BEHAVIOR OF THE ENERGY FOR LAMÉ SYSTEMS IN BOUNDED DOMAINS WITH NONLINEAR DAMPING AND EXTERNAL FORCE

Transcription

1 Elecronic Journal of Differenial Equaion, Vol. 213 (213, No. 1, pp ISSN: URL: hp://ejde.mah.xae.edu or hp://ejde.mah.un.edu fp ejde.mah.xae.edu BEHAVIOR OF THE ENERGY FOR LAMÉ SYSTEMS IN BOUNDED DOMAINS WITH NONLINEAR DAMPING AND EXTERNAL FORCE AHMED BCHATNIA, MOEZ DAOULATLI Abrac. We udy behavior of he energy for oluion o a Lamé yem on a bounded domain, wih localized nonlinear damping and exernal force. The equaion i e up in hree dimenion and under a microlocal geomeric condiion. More preciely, we prove ha he behavior of he energy i deermined by a oluion o a forced differenial equaion, an i depend on he L 2 norm of he force. 1. Inroducion and aemen of he problem Le be a bounded mooh domain in R 3. Le u conider he Lamé yem wih localized nonlinear damping and exernal force, 2 u e u + a(xg( u = f(, x, in R +, u = on R +, u(, x = ϕ 1 (x, u(, x = ϕ 2 (x in. (1.1 Here e denoe he elaiciy operaor, which i he 3 3 marix-valued differenial operaor defined by e u = µ u + (λ + µ div u, u = (u 1, u 2, u 3, and we aume ha he Lamé conan λ and µ aify he condiion µ >, λ + 2µ >. (1.2 Moreover, a(x L ( i a nonnegaive real funcion, f i in (L 2 loc (R +, L 2 ( 3 and g( u = (g 1 ( u 1, g 2 ( u 2, g 3 ( u 3, where g i : R R i a coninuou monoone increaing funcion aifying g i ( = and he following growh aumpion: c 1 2 g i ( c 2 2, 1, for i = 1, 2, 3, (1.3 wih c 1, c 2 >. We can find applicaion for hi yem in geophyic and eimic wave propagaion. In he cae λ + µ = we obain a vecor wave equaion and we aim in hi aricle o generalize ome well known reul for he wave equaion. 2 Mahemaic Subjec Claificaion. 35L5, 35B4. Key word and phrae. Lamé yem; nonlinear damping; bounded domain; exernal force. c 213 Texa Sae Univeriy - San Marco. Submied November 8, 212. Publihed January 7,

2 2 A. BCHATNIA, M. DAOULATLI EJDE-213/1 In hi framework, due o he nonlinear emi-group heory, i i well known ha, for every ϕ = (ϕ 1, ϕ 2 H = (H 1 ( 3 (L 2 ( 3, he yem (1.1 admi a unique global oluion u(, x uch ha u C (R +, (H 1 ( 3 C 1 (R +, (L 2 ( 3. (1.4 The energy of u a ime i defined by E u ( = 1 (µ u 2 + (λ + µ div u 2 + u 2 (, xdx, (1.5 2 and he following energy funcional law hold E u ( + a(xg( u(σ, x u(σ, x dx dσ = E u ( + f(, x u(σ, x dx dσ, (1.6 for every. For he lieraure we quoe eenially he reul of Biognin e al [12] which eablihed ha he oluion of a yem in elaiciy heory wih a nonlinear localized diipaion decay in an algebraic rae o zero uing ome energy ideniie aociaed wih localized muliplier. For more reul on he energy decay for he Lamé yem wih linear or nonlinear damping we refer he reader o Alabau and Komornik [1, 2], Alabau [3], Guemia [14], Horn [16, 17] and reference herein. We noe ha he mehod ued in hee paper i baed on echnical muliplier. In he ame piri, we can alo quoe he work of Guemia [15] for he obervabiliy, exac conrollabiliy and inernal or boundary abilizaion of general elaiciy yem wih variable coefficien depending on boh ime and pace variable. See alo he work of Bellaoued [4] which inveigae he decay propery of he oluion o he iniial-boundary value problem for he elaic wave equaion wih a local ime-dependen nonlinear damping. We noe moreover ha Burq and Lebeau [5] inroduced he microlocal defec meaure aached o equence of oluion of he Lamé yem and proved a propagaion reul when he energy of he longiudinal componen goe o zero. Finally, Daoulali e al [8] adaped he Lax-Philip heory, and under he aumpion (GC, gave he rae of decay of he local energy for oluion of he Lamé yem on exerior domain wih nonlinear localized damping. Le u indicaed ha all he reul above are wihou exernal force and no reul eem o be known when f. We pecially menion he reul of Daoulali [7], which udy he behavior of he energy of oluion of he wave equaion wih localized damping and an exernal force on compac Riemannian manifold wih boundary. The main purpoe of hi work i o give he behavior of he energy of oluion of (1.1. Fir we recall he following definiion. Definiion 1.1. We will call generalized bicharaceriic pah any curve which coni of generalized bicharaceriic of he principal ymbol p (where p(, x; τ, ξ = (µ ξ 2 τ 2 2 ((λ + 2µ ξ 2 τ 2, wih poibiliy of moving from a characeriic manifold o anoher, a each poin of T (, in he way indicaed in [8]. Remark 1.2. A generalized geodeic pah i coniued of egmen living in, ha inerec he boundary ranverally (a hyperbolic poin for p L (, x; τ, ξ = c 2 L ξ 2 τ 2 or p T (, x; τ, ξ = c 2 T ξ 2 τ 2 (where c L = λ + 2µ and c T = µ,

