their uniform discretizations Nonlinear damped partial differential equations and Journal of Functional Analysis

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Nonlinear damped partial differential equations and their uniform discretizations

Fatiha Alabau-Boussouira^a,1, Yannick Privat^b,∗, Emmanuel Trélat^c

a UniversitédeLorraine,IECL,UMR7502,57045MetzCedex1, France

b CNRS,SorbonneUniversités,UPMCUnivParis06,UMR7598, Laboratoire Jacques-LouisLions,F-75005,Paris,France

cSorbonneUniversités,UPMCUnivParis06,CNRSUMR7598,

Laboratoire Jacques-LouisLions,andInstitutUniversitairedeFrance,F-75005, Paris,France

a r t i c l e i n f o a bs t r a c t

Article history:

Received30June2016 Accepted6March2017 Availableonline28March2017 CommunicatedbyF.Béthuel

MSC:

37L15 93D15 35B35 65N22

Keywords:

Stabilization Dissipativesystems Space/timediscretization Optimalweightconvexitymethod

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretizationschemes thatinheritofthesameenergydecay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosityterms.Ourmainargumentsusetheoptimal-weight convexitymethodanduniformobservabilityinequalitieswith respect to the discretization parameters. We establish our results, firstinthe continuoussetting, thenfor spacesemi- discretemodels,andthenfortimesemi-discretemodels.The full discretizationistheninferredfromtheprevious results, byadaptingtheideastodealwithlinearsystems.

Our results cover, for instance, the Schrödinger equation with nonlinear damping, the nonlinear wave equation, the

* Correspondingauthor.

E-mailaddresses:fatiha.alabau@univ-lorraine.fr(F. Alabau-Boussouira),yannick.privat@upmc.fr (Y. Privat),emmanuel.trelat@upmc.fr(E. Trélat).

1 EndélégationCNRSauLaboratoireJacques-LouisLions,UMR7598.

http://dx.doi.org/10.1016/j.jfa.2017.03.010 0022-1236/©2017ElsevierInc.Allrightsreserved.

nonlinearplate equation, thenonlinear transport equation, aswellascertainclassesofequationswithnonlocalterms.

©2017ElsevierInc.Allrightsreserved.

Contents

1. Introduction . . . . 353

2. Continuoussetting . . . . 354

2.1. Mainresult . . . . 354

2.2. Examples . . . . 359

2.2.1. Schrödingerequationwithnonlineardamping . . . . 359

2.2.2. Waveequationwithnonlineardamping . . . . 360

2.2.3. Plateequationwithnonlineardamping . . . . 361

2.2.4. Transportequationwithnonlineardamping . . . . 362

2.2.5. Dissipativeequationswithnonlocalterms . . . . 362

2.3. ProofofTheorem 1 . . . . 364

3. Discretizationissues:uniformdecayresults . . . . 372

3.1. Semi-discretizationinspace . . . . 372

3.1.1. Mainresult . . . . 373

3.1.2. ProofofTheorem 2 . . . . 378

3.2. Semi-discretizationintime . . . . 384

3.2.1. Mainresult . . . . 385

3.2.2. ProofofTheorem 3 . . . . 386

3.3. Fulldiscretization . . . . 396

4. Conclusionandperspectives . . . . 398

Acknowledgments . . . . 401

References . . . . 401

1. Introduction

LetX be aHilbertspace. Throughout thepaper, we denote by · X the norm on X and by ·,·X the corresponding scalar product. Let A : D(A) → X be a densely definedskew-adjoint operator,and letB :X →X be anontrivial bounded selfadjoint nonnegativeoperator.LetF :X →Xbea(nonlinear)mapping,assumedtobeLipschitz continuousonboundedsubsetsofX.Weconsiderthedifferentialsystem

u(t) +Au(t) +BF(u(t)) = 0. (1)

If F = 0 then the system (1) is conservative, and for every u₀ ∈ D(A), there exists auniquesolution u(·)∈C⁰(0,+∞;D(A))∩C¹(0,+∞;X) such that u(0) =u₀, which satisfiesmoreoveru(t)X =u(0)X,foreveryt≥0.

IfF = 0 then the system(1) isexpected to be dissipativeifthe nonlinearity F has

“thegoodsign”.Definingtheenergyofasolutionuof(1)by E_u(t) = 1

2u(t)²X, (2)

wehave,aslongasthesolutioniswelldefined,

E_u(t) =−u(t), BF(u(t))X. (3) In the sequel, we will make appropriate assumptions on B and on F ensuring that E_u(t)≤0.Itisthenexpectedthatthesolutionsaregloballywelldefinedandthattheir energydecaysasymptoticallyto0 ast→+∞.

Theobjective ofthispaperistwofold.

Firstofall,inSection2weprovide adequateassumptionsunderwhichthesolutions of (1) have their norm decayingasymptotically to 0 in aquasi-optimal way. This first result, settled in an abstract continuous setting, extends former results of [7] (estab- lishedfordampedwaveequations)tomoregeneralequationsanddampingoperatorsfor stabilization issuesbasedonindirectarguments(fordirectargumentsseee.g.[4,6]).

Then, inSection3, weinvestigatediscretizationissues, withtheobjectiveofproving that, for appropriate discretization schemes, the discrete approximate solutions have a uniform decay. In Section 3.1, we first consider spatial semi-discrete approximation schemes,andinSection3.2wedealwithtimesemi-discretizations.Thefulldiscretization is doneinSection3.3.Inallcases,we establishuniformasymptotic decaywithrespect to themeshsize,byadding adequateviscositytermsintheapproximationschemes.

