Stabilization for the semilinear wave equation with geometric control condition

Description

Stabilization for the semilinear wave equation with geometric control condition Romain Joly? & Camille Laurent†‡ May 11, 2012 Abstract In this article, we prove the exponential stabilization of the semilinear wave equa- tion with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hormander. Some other consequences are also given for the controllability and the existence of a compact attractor. Key words: damped wave equation, stabilization, analyticity, unique continuation property, compact attractor. AMS subject classification: 35B40, 35B60, 35B65, 35L71, 93D20, 35B41. Resume Dans cet article, on prouve la decroissance exponentielle de l'equation des on- des semilineaires avec un amortissement actif dans une zone satisfaisant seulement la condition de controle geometrique. La nonlinearite est supposee sous-critique, defocalisante et analytique. La principale nouveaute par rapport aux resultats pre- cedents est la preuve d'un resultat de prolongement unique en grand temps pour une solution non amortie.

• wave equation


• topology generated


• nonlinear wave


• infinite time


• whitney topology


• time since


• equation


• decay can


• equation des ondes

Sujets

• Laurent
• Wave equation
• Whitney topologies
• Équation
• maths

Informations

• Publié par: pefav
• Nombre de lectures: 38
• Langue: English

Extrait

CgDEPTPzDgPbaIbegNHfHZPTPaHDejDiH HdhDgPba jPgN LHbZHgePF FbagebT FbaGPgPba

HdbS“c

fl 7Sb“aaWecubSJA

ESn ++” ,)+,

y

A´RaAcaa DˆfTieJdfiMUPffiPbdahPfTPPlbaˆPˆfiJUefJLiUizJfiaˆaRfTPePZiUiˆPJdiJhPPcgJ` fiaˆiifTJNJZbiˆSPPMfihPiˆJzaˆPeJfieRmiˆSfTPSPaZPfdiMMaˆfdaUMaˆNifiaˆ aˆUm´ ITP ˆaˆUiˆPJdifm ie JeegZPN fa LP egLMdifiMJUff NPRaMgeiˆS JˆN JˆJUmfiM´ ITP ZJiˆˆahPUfmMaZbJdPNfabdPhiagedPegUfeffiefTPbdaaRaRJgˆicgPMaˆfiˆgJfiaˆ dPegUf iˆ UJdSP fiZP Rad eaZP gˆNJZbPN PcgJfiaˆ´ ITP iNPJ ie fa geP Jˆ JemZbfafiM eZaafTiˆSPPMfbdahPNLmCJUPJˆNGJgSPUiˆfTPMaˆfPlfaRNmˆJZiMJUemefPZe´ ITPˆffaˆMPfTPJˆJUmfiMifmiˆfiZPiebdahPNffiPJbbUmJgˆicgPMaˆfiˆgJfiaˆdPegUf iifTbJdfiJUJˆJUmfiMifmNgPfaGaLLiJˆaffZgiUmffIJfJdgJˆNCadZJˆNPd´HaZPafTPd MaˆePcgPˆMPeJdPJUeaSihPˆRadfTPMaˆfdaUUJLiUifmJˆNfTPPliefPˆMPaRJMaZbJMf JffdJMfad´

:Sy owcMd:aJgifgˆicifmffˆfiˆgPMaffˆaifJziMifmUJˆJiaJfcgPPiUJLefˆffbPJZJhNiN ˆ bdabPdfmff MaZbJMf JffdJMfad´ SMAduKjSLeLlJddafaLJeaob:/160,ff /162,ff /1621ff /1E3-ff 5/8.,ff /160-´

e eRmu R 8JˆeMPfJdfiMUPffaˆbdaghPUJPNMdaieeJˆMPPlbaˆPˆfiPUUPNPPU'cgJfiaˆNPeaˆ` NPeePZiUiˆPJidPeJhPMgˆJZadfieePZPˆfJMfiRNJˆegPzaˆPeJfieRJieJˆfePgUPZPˆf ˆ UJMaˆNifiaˆNPMaˆfad^UPSPaZPfdicgP´EJˆaˆUiˆPJdiPfPefegbbaPePeage`MdificgPff PNRaMJUieJˆfPPfJˆJUmficgP´EJbdiˆMibJUPˆaghPJgPfbJddJbbadfJglPdegUfJfebPd` PMNPˆfePefUJbdPghPN'gˆPdegUfJfNPbdaUaˆSPZPˆfgˆicgPPˆSdJˆNfPZbebagd gˆPeaUgfiaˆˆaˆJZadfiP´Ei'PNPPefN'gfiUiePdgˆPPfPdSgUJdieJˆfJemZbfaficgP