3 EJDE-213/1 BEHAVIOR OF THE ENERGY 3 or angenially (a diffracive poin. Thee egmen may be conneced o arc of curve living on which are projecion of glancing ray aociaed o p L or p T. The projecion of uch a generalized bicharaceriic pah on will be called a generalized geodeic pah. Definiion 1.3. Le ω be an open ube of, T > and conider he following aumpion: (GC every generalized geodeic pah of, iued a =, mee R + ω beween he limi and T. We hall relae he open ube ω wih he damper a by ω = {x : a(x > µ > }. Before aing he main reul of hi paper, we will define ome funcion. According o [18] here exi a concave coninuou, ricly increaing funcion h i (i = 1, 2, 3, linear a infiniy wih h i ( = uch ha h i (g i ( ɛ ( 2 + g i ( 2, η, (1.7 for ome ɛ, η >. For example when g i i uperlinear, odd and he funcion g i ( i convex, hen h 1 i ( = g i ( when η. For furher informaion on he conrucion of a uch funcion we refer he inereed reader o [6, 9, 18]. Wih hi funcion, we define h( = + h (, where h ( = 3 m a ( T h i (, (1.8 m a ( T for, dm a = a(x dx d and T = (, T. In hi aricle, we how ha under he aumpion (GC we obain he following obervabiliy inequaliy: Non-auonomou obervabiliy inequaliy: There exi a conan T > uch ha he oluion u(, x o he nonlinear problem (1.1 wih iniial daa ϕ = (ϕ, ϕ 1 aifie ( +T E u ( C T h a(xg( u u + f(σ, x 2 dx dσ, for every. From he obervabiliy inequaliy above, we infer ha he behavior of he energy depend on f(, x L2 (. More preciely, we will prove ha hi behavior i governed by a forced differenial equaion and depend on ( Γ( = 2 f(,. 2 L 2 ( + ψ ( f(,. L 2 (, where ψ i he convex conjugae of he funcion ψ, defined by { 1 ψ( = 2T h 1 ( 2 8C T R e T +, + R, wih C T 1 and T >. More preciely we have he following heorem. Theorem 1.4. Le he funcion h be defined by (1.8. We aume ha he aumpion (GC hold and ( Γ( = 2 f(,. 2 L 2 ( + ψ ( f(,. L2 ( L 1 loc(r +. i=1

4 4 A. BCHATNIA, M. DAOULATLI EJDE-213/1 Le u( be he oluion o (1.1 wih iniial condiion (ϕ, ϕ 1 H. Then E u ( 2e T (S( T + Γ(d, T, (1.9 where S( i he poiive oluion of he ordinary differenial equaion d + 1 4T h 1 ( 1 K S = Γ(, S( = E u(, (1.1 wih K 2C T. Moreover, If here exi C >, uch ha Γ(τdτ C, for every T. Then E u ( i bounded. If Γ(τdτ a +, and if E u( admi a limi a infiniy, hen he limi i zero. If Γ L 1 (R +, hen E u ( a +. If Γ(τdτ + a +, hen S( + a +. We dicu now he mehod ued for eablihing he main reul. We noe ha he preen work i compared o he work of [7] and [8]. Here, we follow he ame program and we udy he behavior of he energy for he Lamé yem wih Dirichle boundary condiion in a bounded domain and by adding he exernal force. We conider he noion of bicharaceriic pah and we adap for our conex a propagaion reul for he microlocal defec meaure aached o equence of oluion of (1.1. We deduce hen a nonlinear obervabiliy eimae which i needed o prove Theorem Proof of he main reul Before preening he proof of our main heorem, we inroduce ome noaion and recall ome reul from he lieraure. Propoiion 2.1. Le u be a oluion of (1.1 wih iniial daa in he energy pace. Then E u ( (1 + 1 ɛ eɛ( ( E u ( + 1 f(σ, x 2 dx dσ, (2.1 ɛ for every ɛ > and for every. Proof. Le. From he energy ideniy (1.6, we infer ha E u ( E u ( + f(, x u(σ, x dx dσ. Uing Young inequaliy, we obain E u ( E u ( + 1 ɛ f(σ, x 2 dx dσ + ɛ for every ɛ >. Now Gronwall inequaliy give E u ( e ɛ( ( E u ( + 1 ɛ E u (σdσ, f(σ, x 2 dx dσ. By analogy wih [8, Propoiion 5.1], we obain he following reul.

