ON THE PERSISTENCE OF SPATIAL ANALYTICITY FOR THE BEAM EQUATION

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

On the persistence of spatial analyticity for the beam equation

TamiratT. Dufera^b, Sileshi Mebrate^b, Achenef Tesfahun^a,∗

a DepartmentofMathematics,NazarbayevUniversity,QabanbaiBatyrAvenue53,010000Nur-Sultan, Kazakhstan

bDepartmentofMathematics,AdamaUniversityofScienceandTechnology,Ethiopia

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received11September2021 Availableonline11January2022 Submittedby H.Frankowska

Keywords:

Beamequation

Globalwell-posednesslowerbound Radiusofanalyticity

Gevreyspaces

Persistenceofspatialanalyticityisstudiedforsolutionofthebeamequationu_tt+ (m+ Δ²)u+|u|^p−1u= 0 on Rⁿ×R.Inparticular,fora classof analyticinitial datawithauniformradiusofanalyticityσ0,weobtainanasymptoticlowerbound σ(t)c/√

tontheuniformradiusofanalyticityσ(t) ofsolutionu(·,t),ast→ ∞.

©2022ElsevierInc.Allrightsreserved.

1. Introduction

ThispaperisconcernedwithpersistenceofspatialanalyticityofsolutionsfortheCauchyproblemofthe fourthorderwaveequation

utt+

m+ Δ²

u+|u|^p−1u= 0,

(u, ut)|t=0= (u0, u1), (1)

whereu:Rⁿ×R−→R,p1,andm>0.

Equationslike(1) arealsoreferredtoasBretherton’stypeequationsorthebeamequation.Theoriginal Brethertonequation,writtendownforn= 1 by Bretherton[3],arised inthestudyofweakinteractionsof dispersivewaves.Asimilarequationforn= 2 wasproposedinLove[17] forthemotionofaclampedplate.

Recentdevelopmentsinarbitrarydimensionwereestablishedin[13–15,21,22].

Questionssuchaswell-posedness,blow-upinfinitetime,longtimeexistence,andtheexistenceofuniform bounds for global solutions of (1) are addressed by several authors. For instance, local well-posedness, scattering, and stability inthe energy spaceH²(Rⁿ)×L²(Rⁿ) was studiedby Levandoskyin[13,14] and

* Correspondingauthor.

E-mailaddresses:tamirat.temesgen@astu.edu.et(T.T. Dufera),silenaty2005@gmail.com(S. Mebrate),achenef@gmail.com (A. Tesfahun).

https://doi.org/10.1016/j.jmaa.2022.126001 0022-247X/©2022ElsevierInc. Allrightsreserved.

LevandoskyandStrauss[15], resultswhichwereextendedbyPausader [23,24].Low-regularityglobalwell- posedness was also shownby Zhang [32] in dimensions 3 n 7 in the cubic case for data (u₀,u₁) ∈ H^s(Rⁿ)×H^s−2(Rⁿ) satisfying

s >min

n−2 2 , n

4

.

Weremark thattheenergy

E(t) =1 2

Rⁿ

u²_t+ (Δu)²+mu²+ 2

p+ 1|u|^p+1 dx is conservedbytheflowof(1),i.e., E(t)= const. forallt.

Inthispaper,weshallstudythepersistenceofspatialanalyticityforthesolutionoftheCauchyproblem (1),giveninitialdatainaclassofanalyticfunctions.BythePaley-WienerTheorem,theradiusofanalyticity ofafunctioncanberelatedtodecaypropertiesofitsFouriertransform.Itisthereforenaturaltotakedata for(1) intheGevreyspaceG^σ,s(Rⁿ),definedbythenorm

fG^σ,s(Rⁿ)=exp(σ|ξ|)ξ^sfˆL²_ξ(Rⁿ) (σ0), where ξ=

1 +|ξ|².Whenσ= 0,thisspacecoincideswith theSobolevspaceH^s(Rⁿ),withnorm fH^s(Rⁿ)=ξ^sfˆL²_ξ(Rⁿ),

whileforanyσ >0,anyfunctioninG^σ,s(Rⁿ) hasaradiusofanalyticityofatleastσateachpointx∈Rⁿ. This fact is contained in thefollowing theorem, whose proof can be found in[12] in the case s = 0 and n= 1;thegeneralcasefollows fromasimplemodification.

