ/ 2 2 ) , wher e ≍ s t ands for the e quivalenc e b etwe en norms. 5. Appli cations to PDEs . In this section we describ e some examples of systems of partial differential equations that can b e studied by the r esults of this pape r , but fail to satisfy the compatibility condition ( 10 ). W e will her eafter denote by Ω a bo unded do main in R N with a sufficiently smo oth b oundary Γ. F or i = 1 , . . . , N w e will denote by ∂ i the partia l der iv ative with resp ect to x i and by ∂ t the deriv a tive with resp ect to the time v a riable. W e will also use the nota tion H k (Ω) , H k 0 (Ω) for the usua l Sob olev s paces with norm k u k k, Ω = h Z Ω X | p |≤ k | D p u | 2 dx i 1 2 , 16 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI where we hav e set D p = ∂ p 1 1 · · · ∂ p N N for any m ulti-index p = ( p 1 , . . . , p N ). Finally , we will denote b y C Ω > 0 the larges t constant s uch that Poincar´ e’s ine q uality C Ω k u k 2 0 , Ω ≤ k∇ u k 2 0 , Ω (59) holds true for any u ∈ H 1 0 (Ω). T o avoid to o many notation, we denote in the sa me wa y the constant C Ω such that C Ω k u k 2 0 , Ω ≤ k∇ u k 2 0 , Ω + k u k 2 0 , Γ , (60) for all u ∈ H 1 Ω. In the following examples we take H = L 2 (Ω) , B = β I . Example 5.1. Let β , λ > 0 , α ∈ R , and cons ide r the problem ( ∂ 2 t u − ∆ u + β ∂ t u + λu + αv = 0 ∂ 2 t v − ∆ v + αu = 0 in Ω × (0 , + ∞ ) (61) with b oundar y conditions ∂ u ∂ ν ( · , t ) = 0 on Γ , v ( · , t ) = 0 on Γ ∀ t > 0 (62) and initial conditions ( u ( x, 0) = u 0 ( x ) u ′ ( x, 0) = u 1 ( x ) v ( x, 0) = v 0 ( x ) v ′ ( x, 0) = v 1 ( x ) x ∈ Ω . (63) The ab ov e system ca n b e rewritten in abstra c t form taking D ( A 1 ) =  u ∈ H 2 (Ω) : ∂ u ∂ ν = 0 on Γ  , A 1 u = − ∆ u + λu , D ( A 2 ) = H 2 (Ω) ∩ H 1 0 (Ω) , A 2 v = − ∆ v . (64) Notice that, in o rder to verify a ssumption ( H 3), we shall choo se α such that 0 < | α | < ( C Ω ( C Ω + λ )) 1 / 2 . Then, |h A 1 u, v i| =     Z Ω ∇ u ∇ v dx + λ Z Ω uv dx     ≤  Z Ω |∇ u | 2 dx  1 / 2  Z Ω |∇ v | 2 dx  1 / 2 + λ  Z Ω u 2 dx  1 / 2  Z Ω v 2 dx  1 / 2 ≤ c h A 1 u, u i 1 / 2 | A 2 v | , where we hav e used the co ercivity of A 2 and the well-kno wn inequality Z Ω v 2 + |∇ v | 2 dx ≤ c Z Ω | ∆ v | 2 dx ∀ v ∈ H 2 (Ω) ∩ H 1 0 (Ω) . Since condition ( 30 ) is fulfilled, w e get the fo llowing conclusions . i ) If ( u 0 , u 1 , v 0 , v 1 ) ∈ D ( A 1 ) × D ( A 1 / 2 1 ) × D ( A 2 ) × D ( A 1 / 2 2 ), then the solutio n U of problem ( 61 )-( 62 )-( 63 ) s atisfies E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c t 1 / 4 k U 0 k 2 D ( A ) ∀ t > 0 (65) for so me constant c > 0. Moreover, there exists c 1 > 0 such that k U 0 k 2 D ( A ) ≤ c 1  k u 0 k 2 2 , Ω + k u 1 k 2 1 , Ω + k v 0 k 2 2 , Ω + k v 1 k 2 1 , Ω  . INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 