5 EJDE-213/1 BEHAVIOR OF THE ENERGY 5 Propoiion 2.2. Le (u n be a bounded equence of oluion of he linear Lamé yem 2 u n e u n = in R +, u n = on R +, (u n (, x, u n (, x = ϕ n (x in. (2.2 wih iniial daa in H, weakly converging o in H. We aume ha (GC hold and ha u n in (L 2 loc (], T [ ω3. Then here exi a ubequence (ill denoed (u n uch ha u n in (H 1 loc (], T [, H1 ( 3. Before giving he proof of Propoiion 2.2, we recall ome fac on microlocal defec meaure aociaed o bounded equence of oluion o he linear Lamé yem wih Dirichle boundary condiion. We give hem wihin heir original aemen [8], and we noe ha (wih obviou modificaion of heir proof all hee reul remain valid in our iuaion. We conider he linear Lamé yem on R. 2 u e u =, in R, u = on R, (u(, x, u(, x = (ϕ 1 (x, ϕ 2 (x (H 1 ( 3 (L 2 ( 3. We decompoe fir he oluion of yem (2.3 ino (2.3 u = u L + u T, (2.4 where he longiudinal wave u L and he ranveral wave u T, repecively, aifie he wave yem ( 2 c 2 L u L =, ro u L =, ( 2 c 2 T u T =, div u T =, u = u L + u T = on R, (2.5 wih c L = λ + 2µ and c T = µ. Moreover, if (u n n i a bounded equence of oluion of (2.3 weakly converging o in (H 1 loc (R, H 1 ( 3, he equence (u nl and (u nt are alo of bounded energy and weakly converging o in (H 1 loc (R, H 1 ( 3. In hi way, according o [5], we can aach o (u nl (rep. (u nt a microlocal defec meaure ν L (rep. ν T. Thee meaure are orhogonal in he meaure heory ene (ee [5, Propoiion 4.4] or [11, Lemme 3.3]. In addiion, ν L i uppored in he characeriic e Char L = (Char L (Char L and ν T i uppored in = {(, x, τ, ξ : x, >, c 2 L ξ 2 τ 2 = } {(, y, τ, η : y, >, r L := τ 2 c 2 L η 2 }, Char T ={(, x, τ, ξ; x, >, c 2 T ξ 2 τ 2 = }. Thi fac i known a he ellipic regulariy heorem for he m.d.m. Le u now analyze he propagaion properie of he meaure ν L and ν T. In he inerior, i.e. in T (R, we are in preence of wo wave which propagae independenly, o we have a our dipoal he claical meaure propagaion heorem of [13]. Near he boundary, we have o ake ino accoun, he naure of he bicharaceriic hiing.

6 6 A. BCHATNIA, M. DAOULATLI EJDE-213/1 Take ρ in Char P = {(, y, τ, η; y, >, r T := τ 2 c 2 T η 2 }; for r L,T = r L,T (ρ, we denoe γ L,T (rep. γ+ L,T he (longiudinal/ranveral incoming (rep. ougoing bicharaceriic o (rep. from ρ (hi half bicharaceriic doe no conain ρ. Following hen word by word he argumen developed in [5, proof of Theorem 4], we have Propoiion 2.3. Wih he noaion above, we have (1 r L <, ρ i an ellipic poin for he longiudinal wave. Hence, ν L = near ρ and (a ν T = near ρ if r T <, (b ν T propagae from γ T o γ+ T if r T. (2 < r L r T, ρ i a hyperbolic poin for he longiudinal and he ranveral wave. In hi cae, we obain: If γ L,T uppor(ν L,T =, hen ν T,L propagae from γ T,L o γ+ T,L. (3 = r L < r T, ρ i a glancing poin for he longiudinal wave. Here we have: If γ L uppor (ν L =, hen ν T propagae from γ T o γ+ T. A a conequence, uing he conervaion of he oal ma (ee [5], we obain he following reul. Corollary 2.4. For r L, we have he following equivalence: if and only if (γ L uppor(ν L (γ T uppor(ν T = (γ + L uppor(ν L (γ + T uppor(ν T =. Proof of Propoiion 2.2. Under he decompoiion (2.4, i uffice o prove ha u n,l,t in (H 1 loc (], T [, H1 ( 3, and hank o he orhogonaliy propery of he meaure ν L and ν T and he ellipic regulariy heorem, we have u n,l,t in (L 2 loc (], T [ ω3 and hen ν L = ν T on ], T [ ω. Therefore, o prove Propoiion (2.2, we have o eablih he following implicaion: ν L = ν T = on ], T [ ω ν L = ν T = on ], T [. We argue by conradicion. Le (u n be a bounded equence of oluion of (2.2 wih iniial daa in H, and ν L,T he microlocal defec meaure aociaed o (u n,l,t. Le q T (], T [ uch ha q uppor(ν L uppor(ν T and γ a generalized bicaraceriic pah aring a q. The geomeric aumpion aying ha any raigh line in ha only finie order conac wih, we may aume ha q i an inerior poin. In hi way one can find a bicharaceriic γ (γ L or γ T iued from q and raced backward in ime, conained in he uppor of he aociaed meaure (i.e γ L uppor(ν L or γ T uppor(ν T. A γ hi he boundary, we have wo poibiliie: (a γ hi, for he fir ime, in ome poin ρ uch ha r L (ρ <. (b γ hi, for he fir ime, in ome poin ρ uch ha r L (ρ. In he fir cae, we are near an ellipic poin for he longiudinal wave, o he meaure i carried by he ranveral componen and propagae along he refleced bicharaceriic. In he econd cae, hank o Propoiion 2.3 and Corollary 2.4, one of he wo incoming bicharaceriic γ L or γ T a ρ i, locally, in uppor(ν L or in uppor(ν T. Thu, we can conruc a bicharaceriic pah Γ iued from q