Paley-WienerTheorem.Letσ >0ands∈R.Iff ∈G^σ,s(Rⁿ),thenf istherestrictiontoRⁿ ofafunction F whichisholomorphicin thestrip

Sσ ={x+iy∈Cⁿ : |y|< σ}. Moreover, thefunctionF satisfiestheestimates

sup

|y|<σF(·+iy)H^s <∞.

Information about thedomain of analyticity of asolution to apartial differential equation (PDE) can be used to gaina quantitative understandingof the structure of the equation, and to obtain insight into underlyingphysicalprocesses.Thestudyofreal-analyticsolutionstononlinearPDEhasdevelopedoverthe lastthreedecades.StartingwiththeworksofKatoandMasuda[11] fordispersivewave-typeequations,and Foias andTemam[6] fortheNavier-Stokesequations, analyticfunctionspaces havebecomepopulartools for the study of avarietyof questionsconnected with nonlinearevolutionary PDE. In particular, theuse of Gevrey-typespaceshasgiven riseto anumberof importantresultsinthestudy oflongtime dynamics of dissipative equation, such as estimating the asymptotic degrees of freedom (e.g., determining nodes) [18], approximating theglobal attractors/inertial manifolds [10] and arigorousestimate of the Reynold’s scale [4].

Given anonlineardispersivePDEintheindependentvariables(t,x),consider theCauchyproblemwith real analytic initial data at t= 0. If these data haveauniform radius of analyticityσ₀, inthesense that

there exists aholomorphic extension to the complex strip of widthσ₀, then we ask whetherthe solution at some later time t >0 also hasa uniform radius of analyticity σ= σ(t) >0, inwhich case we would, moreover,like to haveanexplicit lowerbound onσ(t).Heuristically,thepicture oneshouldhaveinmind isthatσ(t) isthedistancefromthex-axistothenearestcomplexsingularityoftheholomorphicextension ofthe solutionattime t. Ifat sometime tthis singularity actuallyhits thex-axis,thenthesolutionitself suffers a breakdown of regularity. This point of view is thebasis for the widely used singularity tracking method[28] innumericalanalysis,whereaspectralmethodis usedtoobtainanumericalestimateofσ(t).

Thisestimatecanthenbeusedtopredicteithertheformationofasingularityinfinitetimeoralternatively global regularity. Even in cases where singularity formation does not occur (as is the case for the beam equation), it is still of interest to obtain lower bounds on σ(t), as this has implications for the rate of convergenceofspectral methodsfortheequationoneislookingat (see[2] foranexampleofthis).

The spaces G^σ,s(Rⁿ) were introduced by Foias and Temam [6] (see also [11]) in the study of spatial analyticityofsolutionstotheNavier-Stokesequations,andvarious refinementsoftheirmethodhavesince beenappliedtoprovelowerboundsontheradiusofspatialanalyticityforanumberofnonlinearevolution equations [5,7–9,16,19,20,25–27,29–31]. The method used here for proving lower bounds on the radius of analyticity was introduced in [27] in the context of the 1D Dirac-Klein-Gordon equations. This method is based on an approximate conservation laws, and has been applied to prove an algebraic lower bound (decayrate)of ordert^−1/αforsomeα∈(0,1] ontheradiusofspatialanalyticityofsolutionstoanumber ofnonlinear dispersiveandwave equations(seee.g.,[1,25–27,29–31]). Theoptimaldecayratethatcanbe obtainedinthissettingis1/t,whichcorresponds toα= 1 (seee.g.,[1,29,31]).Thisdecayrateisrelatedto thebehaviorof theexponentialweight,exp(σ|ξ|),thatsitsintheGevreynorm. Morespecifically,itstems from thesimpleestimate

exp(σ|ξ|)−1(σ|ξ|)^α·exp(σ|ξ|) (0α1),

whichfollowsfrom aninterpolationbetweenexpr−1expr andexpr−1rexpr forr0.

Inthe presentwork, inanattemptto improve thedecay rateobtainedso far, weintroduce amodified Gevreynorm¹ by

fH^σ,s(Rⁿ)=cosh(σ|ξ|)ξ^sfˆL²_ξ(Rⁿ) (σ0),

wheretheexponentialweightexp(σ|ξ|) intheGevreynormisnowreplacedbyahyperbolicweightcosh(σ|ξ|).