2. Continuoussetting

2.1. Mainresult

Assumptions andnotations.Firstofall,weassumethat

u, BF(u)X ≥0, (4)

for everyu∈X. Using(3), thisfirst assumptionensuresthattheenergyE_u(t) defined by(2)is nonincreasing.

SinceB isbounded,nonnegativeandselfadjointonX,itfollowsfromthewell-known spectral theoremthatB isunitarilyequivalenttoamultiplication(see,e.g.,[26]).More precisely, thereexist aprobabilityspaceΩ withmeasure μ,areal-valuedboundednon- negative measurable function b defined on Ω (satisfying bL^∞(Ω,μ) = B), and an isometry U fromL²(Ω,μ) intoX, suchthat

(U⁻¹BU f)(x) =b(x)f(x),

foralmost everyx∈Ω andforeveryf ∈L²(Ω,μ).Anotherusual wayofwritingB is

B=

+∞

0

λ dE(λ),

wherethefamilyofE(λ) isthefamilyofspectralprojectionsassociatedwithB.Werecall thatthespectralprojectionsareobtainedasfollows. Definingtheorthogonalprojection

operatorQλ onL²(Ω,μ) by(Qλf)(x)=χ_{b(x)≤λ}(x)f(x), for every f ∈ L²(Ω,μ) and foreveryx∈Ω,wehaveE(λ)=U QλU⁻¹.

Wedefinethe(nonlinear)mappingρ:L²(Ω,μ)→L²(Ω,μ) by ρ(f) =U⁻¹F(U f),

foreveryf ∈L²(Ω,μ).Inotherwords,themappingρisequaltothemappingF viewed throughtheisometry U.

Note that,setting f =U⁻¹u, the equation (1) is equivalent to ∂tf + ¯Af+bρ(f)= 0, with A¯ = U⁻¹AU, densely defined skew-adjoint operator on L²(Ω,μ), of domain U⁻¹D(A).

Weassumethatρ(0)= 0 andthat

f ρ(f)≥0, (5)

foreveryf ∈L²(Ω,μ).Following [4,6,7], weassumethatthere exist c₁ >0 and c₂ >0 suchthat,foreveryf ∈L^∞(Ω,μ),

c₁g(|f(x)|)≤ |ρ(f)(x)| ≤c₂g⁻¹(|f(x)|) for almost everyx∈Ω such that|f(x)| ≤1, c1|f(x)| ≤ |ρ(f)(x)| ≤c2|f(x)| for almost everyx∈Ω such that|f(x)| ≥1, (6) whereg isanincreasingoddfunctionofclass C¹ suchthat

g(0) =g(0) = 0, sg(s)²

g(s) −−−→_s→0 0 andsuchthatthefunctionH definedbyH(s)=√

sg(√

s),foreverys∈[0,1],isstrictly convexon[0,s²₀] forsomes₀∈(0,1] (chosensuchthatg(s₀)<1).

WedefinethefunctionHonRbyH(s)=H(s) foreverys∈[0,s²₀] andbyH(s) = +∞ otherwise. Using the convex conjugatefunction H^∗ of H, we define the function L on [0,+∞) byL(0)= 0 and,forr >0,by

L(r) =H^∗(r) r =1

rsup

s∈R

rs−H(s)

. (7)

By construction, the function L : [0,+∞) → [0,s²₀) is continuous and increasing. We define the function ΛH : (0,s²₀] → (0,+∞) by ΛH(s) = H(s)/sH(s), and for s ≥ 1/H(s²₀) weset

ψ(s) = 1 H(s²₀)+

H(s²₀) 1/s

1

v²(1−ΛH((H)⁻¹(v)))dv. (8) Thefunctionψ: [1/H(s²₀),+∞)→[0,+∞) iscontinuousand increasing.

Throughout the paper, we use the notations and inmany estimates, with the following meaning. Let S be aset, and let F and G be nonnegative functions defined on R×Ω× S. The notation F G (equivalently, G F) means that there exists a constant C > 0, only depending on the function g or on the mapping ρ, such that F(t,x,λ)≤CG(t,x,λ) forall (t,x,λ)∈R×Ω× S.The notationF₁F₂ meansthat F1F2 andF1F2.

In the sequel, we choose S = X, or equivalently, using the isometry U, we choose S =L²(Ω,μ),sothatthenotationdesignatesanestimateinwhichtheconstantdoes notdependonu∈X,or onf ∈L²(Ω,μ),butdependsonlyonthemappingρ.Wewill use these notations to provide estimateson thesolutionsu(·) of (1), meaningthat the constants intheestimatesdonotdependonthesolutions.

Forinstance,theinequalities(6)canbe writtenas

g(|f|)|ρ(f)|g⁻¹(|f|) on the set |f|1,

|ρ(f)| |f| on the set |f|1.

Themain resultofthissectionisthefollowing.

Theorem1.In additiontotheabove assumptions,weassumethatthere existT >0and C_T >0suchthat

CTφ(0)²X ≤ T

0

B^1/2φ(t)²Xdt, (9)

foreverysolutionofφ(t)+Aφ(t)= 0(observabilityinequalityforthelinearconservative equation).