17 ii ) By p oint ii ) of Corolla ry 4 , if U 0 ∈ ( H , D ( A n )) θ , 2 for so me 0 < θ < 1, n ≥ 1 , then the solution o f ( 61 )-( 62 )-( 63 ) satisfies E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c n,θ t nθ / 4 k U 0 k 2 ( H ,D ( A n )) θ, 2 (66) for every t > 0 and some co nstant c n,θ > 0. Moreov er , point iii ) of Cor ollary 4 ensures that, if U 0 ∈ D (( −A ) θ ) for some 0 < θ < 1, then E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c θ t θ / 4 k U 0 k 2 D (( −A ) θ ) ∀ t > 0 (67) for so me constant c θ > 0. Of in terest is the case when an op erator fulfills different b oundary conditions on prop er subsets of Γ. F or instance, let Γ 0 be an op en subset o f Γ (with resp ect to the top olo gy of Γ ) and s et Γ 1 = Γ \ Γ 0 . W e a ssume that Γ 0 ∩ Γ 1 = ∅ . Consider the system ( 61 ) with b oundar y conditions u ( · , t ) = 0 on Γ 0 , ∂ u ∂ ν ( · , t ) = 0 on Γ \ Γ 0 v ( · , t ) = 0 on Γ ∀ t > 0 (68) and initial conditions ( 63 ). Let us se t D ( A 1 ) =  u ∈ H 2 (Ω) : u = 0 on Γ 0 , ∂ u ∂ ν = 0 on Γ \ Γ 0  , A 1 u = − ∆ u . Then, |h A 1 u, v i| ≤ c h A 1 u, u i 1 / 2 | A 2 v | . So , for 0 < | α | < ( C Ω ( C Ω + λ )) 1 / 2 , co ndition ( 29 ) is fulfilled, and the same conclus io ns i ) − ii ) ho ld for problem ( 6 1 )-( 68 )-( 63 ). Example 5 .2. Another interesting s ituation o ccurs while coupling tw o equations of different orders . Let β , λ > 0 , α ∈ R , a nd conside r the sys tem ( ∂ 2 t u + ∆ 2 u + λu + β ∂ t u + αv = 0 ∂ 2 t v − ∆ v + αu = 0 in Ω × (0 , + ∞ ) (69) with b oundar y conditions ∆ u ( · , t ) = 0 = ∂ ∆ u ∂ ν ( · , t ) on Γ , v ( · , t ) = 0 on Γ ∀ t > 0 (70) and initial conditions ( 63 ). Define D ( A 1 ) =  u ∈ H 4 (Ω) : ∆ u = 0 = ∂ ∆ u ∂ ν on Γ  , A 1 u = ∆ 2 u + λu , D ( A 2 ) = H 2 (Ω) ∩ H 1 0 (Ω) , A 2 v = − ∆ v . Suppo se 0 < | α | < λ 1 / 2 C 1 / 2 Ω , as requir ed by ( H 3). Observing that, for any u ∈ D ( A 1 ) and v ∈ D ( A 2 ), |h A 1 u, v i| =     Z Ω ∆ u ∆ v dx + λ Z Ω uv dx     ≤ c h A 1 u, u i 1 / 2 | A 2 v | , we conclude that co ndition ( 30 ) is fulfilled. So, for every U 0 ∈ D ( A ), the s olution U of pr oblem ( 69 )-( 70 )-( 63 ) satisfie s E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c t 1 / 4 k U 0 k 2 D ( A ) ∀ t > 0 (71) for so me constant c > 0. Moreover, there exists c 1 > 0 such that k U 0 k 2 D ( A ) ≤ c 1  k u 0 k 2 4 , Ω + k u 1 k 2 2 , Ω + k v 0 k 2 2 , Ω + k v 1 k 2 1 , Ω  . 