Thesetwo weightsareequivalent inthesensethat 1

2exp(σ|ξ|)cosh(σ|ξ|)exp(σ|ξ|). (2) Thus,theassociatedG^σ,s andH^σ,s–normsareequivalent, i.e.,

fH^σ,s(Rⁿ)∼ fG^σ,s(Rⁿ), (3) andso thestatementofPaley-WienerTheorem stillholdsforfunctionsinH^σ,s.

Note alsothatH^0,s =G^0,s =H^s. Thespace H^σ,s, however,hasanadvantage sincecosh(σ|ξ|) satisfies theestimate

cosh(σ|ξ|)−1(σ|ξ|)^2α·cosh(σ|ξ|) (0α1). (4) Thisfollows from

1 Asfarasweknowthe spaceH^σ,sisnewtothispaperandwasnotusedbefore.

coshr−1coshr and coshr−1r²coshr (r∈R).

Therefore, inviewof (4),an applicationofourmethod inthe H^σ,s-set upcanyield adecayrateof order t^−1/2α for some α ∈ (0,1] provided that the nonlinear estimates in the derivation of the approximate conservation lawcanabsorbtheweight|ξ|^2α.Inthisworkwemanaged toobtaintheoptimaldecayrateof t^−1/2 (whichcorrespondsto α= 1) fortheCauchyproblem (1).

Weremark thatas aconsequencetheembedding

H^σ,s⊂H^s (σ0) (5)

andtheexistingwell-posednesstheoryinH²(Rⁿ)×L²(Rⁿ),onecanconcludethattheCauchyproblem(1), with 1n3 andp1,hasaunique,global-in-timesolution,giveninitial data(u0,u1)∈H^σ0,2(Rⁿ)× H^σ0,0(Rⁿ) forsomeσ00.

Wenow stateourmain theorem.

Theorem 1(Lowerbound fortheradiusofanalyticity). Let1n3,p1isan odd integerandσ₀>0.

If (u₀,u₁)∈H^σ0,2(Rⁿ)×H^σ0,0(Rⁿ),thenforany T >0thesolutionof (1) satisfies (u, ut)∈C

[0, T];H^σ,2(Rⁿ)

×C¹

[0, T];H^σ,0(Rⁿ) with

σ:=σ(T) = min

σ0, cT⁻¹² ,

where c>0isaconstant depending ontheinitialdata norm.

In viewof the Paley-Wiener theorem and (3), this result impliesthat thesolution u(·,t) has radius of analyticityatleastσ(t) foreveryt>0.

Remark 1.The resultof Theorem 1can be extendedto dimension n= 4 if oneuses Strichartzestimates (see[21])insteadofjustSobolevembeddingsaswedointhispaper.It isalsopossible toextendtheresult forn5 butwith someupperboundrestrictiononp.However,wewillnotpursuethese issueshere.

The first step is to prove the following local-in-time result, where the radius of analyticity remains constant.

Theorem 2(Local well-posedness). Let 1 n 3, p 1 is an odd integer and σ >0. Given (u0,u1) ∈ H^σ,2(Rⁿ)×H^σ,0(Rⁿ),thereexistsatimeδ >0and auniquesolution

(u, u_t)∈C

[0, δ];H^σ,2(Rⁿ)

×C¹

[0, δ];H^σ,0(Rⁿ) of theCauchy problem(1)on Rⁿ×[0,δ].Moreover,theexistencetime isgivenby

δ=c0(u0H^σ,2+u1H^σ,0)^−(p−1), (6) forsome constant c₀>0.

Thesecond stepintheproofofTheorem1istoproveanapproximateconservation lawforthenormof thesolution,thatinvolvesasmallparameterσ >0 andwhichreducestotheexactenergyconservationlaw inthelimitasσ→0.Toderivethis approximateconservation law,weset

v_σ(x, t) := cosh(σ|D|)u(x, t),

whereuisthesolutionto(1).Define amodified energyassociatedwithv_σ by Eσ(t) = 1

2

Rⁿ

(∂_tv_σ)²+ (Δv_σ)²+mv_σ²+ 2

p+ 1|v_σ|^p+1 dx.

Theorem 3(Approximate conservation law). Let 1 n 3, p 1 is an odd integer and σ > 0. Given (u0,u1)∈H^σ,2×H^σ,0,letubethelocal solutionof (1)on Rⁿ×[0,δ]that isobtained inTheorem2.Then

sup

0tδEσ(t) =Eσ(0) +δσ²· O

(Eσ(0))^p+12

. (7)

Observethatinthelimit as σ→0,werecover the conservationE0(t)=E0(0) for0tδ(note that v₀ =u).Applyingthe lasttwo theorems repeatedly,and thenbytaking σ >0 smallenoughwe cancover anytime interval[0,T] andobtainthemainresult,Theorem 1.