Then, for every u0 ∈ X, there exists a unique solution u(·) ∈ C⁰(0,+∞;X)∩ C¹(0,+∞;D(A)) of (1) such that u(0) = u0.² Moreover, the energy of any solution satisfies

E_u(t)Tmax(γ₁, E_u(0))L 1

ψ⁻¹(γ₂t)

, (10)

forevery timet≥0,with γ1 B/γ2 andγ2CT/(T³B^1/24+T).If moreover lim sup

s0 ΛH(s)<1, (11)

then wehave thesimplifieddecay rate

2 Here, thesolutionis understoodin theweaksense,see[13,17], andD(A) is thedual ofD(A) with respecttothepivotspaceX.Ifu0∈D(A),thenu(·)∈C⁰(0,+∞;D(A))∩C¹(0,+∞;X).

Eu(t)Tmax(γ1, Eu(0)) (H)⁻¹ γ₃

t

, foreverytime t>0,forsome positiveconstant γ31.

Theorem 1improvesandgeneralizestoawideclassofequationsthemainresultof[7]

inwhichthe authors dealtwithlocally damped waveequations. Thecase ofboundary dampings is also treated in [6] (see also [4]) by a direct method, which provides the sameenergydecayrates.TheresultgivesthesharpandgeneraldecayrateL(1/ψ⁻¹(t)) oftheenergyat infinity(forgettingabouttheconstants), andthesimplifieddecayrate (H)⁻¹(1/t) underthecondition(11).Itisalso provedin[6]that,underthiscondition, theresultingdecayrateisoptimalinthefinitedimensionalcase,andforsemi-discretized nonlinearwaveorplateequations.Henceourestimatesaresharpandtheyareexpected to be optimal in the infinite dimensional case. The proof of optimality relies on the derivation of a one-step decay formula for general damping nonlinearity, on a lower estimatebasedonanenergycomparisonprinciple,andonacomparisonlemmabetween time pointwise estimates (such our upper estimate) and lower estimates which are of energytype.Notealsothatρhasalineargrowthwheng(0)= 0.Inthiscasetheenergy decaysexponentiallyatinfinity.Moreover,eveninthefinitedimensionalcase,optimality cannot be expectedwhen lim sup

s0 ΛH(s)= 1 (functionsρ leadingto thatcondition are closeto alineargrowth in aneighborhood of 0),and it ispossible to designexamples (linearfeedbackcase)withtwobranchesofsolutionsthatdecayexponentiallyatinfinity butdonothavethesameasymptoticbehavior.

Let us recall some previouswell-known results of the literature. The first examples of nonlinear feedbacks were only concerning feedback functions having a polynomial growthinaneighborhoodof0 (see e.g.[24,39,27,47]and thereferencestherein).Asfar as we know, the first paper considering the case of arbitrary growing feedbacks (in a neighborhoodof 0)is [29].Inthis paper, theanalysisis basedontheexistence(always true)ofaconcave function hsatisfyingh(sρ(s))s²+ρ²(s) for all|s| N (see (1.3) in[29]). Thepaper is veryinteresting but provides only two examples of construction ofsuchfunctionhinCorollary 2,namely thelinearand polynomialgrowing feedbacks.

The results use only the Jensen’s inequality (not the Young’s inequality), and allow the authors to compare the decay of the energy with the decay of the solution of an ordinary differential equation S(t)+q(S(t)) = 0 where q(x) = x−(I+p)⁻¹(x) and p(x)= (cI+h(Cx))⁻¹(Kx) wherec,C,Karenon-explicit constantsandf⁻¹standsfor theinversefunctionoff.Inthegeneralcase,these resultsdonotgivethewaystobuild anexplicit concave function satisfying h(sρ(s))s²+ρ²(s). No generalenergy decay ratesare givenin anexplicit,simple and generalformula, whichbesides this, couldbe showntobe“optimal”.Due tothislackofexplicitexamplesofdecayratesforarbitrary growingfeedbacks inothersituationsthanthelinearor polynomialcases,otherresults were obtained, also based on convexity arguments but through other constructions in [35,36](seealso[40])throughlinearenergyintegralinequalitiesandin[33]throughthe comparisonwith adissipativeordinarydifferential inequality.In bothcases,optimality

Table 1 Examples.

g(s) ΛH(s) decay ofE(t)

s/ln^−p(1/s),p >0 lim sup

x0 ΛH(s) = 1 e^−t1/(p+1)/t^1/(p+1) s^pon [0, s²₀],p >1 ΛH(s)≡p+1² <1 t^−2/(p−1)

e^−1/s2 lim

s0ΛH(s) = 0 1/ln(t)

s^pln^q(1/s),p >1,q >0 lim

s0ΛH(s) =_p+1² <1 t^−2/(p−1)ln^{−2q/(p−1)}(t)

e^−lnp(1/s),p >2 lim

s0ΛH(s) = 0 e^{−2 ln1/p(t)}

isnotguaranteed.Inparticular,[35,36]donotallowtorecoverthewell-knownexpected

“optimal”energydecayratesinthecaseofpolynomiallygrowingfeedbacks.Optimality canbe showninparticulargeometrical situations,inonedimension whenthefeedback is very weak (as for ρ(s) = e^−1/s for s > 0 close to 0 for instance), see e.g. [45,4].

Hencethe challengingquestionsarenotonly toderive energydecayratesfor arbitrary growing feedbacks, butto determine whetherif these decayrates are optimal, at least infinite dimensionsandinsomesituationsintheinfinite dimensionalcase, andalsoto derive one-step, simple and semi-explicit formula which are valid in the general case.

This is the main contribution of [4,6] for directmethods and of the present paper for indirect methods for the continuous as well as the discretized settings (see also [7]for thecontinuoussetting).Notealsothatthedirectmethodisvalidforboundedaswellas unboundedfeedbackoperators.