18 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Note that we give in Example 6.3 a no ther set o f b oundar y conditions for the same symbols fo r the op era tors. It is interesting to see tha t b oth examples ar e treated for different class e s of compatibility conditions, namely the pr esent exa mple satisfies the compatibility condition ( 11 ), w her eas the example ( 6.3 ) satis fie s the co mpa tibilit y condition ( 10 ). Example 5.3. Let β > 0, α ∈ R , and consider the problem ( ∂ 2 t u − ∆ u + β ∂ t u + αv = 0 ∂ 2 t v − ∆ v + αu = 0 in Ω × (0 , + ∞ ) (72) with b oundar y conditions  ∂ u ∂ ν + u  ( · , t ) = 0 on Γ v ( · , t ) = 0 on Γ ∀ t > 0 (73) and initial conditions ( 63 ). Let us define D ( A 1 ) =  u ∈ H 2 (Ω) : ∂ u ∂ ν + u = 0 on Γ  , A 1 u = − ∆ u , D ( A 2 ) = H 2 (Ω) ∩ H 1 0 (Ω) , A 2 v = − ∆ v , (74) and assume 0 < | α | < C Ω . Obser ve that |h A 1 u, v i| =     Z Ω ∇ u ∇ v dx     ≤  Z Ω |∇ u | 2 dx  1 / 2  Z Ω |∇ v | 2 dx  1 / 2 ≤ c h A 1 u, u i 1 / 2 | A 2 v | , since h A 1 u, u i = Z Ω |∇ u | 2 dx + Z Γ | u | 2 dS , Z Ω |∇ v | 2 dx ≤ c Z Ω | ∆ v | 2 dx . Thu s, condition ( 29 ) is fulfilled. So, the energ y of the so lution of proble m ( 72 )- ( 73 )- ( 63 ) sa tis fie s E 1 ( u ( t ) , u ′ ( t )) + E 2 ( v ( t ) , v ′ ( t )) ≤ c t 1 / 4 k U 0 k 2 D ( A ) ∀ t > 0 (75) for so me constant c > 0. Moreover, there exists c 1 > 0 such that k U 0 k 2 D ( A ) ≤ c 1  | A 1 u 0 | 2 + | A 1 / 2 1 u 1 | 2 + | A 2 v 0 | 2 + | A 1 / 2 2 v 1 | 2  . Our next result show that the op era tors in E xample 5.3 do not fulfill the com- patibilit y condition ( 10 ). Prop ositi o n 2. L et A 1 , A 2 b e define d as in ( 74 ) . Then for every k ∈ N , k ≥ 2 , D ( A k/ 2 2 ) is n ot include d in D ( A 1 ) . Pr o of. Since D ( A k 2 ) ⊂ D ( A k/ 2 2 ) for every k ∈ N , it is sufficien t to prov e that D ( A k 2 ) is not included in D ( A 1 ) for every k ∈ N , k ≥ 1 . F or this purp ose, let us fix k ∈ N , k ≥ 1, a nd co nsider the pr oblem ( ( − ∆) k v 0 = 1 v 0 = 0 = ∆ v 0 = · · · = ∆ k − 1 v 0 on Γ . (76) INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 19 Define the sequence v 1 , v 2 , . . . , v k − 1 by ( − ∆ v 0 = v 1 v 0 | Γ = 0 . . . ( − ∆ v k − 2 = v k − 1 v k − 2 | Γ = 0 ( − ∆ v k − 1 = 1 v k − 1 | Γ = 0 . (77) W e will arg ue by co ntradiction, assuming D ( A k 2 ) ⊂ D ( A 1 ). Since v 0 belo ngs to D ( A 2 ) ∩ D ( A 1 ), we hav e v 0 | Γ = 0 = ∂ v 0 ∂ ν | Γ . Moreov er, from the fir st system in ( 77 ), it follows that Z Ω v 1 dx = Z Ω ( − ∆ v 0 ) dx = − Z Γ ∂ v 0 ∂ ν dS = 0 . Hence, Z Ω v 1 dx = 0. Let us pr ove by induction tha t Z Ω ∇ v k − i ∇ v i dx = 0 ∀ i = 1 , 2 , . . . , k − 1 . (78) F or i = 1 we hav e Z Ω ∇ v k − 1 ∇ v 1 dx = Z Ω ( − ∆ v k − 1 ) v 1 dx = Z Ω v 1 dx = 0 , since v k − 1 | Γ = 0 = v 1 | Γ . Now, let i > 1 a nd supp ose Z Ω ∇ v k − i ∇ v i dx = 0 . Then, 0 = Z Ω v k − i ( − ∆ v i ) dx = Z Ω v k − i v i +1 dx = Z Ω ( − ∆ v k − i − 1 ) v i +1 dx = Z Ω ∇ v k − ( i +1) ∇ v i +1 dx . Thu s, ( 78 ) holds for i + 1. Moreover, from ( 78 ) follows that Z Ω v k − i v i +1 dx = 0 ∀ i = 1 , 2 , . . . , k − 1 , (79) since Z Ω v k − i v i +1 dx = Z Ω v k − i ( − ∆ v i ) dx = Z Ω ∇ v k − i ∇ v i dx = 0 . Now, let k be even, say k = 2 p , p ∈ N ∗ . Then, by ( 78 ) with i = p we obtain Z Ω |∇ v p | 2 dx = 0 , whence v p = 0 . So, by a casca de effect, v p +1 = − ∆ v p = 0 ⇒ v p +2 = − ∆ v p +1 = 0 ⇒ · · · ⇒ v k − 1 = − ∆ v k − 2 = 0 . Since − ∆ v k − 1 = 1, we g et a contradiction. If, o n the contrary , k is o dd, i.e. k = 2 p + 1, then, applying ( 79 ) with i = p , we co nclude that Z Ω | v p +1 | 2 dx = 0 , whence v p +1 = 0 . Finally , we have that v p +1 = v p +2 = · · · = v k − 1 = 0. Since − ∆ v k − 1 = 1, we get a contradiction again. Therefore , D ( A k 2 ) is not included in D ( A 1 ). 20 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI Example 5.4. Given β > 0, α ∈ R , let us now consider the undamp ed Petro w s ky equation co upled with the damp ed wa ve equation, ( ∂ 2 t u − ∆ u + β ∂ t u + αv = 0 ∂ 2 t v + ∆ 2 v + αu = 0 in Ω × (0 , + ∞ ) (80) with Robin bo undary co nditions  ∂ u ∂ ν + u  ( · , t ) = 0 on Γ ∀ t > 0 (81) on u and either v ( · , t ) = ∆ v ( · , t ) = 0 o n Γ ∀ t > 0 (82) or v ( · , t ) = ∂ v ∂ ν ( · , t ) = 0 on Γ ∀ t > 0 (83) on v , with initial co nditions ( 63 ). Define D ( A 1 ) =  u ∈ H 2 (Ω) : ∂ u ∂ ν + u = 0 on Γ  , A 1 u = − ∆ u , D ( A 2 ) =  v ∈ H 4 (Ω) : v = ∆ v = 0 on Γ  , A 2 v = ∆ 2 v (with b oundar y conditions ( 82 ) on v ), or ˜ D ( A 2 ) =  v ∈ H 4 (Ω) : v = ∂ v ∂ ν = 0 on Γ  , A 2 v = ∆ 2 v (with b oundar y conditions ( 83 ) on v ). Once ag ain, we hav e |h A 1 u, v i| =     Z Ω ∇ u ∇ v dx     ≤  Z Ω |∇ u | 2 dx  1 / 2  Z Ω |∇ v | 2 dx  1 / 2 ≤ c h A 1 u, u i 1 / 2 | A 2 v | . Thu s, condition ( 29 ) is fulfilled a nd, for 0 < | α | < C 3 / 2 Ω , the p olynomia l decay of the energ y of so lution to ( 80 )-( 81 )-( 82 )-( 63 ) and ( 8 0 )-( 81 )-( 83 ) - ( 63 ) follows as in Example 5.1 . 6. Improv ement of previo us resul ts. In this section we apply int erp olation theory to extend the p oly nomial stability res ult o f [ 4 ] to a larg er c la ss o f initial data. W e will denote by j ≥ 2 the integer for which ( 10 ) is satisfie d. As is shown in [ 4 , Theo rem 4.2], under a ssumptions ( H 1) , ( H 2) , ( H 3) a nd ( 10 ), if U 0 ∈ D ( A nj ) for so me integer n ≥ 1 , the solution U of pro blem ( 9 ) s atisfies E ( U ( t )) ≤ c n t n nj X k =0 E ( U ( k ) (0)) ∀ t > 0 (84) for some constant c n > 0. W e recall that a s sumption ( 10 ) cov ers many situa tions of int erest for a pplications to systems of evolution eq uations. Indeed (see [ 4 , Sectio n 5] for further details), this is the cas e for i) ( A 1 , D ( A 1 )) = ( A 2 , D ( A 2 )), where ( 10 ) is fulfilled with j = 2; ii) D ( A 1 ) = D ( A 2 ), with j = 2 ; iii) ( A 2 , D ( A 2 )) = ( A 2 1 , D ( A 2 1 )), ag ain with j = 2; iv) ( A 1 , D ( A 1 )) = ( A 