7 EJDE-213/1 BEHAVIOR OF THE ENERGY 7 (he union of all hee ucceive ray γ L or γ T charged by he meaure ν L or ν T conained in uppor(ν L uppor(ν T. According o aumpion (GC Γ mee ], T [ ω a < T, and hi conradic he fac ha Γ uppor(ν L uppor(ν T, ince ν L = ν T = on ], T [ ω. The proof of Propoiion 2.2 i complee. Now, we prove he obervabiliy eimae which coniue wih he lemma 2.7 below he main ingredien of he proof of Theorem 1.4. Propoiion 2.5. Le he funcion h be defined by (1.8. We aume ha he aumpion (GC hold. Then here exi C T >, uch ha he following inequaliy hold: ( +T E u ( C T h a(xg( u u + f(σ, x 2 dx dσ, (2.6 for every,for every oluion u of (1.1 wih iniial daa in he energy pace H, and for every fin (L 2 loc (R +, L 2 ( 3. Proof. To prove hi reul we argue by conradicion. We aume ha here exi a equence (u n n oluion of (1.1 wih iniial daa in he energy pace, a non-negaive equence ( n n and f n in (L 2 loc (R +, L 2 ( 3, uch ha ( n+t E un ( n nh n Moreover, u n ha he following regulariy: a(xg( u n u n + f n (σ, x 2 dx dσ. u n C ( R +, (H 1 ( 3 C 1( R +, (L 2 ( 3. Seing α n = (E un ( n 1/2 > and v n (, x = un(n+,x α n. Then v n aifie 2 v n e v n + 1 a(xg(α n v n = 1 f n ( n +, x, in R +, α n α n v n = on R +, (v n (, x, v n (, x = 1 α n (u n ( n, x, u n ( n, x, in. Moreover E vn ( = 1 and 1 n ( T αn 2 h a(xg(α n v n α n v n + f n ( n +, x 2 dx d. (2.7 Since h = I + h and h i non-negaive and increaing funcion and from he inequaliy above, we infer ha T 1 f n ( n +, x 2 dx d 1 α n n (2.8 n + and [ I + 3 I ]( T m a ( T h i a(xg(α n v n α n v n dx d α2 n m a ( T n. (2.9 i=1 Re-uing he fac ha he funcion h i non-negaive give T a(xg(α n v n v n dx d α 1 n (2.1 n +

8 8 A. BCHATNIA, M. DAOULATLI EJDE-213/1 and ( 1 h i m a ( T T αn 2 a(xg i (α n v n α n ( v n i dx d, i = 1, 2, 3. nm a ( T (2.11 Denoe 1,i = {(, x [, T ] : α n ( v n i (, x < µ} and 2,i = T \ 1,i. Since g i ha a linear behavior on { η}, uing (2.1, we infer ha T a(x( v n i 2 L 2 ( c 2,i 1αn 1 a(xg(α n v n v n dx dτ Moreover, h i i concave, hen uing (he revere Jenen inequaliy ( 1 T h i a(xg i (α n v n α n ( v n i dx dτ m a ( T 1 h i (g i (α n v n α n ( v n i dm a, m a ( T T which give αn 2 h i (g i (α n v n α n ( v n i dm a 1 1,i n. Therefore, from (1.7 we obain a(x[αn 2 g i (α n ( v n i 2 + ( v n i 2 ] dx d. n + 1,i Combining he eimae above wih (2.12 we obain and we conclude ha a(x v n (L 2 ( T 3. (2.12 n + (2.13 n + 1 a(xg(α n v n (L2 ( α T 3. (2.14 n n + Hence, paing o he limi in (2.7, we ee ha he weak limi v (H 1 ([, T ] 3 aifie he yem 2 v e v = in ], T [, Moreover, we obain v = on ], T [, (v(, x, v(, x = ψ(x, in. Now, le w n be he oluion of he yem (2.15 a(x v =, on T. ( w n e w n =, in R +, w n =, on R +, (w n (, x, w n (, x = 1 α n (u n ( n, x, u n ( n, x, in R +. (2.17 I i clear ha he equence (w n n i bounded in (H 1 loc ([, T ] 3 ; moreover, by he hyperbolic energy inequaliy, (2.8 and (2.14 we infer ha up E vn w n ( C(T 1 a(xg( v n 1 f n ( n +, x 2 L T α n α 2 ( T. (2.18 n n +