Severalexamplesoffunctionsg(see(6))aregiveninTable 1,withthecorresponding Λ_H and decay rates. Note that other examples canbe easily built, sincethe optimal- weightconvexitymethodgivesageneralandsomehowsimplewaytoderivequasi-optimal energydecayrates.

In the four last examples listed inTable 1, theresulting decay ratesare optimal in finite dimension and forthe semi-discretizedwaveand plateequations. Moreover, (11) is satisfied.

Remark1.Italsoisnaturaltowonderwhetherthelinearfeedbackcase,thatisg(s)=s foreveryscloseto0,aswellasthenonlinearfeedbackcaseswhichhavealineargrowth close to 0, such as the example g(s) = arctans for s close to 0, are covered by our approach in a“natural continuous” way. These cases canbe treatedunder a common assumption,whichisindeedg(0)= 0.Noticethatanapparentdifficultyliesinthefact thatthefunctionHistheidentityfunctionandisnotstrictlyconvexanymore,sothatour constructionmayseemtofail.Thislimitcaseisinvestigatedinthefollowingresult,whose proofispostponedattheendofSection2.3.Itisobtainedunderslightmodificationsin theproofofTheorem 1.Thisapproachisalsovalidforthedirectapproachpresentedin [4,6],whichleadstocontinuousnonlinearintegralinequalities.Wewillalsoformulatea generalresultinthisdirectionbelow.

Corollary 1.Let us assume that g(0) = 0. Under the same assumptions as those of Theorem 1(namely,theonesatthebeginningofSection2.1aswellastheobservability inequality(9)),thereholds lim sup_s0ΛH(s)= 1 and

E_u(t)Tmax(γ₁, E_u(0)) exp (−γ₂t),

foreverytime t>0withγ1 B/γ2 and γ2CT/(T³B^1/24+T).

Letus now apply this generalizationto the directoptimal-weightconvexitymethod as introduced in [4,6]. In all the results presented in these two papers, and when the bounded as wellas unboundedfeedback operatorρsatisfies (6) withafunction g such thatg(0) = 0= g(0) we can extendthe given proofs to thecases for which g(0) = 0 whereas g(0) = 0 (i.e. when g has a linear growth close to 0). More precisely, we can extendthe proof of Theorems 4.8, 4.9, 4.10and 4.11 in[6] (but also inthe more general framework as presented in Theorem 4.1) to the case for which g(0) = 0. For this, it is sufficient as for the proof of Corollary 1 to replace g by the sequence of functionsgε definedby gε(s) =s^1+ε for every |s| 1,where ε ∈(0,1). Onecan then apply the optimal-weight convexity method to this sequence, and define the associate optimal-weightfunctionwε(·)=L⁻¹( .

2β) inasuitableinterval(seetheabovereferences formoredetails).Wethenprovethat

T S

w_ε(E(t))E(t)dtM E(S) ∀ 0S T,

where M, β can be chosen independently on ε. We then let ε goes to 0. Thanks to theproofofCorollary 1,we knowthatthesequencewεconvergespointwise on (0,βr²₀) towards1.This leadsto theinequality

T S

E(t)dtM E(S) ∀0S T,

fromwhichwededucethatE decaysexponentiallyatinfinity(seee.g.[27]).

Wenextprovide sometypical examplesofsituationscoveredbyTheorem 1.

2.2. Examples

2.2.1. Schrödinger equation withnonlineardamping

Ourfirsttypical exampleis theSchrödingerequation withnonlinear damping(non- linearabsorption)

i∂tu(t, x) +u(t, x) +ib(x)u(t, x)ρ(x,|u(t, x)|) = 0,

in a Lipschitz bounded subset Ω of Rⁿ, with Dirichlet boundary conditions. In that case, we have X = L²(Ω,C), and the operatorA = −i is the Schrödinger operator defined onD(A)=H₀¹(Ω,C). TheoperatorB isdefinedby (Bu)(x)=b(x)u(x) where b ∈ L^∞(Ω,R) is a nontrivial nonnegative function, and the mapping F is defined by (F(u))(x) = u(x)ρ(x,|u(x)|), where ρ is a real-valued continuous function defined on Ω¯×[0,+∞) suchthatρ(·,0)= 0 onΩ,ρ(x,s)≥0 onΩ¯×[0,+∞),andsuchthatthere exist afunction g ∈C¹([−1,1],R) satisfying all assumptions listed inSection2.1, and constants c1>0 andc2>0 suchthat

c1g(s)≤sρ(x, s)≤c2g⁻¹(s) if 0≤s≤1, c1s≤ρ(x, s)≤c2s ifs≥1,

for everyx∈Ω.Here,the energyof asolutionuis givenby Eu(t)= ¹₂

Ω|u(t,x)|²dx, and we have E_u(t) = −

Ωb(x)|u(t,x)|²ρ(x,|u(t,x)|)dx ≤ 0. Note that, in nonlinear optics,theenergyE_u(t) iscalledthepower of u.

As concerns the observabilityassumption (9),it is well knownthat, ifb(·)≥α >0 onsomeopensubsetω ofΩ,andifthereexists T suchthatthepair(ω,T) satisfiesthe Geometric Control Condition,then thereexistsCT >0 suchthat

CTφ(0,·)²L²(Ω,C)≤ T 0

Ω

b(x)|φ(t, x)|²dxdt,

for everysolution φ of the linear conservative equation ∂tφ−iφ = 0 withDirichlet boundaryconditions(see[31]).