2 2 , D ( A 2 2 )), with j = 4 . INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 21 The following re sult co mpletes the analysis of [ 4 ], taking the initial data in s uita ble int erp olation spaces . Theorem 6.1. A s sume ( H 1) , ( H 2 ) , ( H 3) and ( 10 ) , and let 0 < θ < 1 , n ≥ 1 . Then for every U 0 in ( H , D ( A nj )) θ , 2 , t he solution U of ( 9 ) satisfies k U ( t ) k 2 H ≤ c n,θ t nθ k U 0 k 2 ( H ,D ( A nj )) θ, 2 ∀ t > 0 (85) for some c onstant c n,θ > 0 . Reasoning as in Remar k 3 , one can derive estimate ( 84 ) also for U 0 ∈ D ( A k ), for every k = 1 , . . . , nj − 1, with decay rate k /j . Corollary 6. A ssume ( H 1) , ( H 2) , ( H 3) and ( 10 ) . i) If U 0 ∈ D ( A n ) for some n ≥ 1 , then t he solution of ( 9 ) satisfi es k U ( t ) k 2 H ≤ c n t n/j k U 0 k 2 D ( A n ) ∀ t > 0 (86) for some c onstant c n > 0 . ii) If U 0 ∈ ( H , D ( A n )) θ , 2 for some n ≥ 1 and 0 < θ < 1 , then t he solution of ( 9 ) satisfies k U ( t ) k 2 H ≤ c n,θ t nθ /j k U 0 k 2 ( H ,D ( A n )) θ, 2 ∀ t > 0 (87 ) for some c onstant c n,θ > 0 . iii) If U 0 ∈ D (( −A ) θ ) for some 0 < θ < 1 , then t he solution of pr oblem ( 9 ) satisfies k U ( t ) k 2 H ≤ c θ t θ /j k U 0 k 2 D (( −A ) θ ) ∀ t > 0 (88) for some c onstant c θ > 0 . In particular , the prev io us fractional decay rates can b e achiev ed for initial data in H n or in H n,θ , whenever H n ⊂ D ( A n ), a s in Coro llary 5 . This happ ens, for instance, if any of the following co nditions is satisfied: i) ( A 1 , D ( A 1 )) = ( A 2 , D ( A 2 )); ii) D ( A 1 ) = D ( A 2 ); iii) ( A 2 , D ( A 2 )) = ( A 2 1 , D ( A 2 1 )). Let us apply Corollar y 6 to tw o exa mples from [ 4 ]. Example 6.2. Given β > 0 , κ > 0 , α ∈ R , let us study the problem  ∂ 2 t u − ∆ u + β ∂ t u + κu + αv = 0 ∂ 2 t v − ∆ v + κv + αu = 0 in Ω × (0 , + ∞ ) (89) with b oundar y conditions u ( · , t ) = 0 = v ( · , t ) o n Γ ∀ t > 0 (90) and initial conditions  u ( x, 0) = u 0 ( x ) , u ′ ( x, 0) = u 1 ( x ) v ( x, 0) = v 0 ( x ) , v ′ ( x, 0) = v 1 ( x ) x ∈ Ω . (91 ) Let H = L 2 (Ω), B = β I , and A 1 = A 2 = A b e defined by D ( A ) = H 2 (Ω) ∩ H 1 0 (Ω) , Au = − ∆ u + κu ∀ u ∈ D ( A ) . Notice that ( 10 ) is fulfilled with j = 2, a nd condition 0 < | α | < C Ω + κ = : ω is required in order to fulfill ( H 3 ). 22 F. ALABAU-B OUSSOUIRA, P . CANNARSA AND R. GUGLIELMI As show ed in [ 4 , E x ample 6 .1], if u 0 , v 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω) and u 1 , v 1 ∈ H 1 0 (Ω), then Z Ω  | ∂ t u | 2 + |∇ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c t  k u 0 k 2 2 , Ω + k u 1 k 2 1 , Ω + k v 0 k 2 2 , Ω + k v 1 k 2 1 , Ω  ∀ t > 0 . Moreov er, if u 0 , v 0 ∈ H n +1 (Ω) and u 1 , v 1 ∈ H n (Ω) are such that u 0 = · · · = ∆ [ n 2 ] u 0 = 0 = v 0 = · · · = ∆ [ n 