 GFU`dPidga:fgfifˆeDPU-`ˆPLaPNdBfiPhPde’Uˆi7NGH114.0`P3/4:`.0,dff,,PNPgfJFe6`eT Hf`FJdfiˆ`N'CPdPeff:dJˆMPffPZJiU:[e-mail]-fjromai y 7G3UFHffNGPe`EJMcgEiaˆagieJEaL51ff4diJPJdafdieff1ffPJ31,,eff:`PMd:ˆJ z PZPffU:Jiiaˆeff:`3e`EagieEei:ffJdMˆ,11,PffdJffE5431FGffU,2ieJdPgcMJJPdiafJdaLJiˆPhFPU7U [e-mail]

+

bdagPhbJdCJUPPfGJgSPUNJˆeUPMaˆfPlfPNPeemePfZPeNmˆJZicgPe´9ˆegifPffgˆP RaieU'JˆJUmfiMiPfPˆfPZbebdaghPPffaˆJbbUicgPgˆfPTaPdZPNPbdaUaˆSPZPˆfgˆicgP JhPMJˆJUmfiMiPfbJdfiPUUPNg^JGaLLiJˆaffZgiUmffIJfJdgPfCadZJˆNPd´8PeJbbUiMJ` fiaˆeJUJMaˆfad^UJLiUiPfPfJU'PliefPˆMPN'JffdJMfPgdSUaLJUMaZbJMfbagdPU'cgJfiaˆ NPe aˆNPe eaˆf Jgeei N ˆˆPPe´ a 

) IagebGhFgPba

Ac i[“h Sgi“UaW” lW Udch“VWg i[W hWb“a“cWSg VSbeWV lSkW WfjSi“dc 8

< 2f´β`i)χ H f´αf´R`f) = ) `eσ i + ∙σ f`eσ i) = )eσ i)2R + ∙χ `+’+) ` : `fσ χ H f) = `f Ω σ f ( )2/ Ω( `) ∙∂ ) ` ) l[WgW2=χ H ) H  
l“i[
TW“cZ i[W DSeaSUW&6WaigSb“ deWgSidg l“i[ 8“g“U[aWi TdjcVSgn UdcV“i“dch’ J[W VdbS“c “h S UdccWUiWVC × i[gWW&V“bWch“dcSa H“WbScc“Sc bSc“YdaV l“i[ TdjcVSg“Wh” l[“U[ “h W“i[Wg4 “) UdbeSUi’ ““) S UdbeSUi eWgijgTSi“dc dYR 3 ” i[Si “hR 3 nDl[WgWD“h S TdjcVWV hbddi[ VdbS“c” WcVdlWV l“i[ S hbddi[ bWig“U WfjSa id i[W WjUa“VWSc dcW djih“VW dY S TSaa’ “““) dg S bSc“YdaV l“i[ eWg“dV“U ZWdbWign `Una“cVWg”R 3 l“i[ eWg“dV“U bWig“U WiU ) ’’ J[W cdca“cWSg“inR2 C ( `RσR) “h ShhjbWV id TW VWYdUjh“cZ” WcWgZn hjTUg“i“USa ScV hjU[ i[Si ) “h Sc Wfj“a“Tg“jb ed“ci’ EdgW egWU“hWan” lW ShhjbW i[Si i[WgW Wm“hiϕ >) hjU[ i[Si R`)) = )σ dR`d))σ∈R`d)∈ ∗ϕ`+ ´∈d∈) ∂ ScV∈R − `d)∈ ∗ϕ`+ ´∈d∈) ∂ ( `+’,)

l“i[ +∗a τ/’ WW ShhjbWα) id TW hjU[ i[Si
 α“h S cWZSi“kW VWΩc“iW deWgSidg” i[Si “h i[Si lW Pd“cUSWg “cWfjSa“in R ∈rf∈ ) ´α∈f∈ ) R ∈ ) Ac eSgi“UjaSg” “i b [SkW Sϕ∈fl“i[ϕ >)’ Sn gWfj“gWα >) “Yχ =;dg “Y “h jcTdjcVWV’ J[W VSbe“cZβ2∂ × WW“m[igW[iiSjhWbWWSh“dc’jcUi“kWYSchic&cdiSZW)`cSh“ deWc hWiπ ”2RScV) hjU[ i[Si

8i2 βπ σ`i) >  ) ScVnε i Ω σ )π ρ`+’-) ` EdgWdkWg” lW ShhjbW i[SiπhSi“hΩWh i[W ZWdbWig“U Udcigda UdcV“i“dc “cigdVjUWV “c O.-R ScV O1R

ff955)J[WgW Wm“hih∂ >) hjU[ i[Si Scn ZWcWgSa“oWV ZWdVWh“U dY dY aWcZi[ πl[WgW i[W VSbe“cZ “h WΔWUi“kW’