Notation.For any positive numbers a and b, the notation a b stands for a cb, where c is a positive constantthatmaychangefrom linetoline.Moreover,wedenotea∼b whenabandba.

InthenextsectionsweproveTheorems2,3and1.

2. ProofofTheorem2

Theorem2caneasilybeprovedusingenergyinequality,Sobolevembeddingandastandardcontraction argument.Indeed,consider theCauchyproblem forthelinearbeam equation

u_tt+

m+ Δ² u=F, (u, u_t)|t=0= (u₀, u₁) whosesolutionisgivenbyDuhamel’sformula

u(t) =Sm (t)u0+Sm(t)u1+ t

0

Sm(t−s)F(s)ds, (8)

where

Sm(t) = sin (tΔm) Δm

.

Applyingcosh(σ|D|) to(8) andtakingtheH²-normonbothsidesyieldstheenergyinequality

sup

0tδ

(uH^σ,2+utH^σ,0)u0H^σ,2+u1H^σ,0+ δ 0

F(s)H^σ,0 ds (9)

forsomeδ >0.

Nowconsider theintegralformulationof(1),

u(t) =Sm (t)u0+Sm(t)u1+ t

0

Sm(t−s)u^p(s)ds, (10)

where weusedthefactthat|u|^p−1u=u^p foroddp.

Thenby(9) andastandardcontractionargument,Theorem2reducestoprovingthenonlinearestimate

u^p_H^σ,0u^p_H^σ,2, (11)

whichisalso equivalent,by(3),toproving

u^p_G^σ,0u^p_G^σ,2.

SettingU = exp(σ|D|)u, thisestimatereducesfurtherto

exp(σ|D|) [(exp(−σ|D|)U)^p]_L²_x U^p_H². (12) ByPlancherel,

LHS (12) =Fx{exp(σ|D|) [(exp(−σ|D|)U)^p]}(ξ)_L²_ξ

=

ξ=p j=1ξj

exp

⎛

⎝σ

⎡

⎣|ξ| − p j=1

|ξ_j|

⎤

⎦

⎞

⎠^p

j=1

U(ξˆ _j)dξ₁dξ₂· · ·dξ_p

L²_ξ

ξ=p j=1ξj

p j=1

|Uˆ(ξj)|dξ1dξ2· · ·dξp

L²_ξ

=V^pL²_x

where V =F⁻¹

|U|

. To obtain thethird line weused thefactthat |ξ| p

j=1|ξj|, whichfollows from thetriangleinequality.

Now bySobolevembedding,

V^pL²_x=V^p_L^2p_x V^pH² =U^pH²

forall² p1 if1n4.Thisconcludestheproofof(12),andhence(11).

3. ProofofTheorem 3andTheorem1

Let 1n 3,p 1 is an oddinteger, σ >0, and δ > 0 bethe local existence time forthe solution obtained inTheorem 2,(6).Recall thatvσ(x,t)= cosh(σ|D|)u(x,t), whereuis thesolutionto (1). Thus, u(x,t)= sech(σ|D|)vσ(x,t).

3.1. Proofof Theorem3

Using integrationbypartsandequation (1),weobtain Eσ(t) =

Rⁿ

∂tvσ

∂ttvσ+ Δ²vσ+mv+v_σ^p dx

2 Thisestimatealsoholdsfor1pnⁿ−4 ifn5.

=

Rⁿ

∂tvσ

cosh(σ|D|)

utt+ Δ²u+mu +v^p_σ

dx

=

Rⁿ

∂tvσ[−cosh(σ|D|)u^p+v^p_σ] dx

=

Rⁿ

∂tvσ·Np(vσ) dx,

where

Np(vσ) =v_σ^p−cosh(σ|D|) [sech(σ|D|)vσ]^p. Therefore,

Eσ(t) =Eσ(0) + t 0

Rⁿ

∂tvσ(x, s)·Np(vσ(x, s))dxds. (13)

Wenowstateakeyestimatethatwillbe provedinthenextsection.

Lemma1.For∂_tv_σ∈L²_x and v_σ∈H²,wehavetheestimate

Rⁿ

∂_tv_σ·N_P(v_σ)dx Cσ²∂_tv_σL²_xv_σ^pH² (14)

forsomeconstant C >0.