2.2.2. Wave equationwith nonlineardamping

Weconsider thewaveequation withnonlineardamping

∂ttu(t, x)− u(t, x) +b(x)ρ(x, ∂tu(t, x)) = 0,

inaC²boundedsubsetΩ ofRⁿ,withDirichletboundaryconditions.Thisequationcan be writtenas afirst-order equationoftheform(1), withX=H₀¹(Ω)×L²(Ω) and

A=

0 −id

− 0

definedonD(A)=H₀¹(Ω)∩H²(Ω)×H₀¹(Ω).TheoperatorBisdefinedby(B(u,v))(x)= (0,b(x)v(x))whereb∈L^∞(Ω) isanontrivialnonnegativefunction,andthemappingF isdefinedby(F(u,v))(x)= (0,ρ(x,v(x))),whereρisareal-valuedcontinuousfunction definedonΩ¯×Rsuchthatρ(·,0)= 0 onΩ,sρ(x,s)≥0 onΩ¯×R,andsuchthatthere exist afunction g ∈C¹([−1,1],R) satisfying all assumptions listed inSection2.1, and constants c1>0 andc2>0 suchthat

c₁g(|s|)≤ |ρ(x, s)| ≤c₂g⁻¹(|s|) if|s| ≤1, c1|s| ≤ |ρ(x, s)| ≤c2|s| ifs≥1,

(12)

foreveryx∈Ω.Here,theenergyofasolutionuisgivenbyE_u(t)=¹₂

Ω (|∇u(t,x)|²+ (∂tu(t,x))²

dx,andwehaveE_u(t)=−

Ωb(x)∂tu(t,x)ρ(x,∂tu(t,x))dx≤0.

Theframeworkof thisexampleistheoneof[7].

As concernsthe observability assumption (9), itis well known that,if b(·)≥α >0 onsomeopensubsetω ofΩ,andifthere existsT suchthatthepair(ω,T) satisfiesthe Geometric ControlCondition,thenthereexists CT >0 suchthat

C_T(φ(0,·), ∂_tφ(0,·))²_H0¹_(Ω)×L²_(Ω)≤ T 0

Ω

b(x)(∂_tφ(t, x))²dxdt,

for every solution φ of the linear conservative equation ∂ttφ− φ = 0 with Dirichlet boundaryconditions(see[12]).

2.2.3. Plateequation with nonlineardamping Weconsiderthenonlinearplateequation

∂ttu(t, x) +²u(t, x) +b(x)ρ(x, ∂tu(t, x)) = 0,

inaC⁴boundedsubsetΩ ofRⁿ,withDirichletandNeumannboundaryconditions.This equationcanbewrittenasafirst-orderequationoftheform(1),withX =H₀²(Ω)×L²(Ω) and

A=

0 −id ² 0

defined on D(A) = H₀²(Ω)∩H⁴(Ω)

× H₀²(Ω). The operator B is defined by (B(u,v))(x) = (0,b(x)v(x)) where b ∈ L^∞(Ω) is a nontrivial nonnegative function, andthemappingFisdefinedby(F(u,v))(x)= (0,ρ(x,v(x))),whereρisareal-valued continuousfunction definedonΩ¯×[0,+∞) suchthatρ(·,0)= 0 onΩ, sρ(x,s)≥0 on Ω¯×R,andsuchthatthereexist afunctiong∈C¹([−1,1],R) satisfyingallassumptions listed inSection2.1, and constants c1 >0 andc2 >0 such that (12)holds. Here, the energy of asolution uis given by Eu(t) = ¹₂

Ω (u(t, x))²+ (∂tu(t, x))²

dx, and we haveE_u(t)=−

Ωb(x)∂_tu(t,x)ρ(x,∂_tu(t,x))dx≤0.

Theframeworkof thisexampleistheoneof[5].

A sufficientcondition obtainedin[31], ensuringthe observability assumption (9), is thefollowing:ifb(·)≥α >0 onsomeopensubsetω of Ω forwhichthereexists T such thatthepair(ω,T) satisfiestheGeometricControlCondition,thenthereexistsC_T >0 suchthat

C_T(φ(0,·), ∂_tφ(0,·))²H₀²(Ω)×L²(Ω)≤ T 0

Ω

b(x)(∂_tφ(t, x))²dxdt,

foreverysolutionφofthelinearconservativeequation∂_ttφ+²φ= 0 associatedtothe corresponding boundaryconditions.

2.2.4. Transport equationwith nonlineardamping Weconsider theone-dimensionaltransportequation

∂tu(t, x) +∂xu(t, x) +b(x)ρ(x, u(t, x)) = 0, x∈(0,1),

with periodicity conditions u(t,0) = u(t,1). This equation can be written as a first-order equation of the form (1), with X = L²(0,1) and A = ∂_x defined on D(A) = {u ∈ H¹(0,1) | u(0) = u(1)}. We make on ρ the same assumptions as before. The energy is given by E_u(t) = ¹₂1

0 u(t,x)²dx, and we have E_u(t) =

−

Ωb(x)u(t,x)ρ(x,u(t,x))dx ≤ 0. The observability inequality for the conservative equation issatisfiedas soonastheobservabilitytimeischosenlargeenough.

Onthis example,wenotetwothings.