9 EJDE-213/1 BEHAVIOR OF THE ENERGY 9 Conequenly, hank o (2.13, we deduce ha a(x w n (L2 ( T 3, (2.19 n + o obain a conradicion we ue he following reul for which we popone i proof. Propoiion 2.6. We aume ha he aumpion (GC hold. Then here exi α T >, uch ha he inequaliy ( T E w ( α T w 2 dx d (2.2 hold for every oluion w of 2 w e w =, in R +, w =, ω on R +, (w(, x, w(, x = (w (x, w 1 (x, wih iniial daa in he energy pace H. in Now, uing (2.19 and Propoiion 2.6, we obain T 1 = E vn ( = E wn ( α T w n 2 dx d, n + and hi conclude he Proof of Propoiion 2.5. ω (2.21 Proof of Propoiion 2.6. We argue by conradicion: we uppoe he exience of a equence (w n, oluion of (2.21 uch ha T w n 2 dx d E w n (. n Denoe α n = E wn ( 1/2 and z n = wn α n. Moreover z n aifie ω 2 z n e z n =, in R +, E zn ( = 1, z n =, T in R +, (2.22 z n 2 dx d 1 n. The equence z n i bounded in C ([, T ], (H 1 ( 3 C 1 ([, T ], (L 2 ( 3, hen, i admi a ubequence, ill denoed by z n, ha i weakly-* convergen in he pace L ([, T ], (H 1 ( 3 W 1, ((, T, (L 2 ( 3. In hi way, z n z in (H 1 ([, T ] 3. Paing o he limi in he equaion aified by z n we obain 2 z e z =, z = z = ω in ], T [, in ], T [, on ], T [ ω. (2.23 We need o check ha he rivial oluion, v =, i he only oluion of (2.23 in C ([, T ], (H 1 ( 3 C 1 ([, T ], (L 2 ( 3. For hi, we idenify he funcion z oluion of (2.23 wih i iniial daa φ H, and we conider he pace G = {φ H, z i a oluion of (2.23}. Every z in G i mooh on ], T [ ω; herefore, according o he geomeric conrol condiion and he reul of [21] on propagaion of ingulariie, G i coniued of

10 1 A. BCHATNIA, M. DAOULATLI EJDE-213/1 mooh funcion. Moreover, G i obviouly cloed in H, and we deduce ha i i of finie dimenion. On he oher hand, / operae on G, o i admi an eigenvalue λ, and here exi a nonzero funcion z (x on uch ha e z = λz, z on ω, z = on ; and hi i impoible by unique coninuaion propery of e (ee, for inance, [1]. Now, we muliply E zn ( by ϕ(, wih ϕ C (], T [, ϕ = 1 on ]ε, T ε[, ϕ, and we inegrae. Thi give T = 1 2 ϕ(e zn (d T (µϕ( z n 2 + (λ + µϕ( div z n 2 + ϕ( z n 2 (, x dx d. Propoiion 2.2 and (2.22 imply ha he econd member approache a n +. Uing he fac ha E zn ( = 1, we obain T 2ε a n + and hi give a conradicion. We recall now he following lemma due o [7] which i ueful o deermine he behavior of he energy. Lemma 2.7. Le T > and Γ L 1 loc (R + and non-negaive. Seing δ( = +T Γ(d, for. W ( be a non-negaive funcion for R +. Moreover we aume ha here exi a poiive, monoone, increaing funcion α wih α( 1, uch ha [ ] W ( α( W ( + Γ(σdσ, for every. Suppoe ha l and I l : R + R are increaing funcion wih l( = and W ((m + 1T + l{w (mt + δ(mt } W (mt + δ(mt, (2.24 for m =, 1, 2,..., where l( doe no depend on m. Then ( W ( α(t S( T + Γ(d, T, where S( i he non negaive oluion of he differenial equaion d + 1 l(s = Γ(; S( = W (. (2.25 T Moreover, we aume ha l i coninuou, ricly increaing and lim + l( = + If here exi C >, uch ha Γ(τdτ C, for every T. Then S( i bounded. If Γ(τdτ a +, and if S( admi a limi a infiniy, hen hi limi i zero. If Γ L 1 (R +, hen S( a +. We aume ha lim + (I l( = +, hen if Γ(τdτ + a +, we have S( + a +. We can now proceed he proof of he main reul of hi aricle.