Nowweuse(13) and(14) toobtaintheapriorienergyestimate

0tδsup Eσ(t) =Eσ(0) +δσ²· O

∂tvσ_L^∞_{δ L}²_xvσ^p_L^∞_δH²

, (15)

wherewe usethenotation

L^∞_δ X :=L^∞_t X([0, δ]×Rⁿ) withX =L²_x orH².

AsaconsequenceofTheorem 2weget

v_σL^∞_δH²+∂_tv_σL^∞_δ L²_x =uL^∞_δ H^σ,2+u_tL^∞_δ H^σ,0

C(u₀H^σ,2+u₁H^σ,0)

=C

vσ(·,0)H²+∂tvσ(·,0)L²_x

.

(16)

Since

Eσ(0) = 1 2

Rⁿ

[∂tvσ(x,0)]²+ (Δvσ(x,0))²+m[vσ(x,0)]²+ 2

p+ 1|vσ(x,0)|^p+1 dx

vσ(·,0)_H²+∂tvσ(·,0)_L²_x2

itfollows from(16) that

vσL^∞_δ H²+∂tvσL^∞_δ L²_x [Eσ(0)]¹². (17) Finally,using (17) in(15) weobtainthedesiredestimate(7).

3.2. Proofof Theorem1

Supposethat(u0,u1)∈H^σ0,2×H^σ0,0 forsomeσ0>0.This implies

vσ0(·,0) = cosh(σ0|D)|)u0∈H², ∂tvσ0(·,0) = cosh(σ0|D)|)u1∈L². Then bySobolevembedding

Eσ0(0)vσ0(·,0)²_H²+∂tvσ0(·,0)²_L²_x+vσ0(·,0)^p+1_H² <∞.

Now followingtheargumentin[27] (seealso[25])wecanconstructasolutionon[0,T] forarbitrarilylarge time T by applying the approximate conservation (7), so as to repeat the local result in Theorem 3 on successive shorttimeintervalsofsizeδtoreachT byadjustingthestripwidthparameterσ∈(0,σ0] ofthe solutionaccordingtothesizeof T.

Thegoalistoprovethatforagivenparameterσ∈(0,σ₀] and largeT >0, sup

t∈[0,T]Eσ(t)2Eσ0(0) for σ=c/√

T , (18)

where c>0 dependsonlyontheinitialdatanorm,σ₀andp.ThiswouldimplyEσ(t)<∞forallt∈[0,T], and hence

(u, ut)(·, t)∈H^σ,2×H^σ,0 for σ=c/√

T and t∈[0, T].

Toprove(18) firstobservethatforagivenparameterσ∈(0,σ₀] and0< t₀δwehavebyTheorems2 and 3,

sup

t∈[0,t0]Eσ(t)Eσ(0) +Cδσ²(Eσ(0))^p+12 Eσ0(0) +Cδσ²(Eσ0(0))^p+12 ,

where we also used thefactthe Eσ(0)Eσ0(0) forσ σ₀;this holds sincecoshr is increasing forr0.

Thus,

sup

t∈[0,t0]

Eσ(t)2Eσ0(0), (19)

provided

Cδσ²(Eσ0(0))^p+12 Eσ0(0). (20) ThenwecanapplyTheorem2,withinitialtimet=t0andthetimestepδasin(6) toextendthesolution to [t0,t0+δ].ByTheorem 3,theapproximateconservationlaw, and(19) wehave

sup

[t0,t0+δ]

Eσ(t)Eσ(t₀) +Cδσ²(2Eσ0(0))^p+12 . (21)

Inthisway,wecovertimeintervals [0,δ], [δ,2δ] etc.,andobtain Eσ(δ)Eσ(0) +Cδσ²(2Eσ0(0))^p+12

Eσ(2δ)Eσ(δ) +Cδσ²(2Eσ0(0))^p+12 Eσ(0) +C2δσ²(2Eσ0(0))^p+12

· · ·

Eσ(nδ)Eσ(0) +Cnδσ²(2Eσ0(0))^p+12 . Thiscontinuesas longas

Cnδσ²(2Eσ0(0))^p+12 Eσ0(0) (22) sincethen

Eσ(nδ)Eσ(0) +Cnδσ²(2Eσ0(0))^p+12 2Eσ0(0) sowe cantakecareonemorestep.Note alsothat(20) followsfrom(22).