Firstof all,theaboveexamplecanbeeasily extendedinmulti-Donthetorus Tⁿ = Rⁿ/Zⁿ,byconsideringthefollowing non-lineartransportequation

∂_tu(t, x) + div(v(x)u(t, x)) +b(x)ρ(x, u(t, x)) = 0, t >0, x∈Tⁿ,

wherevisaregularvectorfieldonTⁿsuchthatdiv(v)= 0.Thedivergence-freecondition onthefunctionv ensuresthattheoperatorA definedbyAz= div(v(x)z(x)) on

D(A) ={z∈H¹(Tⁿ) | z(·+ei) =z(·), ∀i∈J1, nK}, where e_i denotesthei-thvectorofthecanonicalbasisofRⁿ,isskew-adjoint.

Second,in1Dwecandroptheassumptionofzerodivergence,byusingasimplechange of variable,whichgoesasfollows. Weconsidertheequation

∂tu(t, x) +v(x)∂xu(t, x) +bρ(x, u(t, x)) = 0, t >0, x∈(0,1),

with v ameasurable functionon (0,1) suchthat0< v₋ ≤v(x)≤v+.Then, using the changeof variablex→x

0 ds

v(s), weimmediately reducethis equationto thecasewhere v= 1.

Henceourresultscanaswell beappliedtothosecases.

2.2.5. Dissipativeequationswith nonlocalterms

Inthethreepreviousexamples,thetermb(·)ρ(·,·) isaviscousdampingwhichislocal.

In other words, thevalue at x of the functionF(u) does only depend on thevalue at x of the function u. To illustrate the potential of our approach and the large family

of nonlinearities thatit covers, we slightly modify herethe examples presented in the sections2.2.1, 2.2.2,2.2.3 and2.2.4, byproviding severalexamples ofviscous damping termscontaininganon-localterm.

Werefer tothe previoussections for theprecisionson theboundaryconditions and thefunctionalsettingassociated toeachsystem.

Weconsiderthenon-linearsystems

i∂_tu(t, x) +u(t, x) +ib(x)u(t, x)ρ(|u|)(t, x) = 0

∂_ttu(t, x)− u(t, x) +b(x)ρ(∂_tu)(t, x) = 0

∂_ttu(t, x) +²u(t, x) +b(x)ρ(∂_tu)(t, x) = 0

∂_tu(t, x) +∂_xu(t, x) +b(x)ρ(u)(t, x) = 0, wherethenon-lineartermρisdefinedby

ρ(f)(x) =ϕ(f(x),N(f)),

where ϕ:R² →Ris acontinuousfunction and N :L²(Ω) →Rstands foranon-local term. We cantypically chooseN(f)=

Ωχ(x)f(x)dx with χ ∈L²(Ω) whenever Ω is boundedorχsmoothwithcompactsupportelse.IntheframeworkofSection2.2.4,one isalsoallowedtochooseN(f)=K f withK∈L²(Tⁿ).Wealsoimposethatϕsatisfies thefollowinguniformLipschitzproperty:thereexists C >0 such that

|ϕ(s, τ)−ϕ(s, τ)|+|ϕ(s, τ)−ϕ(s, τ)| ≤C(|s−s|+|s|.|τ−τ|)

for every (s,s,τ,τ) ∈ R⁴. As aconsequence, oneeasily infers that the mapping ρ is locallyLipschitzfrom L²(Ω) intoL²(Ω).

Moreover, we choose the function ϕ odd with respect to its first variable and such thatϕ(s,τ)≥0 foreverys≥0 andτ∈R.Itfollows thattheassumptionf ρ(f)≥0 is satisfiedbyeveryf ∈L²(Ω).

Finally,weassumethattheassumption(6)is satisfied.

Letusprovideanexampleofsuchafunctionϕ.Noticethat,ifthereexisttwopositive constantsk₁ andk₂ suchthatforeveryτ ∈R,there exist twopositiverealnumbersc_τ andCτ in[k1,k2] suchthat

ϕ(s, τ)∼cτs³ ass→0 and ϕ(s, τ)∼Cτs as s→+∞, thentheassumption(6)issatisfied.Apossible functionϕis givenby

ϕ(s, τ) =ϕ₁(s)ϕ₂(τ) where ϕ₁:Rs→s−sins

and ϕ₂ : R → R denotes any function bounded above and below by some positive constants,forinstance ϕ2(τ)=π+ arctan(τ).

2.3. Proofof Theorem 1

Firstofall,notethattheglobalwell-posednessfollowsfromusualaprioriarguments.

Indeed,in sequelwearegoingtoconsider thesolution,as longas itiswell defined,and establish energy estimates. Since we provethatthe energy (whichis the Hilbertnorm of X)isdecreasing,theglobal existenceof weakand thenstrongsolutionsfollows(see, e.g., [13, Theorem 4.3.4 and Proposition 4.3.9]). Hence, in the sequel, without taking care,wedoas ifthesolutionweregloballywelldefined.

Note that uniqueness follows from the assumption that F is locally Lipschitz on bounded sets.

Theproof goesinfoursteps.

Firststep. Comparisonof thenonlinearequation withthelinear dampedmodel.

In thisfirst step,we aregoing tocompare thenonlinear equation(1) withits linear damped counterpart

z(t) +Az(t) +Bz(t) = 0. (13)

Lemma 1.For every solution u(·) of (1), the solution of (13) such that z(0) = u(0) satisfies

T 0

B^1/2z(t)²Xdt≤2 T 0

B^1/2u(t)²X+B^1/2F(u(t))²X

dt. (14)

Proof. Settingψ(t)=u(t)−z(t),wehave

ψ(t) +Aψ(t) +BF(u(t))−Bz(t), ψ(t)X = 0.