2 ] v 0 on Γ , u 1 = · · · = ∆ [ n − 1 2 ] u 1 = v 1 = · · · = ∆ [ n − 1 2 ] v 1 = 0 on Γ , then Z Ω  | ∂ t u | 2 + |∇ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c n t n  k u 0 k 2 n +1 , Ω + k u 1 k 2 n, Ω + k v 0 k 2 n +1 , Ω + k v 1 k 2 n, Ω  ∀ t > 0 . F urther mo re, applying Coro lla ry 6 , we conclude that if U 0 belo ngs to H n,θ = ( H , D ( A n )) θ , 2 for so me 0 < θ < 1, n ≥ 1, then the solution to ( 89 )-( 90 )-( 91 ) satisfies Z Ω  | ∂ t u | 2 + |∇ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c n,θ t nθ / 2 k U 0 k 2 H n,θ ∀ t > 0 (92) for so me constant c n,θ > 0, with k U 0 k 2 H n,θ ≍ | u 0 | 2 D ( A 1 2 + n 2 θ 1 ) + | u 1 | 2 D ( A n 2 θ 1 ) + | v 0 | 2 D ( A 1 2 + n 2 θ 2 ) + | v 1 | 2 D ( A n 2 θ 2 ) . Example 6.3. T aking β > 0 , 0 < | α | < C 3 / 2 Ω , a nd the same op erator s A 1 and A 2 as in Example 5.2 , but with different b oundary conditio ns , we can consider the system  ∂ 2 t u + ∆ 2 u + β ∂ t u + αv = 0 ∂ 2 t v − ∆ v + αu = 0 in Ω × (0 , + ∞ ) (93) with b oundar y conditions v ( · , t ) = u ( · , t ) = ∆ u ( · , t ) = 0 on Γ ∀ t > 0 (94) and initial conditions a s in ( 91 ). Let us se t H = L 2 (Ω), B = β I , and D ( A 1 ) =  u ∈ H 4 (Ω) : ∆ u = 0 = u o n Γ  , A 1 u = ∆ 2 u , D ( A 2 ) = H 2 (Ω) ∩ H 1 0 (Ω) , A 2 v = − ∆ v . In this cas e , since A 1 = A 2 2 , condition ( 10 ) holds with j = 4. Consequent ly , as is shown in [ 4 , Example 6.4], for initial condition U 0 ∈ D ( A 4 ) Z Ω  | ∂ t u | 2 + | ∆ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ C t k U 0 k 2 D ( A 4 ) ∀ t > 0 , for some c o nstant C > 0. B y p oint i ) of Co rollary 6 , we can gener alize this res ult to initial data U 0 ∈ D ( A n ) for some n ≥ 1 . Indeed, in this case the so lution to ( 93 )-( 94 )-( 91 ) satisfie s Z Ω  | ∂ t u | 2 + | ∆ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c n t n/ 4 k U 0 k 2 D ( A n ) ∀ t > 0 , INDIRECT ST ABILIZA TION WITH HYBRID BOUNDAR Y CONDITIONS 23 for some constant c n > 0. Moreov er, thanks to po in t ii ) o f Corollar y 6 , if U 0 ∈ ( H , D ( A n )) θ , 2 for so me n ≥ 1 and 0 < θ < 1 , then Z Ω  | ∂ t u | 2 + | ∆ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c n,θ t nθ / 4 k U 0 k 2 ( H ,D ( A n )) θ, 2 ∀ t > 0 for some constant c n,θ > 0. F ur ther more, thanks to p oint ii i ) o f Corolla ry 6 , if U 0 belo ngs to H 1 ,θ = D (( −A ) θ ) for s ome 0 < θ < 1 , then the solution to ( 93 )-( 94 )-( 91 ) satisfies Z Ω  | ∂ t u | 2 + | ∆ u | 2 + | ∂ t v | 2 + |∇ v | 2  dx ≤ c θ t θ / 4 k U 0 k 2 D (( −A ) θ ) ∀ t > 0 (95) for so me constant c θ > 0, with k U 0 k 2 D (( −A ) θ ) ≍ | u 0 | 2 D ( A 1 2 + 1 2 θ 1 ) + | u 1 | 2 D ( A 1 2 θ 1 ) + | v 0 | 2 D ( A 1 2 + 1 2 θ 2 ) + | v 1 | 2 D ( A 1 2 θ 2 ) . 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