,

∂bWWih i[W hWi

J[W ShhdU“SiWV WcWgZn.2 C Ω `KσR + ) “h Z“kWc Tn Z Z .`f) 4=.`fσ χ H f) =  +  `∈χ H f∈ ) ´∈rf∈ ) ´α∈f∈ ) ) ´H`f)σ`+’.) , l[WgWH`f) = R u Ω R`d)Pd id 5hhjbei“dc `+’,) ScV i[W IdTdaWk WbTWVV“cZ’ 8jW/ ( ` ),! ∂ 6 Wg”“YVaΩWWcSVVcdbWgkd[“”icWhWn“gZWahl)`fhdakWh`+’+anSagbYd”WkS[Wl”)ihSWaiS” Z χ H .`f`e)= β`i)∈χ H f`iσ e)∈ ) Pi∗)ρ`+’/) J[W hnhiWb “h i[WgWYdgW V“hh“eSi“kW’ WW SgW “ciWgWhiWV “c i[W WmedcWci“Sa VWUSn dY i[W WcWgZndYi[Wcdca“cWSgVSbeWVlSkWWfjSi“dc`+’+)”i[Si“hi[WYdaadl“cZegdeWgin4 ff76):dg Scn. Ω )” i[WgW Wm“hi> >) ScVζ >) hjU[ i[Si” Ydg Saa hdaji“dchfdY `+’+) l“i[.`f`))∗. Ω ” 8e)σ .`f`e)∗>Q  H .`f`))

PgdeWgin `98) bWSch i[Si i[W VSbe“cZ iWgbβχ H fhiST“a“oWh Scn hdaji“dc dY `+’+) id oWgd” l[“U[ “h Sc “bedgiSci egdeWgin Ygdb i[W VncSb“USa ScV Udcigda ed“cih dY k“Wl’

Gjg bS“c i[WdgWb “h Sh Ydaadlh’ TNHbeHZ ).).1ffhˆR gUMg gUR PMˆcVaTβfMgVfRf`+’-)MaP gUR TRbˆRgeVO Obageb] Oba´ PVgVba ffi:33.) ISRfRVfMgPfMaVOVfeRM]MaM]lg`+’,)` gURa gUR RkcbaRagVM] PROMl cebcRegl ffi54) Ub]Pf. J[WdgWb +’+ Seea“Wh Ydg cdca“cWSg“i“WhR UdjghW” i[W GYl[“U[ SgW ZadTSaan ScSani“U’ cdca“cWSg“i“WhR`f) =∈f∈ ∂ ( fSgW cdi ScSani“U “Ya◦2 ⊂+σ-→” Tji lW USc gWeaSUW i[WhW jhjSa cdca“cWSg“i“Wh Tn h“b“aSg dcWh ShR`f) = `fφi[`f) ∂ ( f” l[“U[ SgW ScSani“U Ydg Saaa2O+σ/)’ FdiW i[Si i[W Whi“bSiWh `+’,) SgW dcan gWfj“gWV Ydgd2Ri[Si “i VdWh cdi “bean i[Si” hd R“h edancdb“Sa’ EdgWdkWg” lW Wc[ScUW i[Si `98) [daVh “c YSUi Ydg Sabdhi Saa i[W cdca“cWSg“i“Wh RhSi“hYn“cZ `+’,)” “cUajV“cZ cdc&ScSani“U dcWh’ EdgW egWU“hWan” lW hWi C ( `R) =⊂R2 CR) hjU[ i[Si i[WgW Wm“hiϕ >) ScVa2O+σ/) hjU[ i[Si `+’,) [daVh→ ( ` `+’0) WcVdlWVl“i[W[“icWnidedadZn`dgScndi[WggWShdcSTaWidedadZn)’WWgWUSaai[SiW[“i& cWn idedadZn “h i[W idedadZn ZWcWgSiWV Tn i[W cW“Z[Tdjg[ddVh ≤ ;ε =⊂S2C ( `R)∈ 8f2RσbSm`∈R`f) S`f)∈σ∈R − `f) S − `f)∈)τ `f)→`+’1) l[WgWR“h Scn YjcUi“dc “cC ( `R) ScV“h Scn edh“i“kW Udci“cjdjh YjcUi“dc’ hWi J[WC ( `R) “h S 6S“gW heSUW” l[“U[ bWSch i[Si Scn ZWcWg“U hWi” i[Si “h Scn hWi UdciS“c“cZ S UdjciSTaW “ciWghWUi“dc dY deWc ScV VWchW hWih” “h VWchW “cC ( `R) `hWW Pgdedh“i“dc 1’+)’ egdeWgin 6S“gW WchjgWh i[Si i[W ZWcWg“U“in dY S hWi “cC ( `RS ZddV cdi“dc Ydg Pi[W hWi UdciS“ch Sabdhi) “h Saa cdc&a“cWSg“inR"’

-