11 EJDE-213/1 BEHAVIOR OF THE ENERGY 11 Proof of Theorem 1.4. We aume ha he aumpion (GC hold. Le u be a oluion of (1.1 wih iniial daa in he energy pace. Then according o Propoiion 2.5, we have ( +T +T E u ( C T h a(xg( u u dx dσ + f(, x 2 dx d, (2.26 for ome C T 1. The energy ideniy (1.6 give +T +T a(xg( u u dx dσ E u ( E u ( + T + f(σ, x u dx dσ. Le ψ be defined by ψ( = { 1 2T h 1 ( 2 8C T e T R +, + R. (2.27 I i clear ha ψ convex i and proper funcion. Hence, we can apply Young inequaliy [2] +T +T f(σ, x u dx dσ f(σ,. L 2 u(σ,. L 2dσ +T ψ ( f(σ,. L 2 + ψ( u(σ,. L 2dσ, where ψ i he convex conjugae of he funcion ψ, defined by ψ ( = up y R [y ψ(y] Uing he energy inequaliy (2.1 and he obervabiliy eimae (2.26, we infer ha +T ψ( u(σ,. L 2dσ 1 2 hen (2.27 give +T a(xg( u u dx dσ ( +T +T g( u udm a + f(, x 2 dx d ( +T +T 2 E u ( E u ( + T + f(, x 2 dx d + ψ ( f(σ,. L 2dσ. The inequaliy above combined wih he obervabiliy eimae (2.26 and he fac I h = I + m a ( T h m a( T i increaing, give ( ( E u ( C T h 4 E u ( E u ( + T + 2 Seing Therefore, E u ( + +T wih K 2C T. Seing θ( = +T +T Γ( = 2( f(σ,. 2 L 2 + ψ ( f(,. L 2. f(σ,. 2 L + 2 ψ ( f(σ,. L 2dσ. ( ( +T Γ(d Kh 4 E u ( E u ( + T + Γ( dx d, Γ(d. Thu E u ( + T h 1( 1 K (E u( + θ( E u ( + θ(, (2.28

12 12 A. BCHATNIA, M. DAOULATLI EJDE-213/1 for every. Take = m, m N, E u ((m + 1T h 1( 1 K (E u(mt + θ(mt Seing W ( = E u (, l( = 1 4 h 1 I K and Γ( = 2( f(,. 2 L 2 + ψ ( f(,. L 2. E u (mt + θ(mt. I i clear ha he funcion l and I l are increaing on he poiive axi and l( =. The funcion Γ L 1 loc (R + and non-negaive on R +. According o lemma 2.7, we obain ( E u ( 2e T S( T + Γ(d, T, where S( i he oluion of he following differenial equaion d + 1 l(s = Γ(, S( = W (. T The funcion l i coninuou, ricly increaing and lim + l( = +, herefore uing Lemma 2.7, we infer ha If here exi C >, uch ha Γ(τdτ C for every T. Then S( i bounded, which give E u ( i bounded. We aume ha E u ( α a + and Γ(τdτ a +. Conequenly (2.28 give ( E u ( + l E u ( T + Γ(τdτ E u ( T + Γ(τdτ, (2.29 for every T. Paing o he limi in he inequaliy above, we infer ha l(α =. Which mean α =. Therefore, if E u ( admi a limi a infiniy, hen he limi i zero. If Γ L 1 (R +, hen S( a +, which give E u ( a +. Since h 1 i linear a infiniy, herefore (I l i poiive and linear a infiniy, which give lim + (I l( = +. Thu, if Γ(τdτ + a +, we obain S( + a Applicaion Preliminary reul. In he following propoiion we give a reul on he behavior of he oluion of (1.1 due o [7]. Propoiion 3.1. Le p a differeniable, ricly increaing funcion on R + wih p( =. We aume ha here exi m 1 > uch ha, p(x m 1 x for every x [, η] for ome < η << 1 and ha he propery p(kx mp(kp(x, (3.1 hold, for ome m > and for every (K, x [1, + [ R +. We uppoe ha Γ C 1 (R + and non-negaive.