DenotingEψ(t)=¹₂ψ(t)²X,itfollowsthat

E_ψ(t) +B^1/2z(t)²X=−u(t), BF(u(t))X+B^1/2F(u(t)), B^1/2z(t)X

+B^1/2u(t), B^1/2z(t)X. Using (4),wehaveu(t),BF(u(t))X≥0,andhence

E_ψ(t) +B^1/2z(t)²X≤ B^1/2F(u(t))XB^1/2z(t)X+B^1/2u(t)XB^1/2z(t)X. Using theYounginequalityab≤ ^a_2θ² +θ^b₂² withθ= ¹₂,weget

E_ψ(t) +B^1/2z(t)²X ≤1

2B^1/2z(t)²X+B^1/2F(u(t))²X+B^1/2u(t)²X, and thus,

E_ψ (t) +1

2B^1/2z(t)²X≤ B^1/2F(u(t))²X+B^1/2u(t)²X. Integratingintime,andnotingthatE_ψ(0)= 0,weinferthat

E_ψ(T) +1 2 T 0

B^1/2z(t)²Xdt≤ T 0

B^1/2F(u(t))²X+B^1/2u(t)²X

dt.

SinceEψ(T)≥0,theconclusionfollows. 2

Second step. Comparison of the linear damped equation with the conservative linear equation.

Wenowconsidertheconservativelinearequation

φ(t) +Aφ(t) = 0. (15)

Lemma 2.Forevery solution z(·) of (13),the solution of (15) suchthat φ(0) =z(0) is suchthat

T 0

B^1/2φ(t)²Xdt≤kT

T 0

B^1/2z(t)²Xdt, (16)

withk_T = 8T²B^1/24+ 2.

Proof. Settingθ(t)=φ(t)−z(t),wehave

θ(t) +Aθ(t)−Bz(t), θ(t)X= 0.

Denoting Eθ(t) = ¹₂θ(t)²X, it follows thatE_θ(t) = Bz(t),θ(t)X. Integrating a first timeover[0,t],andasecondtime over[0,T],and notingthatE_θ(0)= 0,weget

T 0

Eθ(t)dt= T 0

t 0

Bz(s), θ(s)Xds dt= T 0

(T−t)Bz(t), θ(t)Xdt.

ApplyingasintheproofofLemma 1theYounginequalitywithθ=¹₂ yields 1

2 T 0

θ(t)²Xdt≤ T

0

T²Bz(t)²Xdt+1 4 T

0

θ(t)²Xdt,

andtherefore,sinceB isbounded,

1 4 T 0

θ(t)²Xdt≤T²B^1/22 T 0

B^1/2z(t)²Xdt.

Now, sinceφ(t)=θ(t)+z(t),itfollows that T

0

B^1/2φ(t)²Xdt≤2 T 0

B^1/2θ(t)²Xdt+ 2 T 0

B^1/2z(t)²Xdt≤kT

T 0

B^1/2z(t)²Xdt.

Thelemma isproved. 2

Thirdstep.Nonlinear energyestimate.

Letβ >0 (tobe chosenlargeenough,later).Following theoptimalweightconvexity methodof[4,6],wedefinethefunction

w(s) =L⁻¹ s

β

, (17)

for everys∈[0,βs²₀).Inthesequel,thefunctionwisaweightintheestimates,instru- mentalinordertoderiveourresult.

Lemma 3.Forevery solutionu(·)of (1),wehave T

0

w(Eφ(0))

B^1/2u(t)²X+B^1/2F(u(t))²X

dt

TBH^∗(w(E_φ(0))) + (w(E_φ(0)) + 1) T

0

Bu(t), F(u(t))Xdt. (18)

Proof. To provethis inequality,we use the isometric representation of B in thespace L²(Ω,μ).Denotingf =U⁻¹u, usingthatU⁻¹BU f =bf, ρ(f)=U⁻¹F(U f),we have, forinstance,B^1/2u²X=f,bfL²(Ω,μ), andhenceitsufficesto provethat

T 0

w(Eφ(0))

Ω

bf²+bρ(f)² dμ dt

T

Ω

b dμ H^∗(w(Eφ(0))) + (w(Eφ(0)) + 1) T 0

Ω

bf ρ(f)dμ dt. (19)

Indeed,this implies(18)(notethat

Ωbdμ≤ Bbythespectral theorem).

Letusprove(19).Firstofall,foreveryt∈[0,T] wesetΩ^t₁={x∈Ω| |f(t,x)|≤ε0}. Ifb= 0 onΩ^t₁thentheforthcomingintegrals(seeinparticulartheleft-handsideof(20))

arezeroand thereisnothingtoprove;hence,withoutloss ofgeneralityweassumethat bis nontrivialonΩ^t₁.Using (6),wechooseε0>0 smallenoughsuchthat _c¹2

2ρ(f)²≤s²₀ almosteverywhereinΩ^t₁,and thereforewehave

1

Ω^t₁b dμ

Ω^t₁

1

c²₂ρ(f)²b dμ∈[0, s²₀].

Using the Jensen inequality with the measure bdμ, and using the fact that H(x) =

√xg(√

x),we get

H

⎛

⎜⎝ 1

Ω^t₁b dμ

Ω^t₁

1

c²₂ρ(f)²b dμ

⎞

⎟⎠≤ 1

Ω^t₁b dμ

Ω^t₁

1

c2|ρ(f)|g 1

c2|ρ(f)|

b dμ.