13 EJDE-213/1 BEHAVIOR OF THE ENERGY 13 (1 Le p be a increaing funcion vanihing a he origin. Le S aify he differenial equaion + p(s = Γ(, d S(. (3.2 Then S( for every. (2 Le S be a non-negaive funcion, aifying he differenial inequaliy + p(s Γ(, S(. d (a If Γ( =, for every, hen S( ψ 1 (, for every where ψ(x = S( d x p(, x ], S(]. (b If Γ( >, for every, and (i There exi c > and κ 1 uch ha d d p 1 (Γ( + cγ(, for every, (3.3 mp(κ κc 1, κp 1 Γ( S(, (3.4 hen S( κψ 1 (c for every, where ψ(x = p 1 Γ( x d p(, x ], p 1 Γ(]. Noing ha in hi cae we have p 1 Γ( ψ 1 (c, for every. d (ii There exi c > and κ 1 uch ha d p 1 (Γ( + cγ(, for every and mp(κ cκ 1, κp 1 Γ( S(, hen S( κp 1 Γ(, for every. Noing ha in hi cae we have p 1 Γ( ψ 1 (c for every, where ψ(x = p 1 Γ( x d p(, x ], p 1 Γ(]. Example. Seing ( Γ( = 2 f(,. 2 L 2 ( + ψ ( f(,. L2 (, where ψ i he convex conjugae of he funcion ψ, defined by { 1 ψ( = 2T h 1 ( 2 8C T R e T + + R, and ψ ( = up[y ψ(y]. To obain he rae of decay, we ue propoiion 3.1. y R g i i linearly bounded. We have h( = 2, hen ψ ( f(,. L 2 (M C 1 f(,. 2 L 2 (M, for ome C 1 >. The ODE (1.1 governing he energy bound reduce o + CS = Γ(, (3.5 d where he conan C > and doe no depend on E u (.

14 14 A. BCHATNIA, M. DAOULATLI EJDE-213/1 (1 If here are conan C > and θ R, uch ha Γ( C e θ. We have { e θ 1 d θ [e θ T 1]e θ θ T θ = for T. Muliply boh ide of (3.5 by exp(c and inegrae from o, o obain (a C > θ, E u ( c(1 + E u (e θ for, (b C = θ, E u ( c(1 + E u ((1 + e θ for, (c C < θ, E u ( c(1 + E u (e C for. (2 If here are conan C > and θ R, uch ha Γ( C (1 + θ, hen we have { (1 + θ T (1 + T θ θ > d T (1 + θ θ for T. Therefore, for T, where c >. E u ( { c(1 + E u ((1 + T θ θ > c(1 + E u (T (1 + θ θ (3.6 The nonlinear cae. The rae of decay of he energy depend only on he behavior of h 1 near zero. To deermine i, we have only o find < N 1, uch ha C 1 h 1 ( i h 1 ( for every N, 2C 2 where C 1 = min(m a ( T, 1 and C 2 = max(m a ( T, 1. (1 If Γ L 1 (R +, we chooe K max(c T, Eu(+ Γ L 1 (R + N. Equaion (1.1 governing he energy bound reduce o wih S( = E u (. (2 If Γ L 1 loc (R + and d + C 1h 1 ( S i Γ( 2KC 2 on [, + [, Γ(τdτ C for every T, hen S( i bounded and herefore here exi A > uch ha S( A, for every. We chooe K max(c T, A N. The ODE (1.1 governing he energy bound reduce o d + C 1h 1 S i ( Γ(, 2KC 2 wih S( = E u (. (3 If Γ L 1 loc (R + and Γ(τdτ +, +

15 EJDE-213/1 BEHAVIOR OF THE ENERGY 15 hen S( + a +. Therefore, here exi > uch ha S( K >> 1 for. Since he funcion h i ricly increaing and linear a infiniy, hen he ODE (1.1 governing he energy bound reduce o wih S( E u ( + Γ(d. d + C K S Γ( on [, + [, Example 1: Sublinear near he origin. Aume g i ( = r 1, < 1, r (, 1. We chooe h 1 i ( = g 1 i ( = 1+r 2r, for 1. We have ψ ( f(,. L 2 ( C ( f(,. r+1 L 2 ( + f(,. 2 L 2 (, for ome C >. The ODE (1.1 governing he energy bound reduce o d + CS(1+r/2r Γ(, where C i poiive and depend on K. (1 If here are conan C > and θ > uch ha Γ( C (1 + θ, hen (1 θ ], 1+r 1 r ] implie where c >. (2 θ 1+r 1 r implie E u ( c(1 + T 2r θ 1+r, T, E u ( c(1 + T 2r 1 r, > T, wih c > and depend on E u (. (2 If here are conan C > and θ >, uch ha Γ( C e θ, hen E u ( c( T + 1 2r 1 r, > T, where c i poiive and depend on E u (. Example 2: Differen behavior. Aume { 2 e 1/2 < 1 g 1 ( = 2 e 1/2 1 < < We chooe We have g 2 ( = r 1, < 1, r > 1 g 3 ( = r 1, < 1, r (, 1. h 1 1 ( = g 1 ( = 3/2 e 1/, < < η << 1, h 1 2 ( = g 2 ( = 1+r 2, η, h 1 3 ( = g 1 3 ( = 1+r 2r, η. ψ ( C( ln( 1/2 + r+1 r + r 1 r , for ome C > and >. The ODE (1.1 governing he energy bound reduce o d + CS3/2 e 1 CS Γ(,

16 16 A. BCHATNIA, M. DAOULATLI EJDE-213/1 where C i poiive and depend on K. If here are conan C > and θ >, uch ha Γ( C (1 + θ, hen E u ( c ln(c + c 1, T, wih c, c, c 1 >. Thee conan depend on E u (. Reference [1] F. Alabau, V. Komornik; Obervabilié, conrôlabilié e abiliaion fronière du yème d élaicié linéaire, C. R. Acad. Sci. Pari, Sér. I Mah., 324 (1997, 5l [2] F. Alabau and V. Komornik; Boundary obervabiliy, conrollabiliy and abilizaion of linear elaodynamic yem, Siam J on Conrol and Opimizaion 37, (1998, [3] F. Alabau; Convexiy and weighed inegral inequaliie for energy decay rae of nonlinear diipaive hyperbolic yem, Applied Mah and Op. 51, (25, [4] M. Bellaoued; Energy decay for he elaic wave equaion wih a local ime-dependan nonlinear damping, Aca Mah. Sinica, Englih Serie, 24 (7, ( [5] N. Burq and G. Lebeau; Meure de Dé fau de compacié, Applicaion au yème de Lamé, Ann. Scien. Ec. Norm. Sup. 4 érie,.34, ( [6] M. Daoulali; Rae of decay for he wave yem wih ime dependen damping. Dicree Conin. Dyn. Sy. 31, No. 2, ( [7] M. Daoulali; Behavior of he energy of oluion of he wave equaion wih damping and exernal force, Journal of Mahemaical Analyi and Applicaion, Volume 389, Iue 1, 1 May 212, [8] M. Daoulali, B. Dehman, M. Khenii; Local energy decay for he elaic yem wih nonlinear damping in an exerior domain. SIAM J. Conrol Opim., Vol. 48, No. 8, (21, [9] M. Daoulali, I. Laiecka and D. Toundykov; Uniform energy decay for a wave equaion wih parially uppored nonlinear boundary diipaion wihou growh rericion. Dicree Conin. Dyn. Sy., Ser. S 2, No.1, ( [1] B. Dehman and L. robbiano; La proprié é du prolongemen unique pour un yème ellipique. Le yème de Lamé, J. Mah. Pure Appl. (9, 72 (1993, [11] T. Duyckaer; Thèe de Docora, Univerié de Pari Sud (24. [12] E. Biognin, V. Biognin, R. Charão; Uniform abiliaion for elaic wave yem wih highly nonlinear localized diipaion, Porugaliae Mahemaica. 6: Iue 1, (23, [13] P. Gérard; Microlocal defec meaure, Com. Par. Diff. Eq. 16, (1991, [14] A. Guemia; On he decay eimae for elaiciy yem wih ome localized diipaion, Aympoic Analyi, 22 (2, [15] A. Guemia; Conribuion à la conrôlabilié exace e la abiliaion de yème d évoluion, Ph. D. Thei, Srabourg I Univeriy, France (2. [16] M. A. Horn; Sabilizaion of he dynamic yem of elaiciy by nonlinear boundary feedback, Inernaional Serie of Numerical Mahemaic, 133, Birkhäuer, Verlag, Bael/Swizerland (1999, [17] M. A. Horn; Nonlinear boundary abilizaion of a yem of anioropic elaiciy wih ligh inernal damping, Conemporary Mahemaic, 268 (2, [18] I. Laiecka, D. Taaru; Uniform boundary abilizaion of he emi-linear wave equaion wih non linear boundary diipaion, Diff. In. Equ. 6, (1993, [19] G. Lebeau and E. Zuazua; Decay rae for he hree-dimenional linear yem of hermoelaiciy, J. Arch. Raion. Mech. Anal. 148, No.3, (1999, [2] R. T. Rockafellar; Convex Analyi, Princeon Univeriy Pre, Princeon, NJ, 197. [21] K. Yamamoo; Singulariie of oluion o he boudary value problem for elaic and Maxwell equaion, Japan J. Mah., 14, (1988, [22] K. Yamamoo; Exponenial energy decay of oluion of elaic wave equaion wih he Dirichle condiion, Mah. Scand. 65, (1989,

17 EJDE-213/1 BEHAVIOR OF THE ENERGY 17 Ahmed Bchania Deparmen of Mahemaic, Faculy of Science of Tuni, Univeriy of Tuni El Manar, Campu Univeriaire El Manar 2, Tuni, Tuniia addre: ahmed.bchania@f.rnu.n Moez Daoulali Deparmen of Mahemaic, Faculy of Science of Bizere, Univeriy of Carhage, 721, Jarzouna, Bizere, Tuniia addre: moez.daoulali@infcom.rnu.n