Using (6), we have |ρ(f)(x)| ≤ c2g⁻¹(|f(x)|) for almost everyx ∈ Ω^t₁, and since g is increasing,we getthatg

1 c2|ρ(f)|

≤ |f|almosteverywhere inΩ^t₁.Sincef ρ(f)≥0 by (5),weget

H

⎛

⎜⎝ 1

Ω^t₁b dμ

Ω^t₁

1

c²₂ρ(f)²b dμ

⎞

⎟⎠≤ 1

Ω^t₁b dμ 1 c2

Ω^t₁

b|f||ρ(f)|dμ≤ 1

Ω^t₁b dμ 1 c2

Ω

bf ρ(f)dμ.

SinceH isincreasing,itfollowsthat

Ω^t₁

bρ(f)²dμ≤c²₂

Ω^t₁

b dμ H⁻¹

⎛

⎝ 1

Ω^t₁b dμ 1 c2

Ω

bf ρ(f)dμ

⎞

⎠,

andtherefore, T

0

w(Eφ(0))

Ω^t₁

bρ(f)²dμ dt≤ T 0

w(Eφ(0))c²₂

Ω^t₁

b dμ H⁻¹

⎛

⎝ 1

Ω^t1b dμ 1 c2

Ω

bf ρ(f)dμ

⎞

⎠dt.

Thanksto theYounginequalityAB≤H(A)+H^∗(B) (where H^∗ isthe convexconju- gate),weinferthat

T 0

w(Eφ(0))

Ω^t₁

bρ(f)²dμ dt≤ T 0

c2

Ω

bf ρ(f)dμ+ T 0

c²₂

Ω^t₁

b dμ H^∗(w(Eφ(0)))dt

≤c2

T 0

Ω

bf ρ(f)dμ dt+c²₂T

Ω

b dμ H^∗(w(Eφ(0))).

(20)

Besides, inΩ\Ω^t₁,using(6) wehave|ρ(f)||f|.Using (5),itfollowsthat T

0

w(Eφ(0))

Ω\Ω^t1

bρ(f)²dμ dt T 0

w(Eφ(0))

Ω\Ω^t1

b|f||ρ(f)|dμ dt

T 0

w(Eφ(0))

Ω

bf ρ(f)dμ dt.

(21)

From (20)and(21),we inferthat T

0

w(E_φ(0))

Ω

bρ(f)²dμ dt

(w(E_φ(0)) + 1) T 0

Ω

bf ρ(f)dμ dt+T

Ω

b dμ H^∗(w(E_φ(0))).

Let us now proceed in a similar way in order to estimate the term T

0 w(Eφ(0))×

Ωbf²dμdt. We set r²₁ = H⁻¹

c1

c2H(s²₀)

and ε₁ = min(s₀,g(r₁)) ≤ 1. For every t ∈ [0,T], we define Ω^t₂ = {x ∈ Ω | |f(t,x)| ≤ ε₁}. As before, without loss of gener- alityweassumethatbisnontrivialonΩ^t₂.From(6),wehavec1g(|f|)≤ |ρ(f)|inΩ^t₂.By construction, wehave

1

Ω^t₂b dμ

Ω^t₂

f²b dμ∈[0, s²₀].

Using theJenseninequalityaspreviously,andusing(6)and (5),weinferthat

H

⎛

⎜⎝ 1

Ω^t₂b dμ

Ω^t₂

f²b dμ

⎞

⎟⎠≤ 1

Ω^t₂b dμ

Ω^t₂

|f||g(f)|b dμ

≤ 1 c1

Ω^t₂b dμ

Ω^t₂

b|f||ρ(f)|dμ≤ 1 c1

Ω^t₂b dμ

Ω

bf ρ(f)dμ

Since H isincreasing, andintegratingintime,weget T

0

w(Eφ(0))

Ω^t₂

f²b dμ dt≤ T 0

w(Eφ(0))

Ω^t₂

b dμ H⁻¹

⎛

⎝ 1 c₁

Ω^t₂b dμ

Ω

bf ρ(f)dμ

⎞

⎠ dt

Itthenfollows fromtheYounginequalitythat T

0

w(E_φ(0))

Ω^t₂

f²b dμ dt≤T

Ω

b dμ H^∗(w(E_φ(0))) + 1 c1

T 0

Ω

bf ρ(f)dμ dt

TheestimateinΩ\Ω^t₂ isobtainedinasimilarway.

Thelemmaisproved. 2 Fourthstep. Endof theproof.

Lemma4.Wehave

E_u(T)≤E_u(0)

1−ρ_TL⁻¹

E_u(0) β

, (22)

forsomepositivesufficiently smallconstant ρT.

Proof. Usingsuccessivelytheobservabilityinequality(9),theestimate(16)ofLemma 2 andtheestimate(14)ofLemma 1,wefirstgetthat

2CTEφ(0)≤ T 0

B^1/2φ(t)²Xdt≤kT

T 0

B^1/2z(t)²Xdt

≤2kT

T 0

B^1/2u(t)²X+B^1/2F(u(t))²X

dt.

Multiplyingthisinequalitybythe constantw(E_φ(0)), itfollows from theestimate(18) ofLemma 3that

C_Tw(E_φ(0))E_φ(0)

k_TTBH^∗(w(E_φ(0))) +k_T(w(E_φ(0)) + 1) T 0

Bu(t), F(u(t))Xdt. (23)

From (7) and (17), we haveL(w(s))= _β^s = ^H∗_w(s)^(w(s)) forevery s∈ [0,βs²₀), and hence βH^∗(w(s)) = sw(s). We choose β large enough such that E_φ(0) < βs²₀, and thus in particularweget

H^∗(w(Eφ(0))) = w(E_φ(0))E_φ(0)

β . (24)