Thus,theinductionstopsat thefirstintegernforwhich

Cnδσ²(2Eσ0(0))^p+12 >Eσ0(0) andthenwehavereachedthefinaltime

T =nδ, when

CT σ²(2Eσ0(0))^p−12 >1.

NotethatT willbearbitrarily largeforσ >0 smallenough.Moreover, σ²> C⁻¹(2Eσ0(0))^−p+12 ·T⁻¹ proving

σcT⁻¹²,

asclaimed.

4. ProofofLemma1

Firstweprovethefollowingstwo LemmaswhicharecrucialintheproofofLemma1.

Lemma2.Fora,b∈R,wehave

|coshb−cosha| 1 2

b²−a² (coshb+ cosha). (23)

Proof. Note thatcoshris an increasingfunction forr0.Since coshr is even,i.e., coshr = cosh|r|, we mayassumea,b0.Bysymmetrywemayalsoassumeba.Then

coshb−cosha= b a

s 0

coshr dr dscoshb b a

s 0

dr ds=1 2

b²−a²

coshb.

Lemma 3.Letξ=p

j=1ξj forξj∈Rⁿ,wherep1isaninteger.Then

1−cosh|ξ| p j=1

sech|ξj| 2^p

p j=k=1

|ξj||ξk|. (24)

Proof. Firstobserve that p j=1

cosh|ξ_j|= 2^1−p

s2,s3···,sp

cosh

⎛

⎝|ξ₁|+ p j=2

s_j|ξ_j|

⎞

⎠, (25)

where s2,s3,· · ·,spareindependentsigns(+ or−).Indeed,thecasep= 1 isobvious, whilethecasep= 2 follows fromtheidentity

2 cosh|ξ1|cosh|ξ2|= cosh(|ξ1| − |ξ2|) + cosh(|ξ1|+|ξ2|), whichalso implies(25) forp= 3.Thegeneralcasefollowsbyinduction.

It followsfrom(25) that

cosh

⎛

⎝|ξ1|+ p j=2

sj|ξj|

⎞

⎠+ cosh|ξ|2^p p j=1

cosh(|ξj|). (26)

Observealsothat

⎛

⎝|ξ1|+ p j=2

sj|ξj|

⎞

⎠

2

− |ξ|² 2

p j=k=1

|ξj||ξk|. (27)

Applying(25),(23),(26) and(27) weobtain

p j=1

cosh|ξ_j| −cosh|ξ| =

2^1−p

s2,s3···,sp

cosh

⎛

⎝|ξ₁|+ p j=2

s_j|ξ_j|

⎞

⎠−cosh|ξ|

=

2^1−p

s2,s3···,sp

⎡

⎣cosh

⎛

⎝|ξ₁|+ p j=2

s_j|ξ_j|

⎞

⎠−cosh|ξ|

⎤

⎦

2^1−p

s2,s3···,sp

1 2

⎛

⎝|ξ1|+ p j=2

sj|ξj|

⎞

⎠

2

− |ξ|²

⎛

⎝cosh

⎛

⎝|ξ1|+ p j=2

sj|ξj|

⎞

⎠+ cosh|ξ|

⎞

⎠

2^1−p

s2,s3···,sp

⎛

⎝ ^p

j=k=1

|ξj||ξk|

⎞

⎠·2^p p j=1

cosh(|ξj|)

= 2^p

⎛

⎝ ^p

j=k=1

|ξj||ξk|

⎞

⎠^p

j=1

cosh(|ξj|).

Dividingby!p

j=1cosh(|ξ_j|) yieldsthedesiredestimate (24).

NowweproveLemma1.Recallthat

N_p(v_σ) =v_σ^p−cosh(σ|D|) [sech(σ|D|)v_σ]^p. ByCauchy-Schwarzinequality

Rⁿ

∂tvσ·NP(vσ)dx ∂tvσL²_xNP(vσ)L²_x,

andso wearereducedtoprove

NP(vσ)_L²_x σ²vσ^p_H². (28) TakingtheFouriertransformwehave

Fx[Np(vσ)](ξ) =

ξ=p j=1ξj

⎡

⎣1−cosh(σ|ξ|) p j=1

sech(σ|ξj|)

⎤

⎦^p

j=1

"

vσ(ξj)dξ1dξ2· · ·dξp (29)

Bysymmetry,wemayassume|ξ1||ξ2|· · ·|ξp|.ByLemma3,

1−cosh(σ|ξ|) p j=1

sech(σ|ξj|) 2^p

p j=k=1

|σξj||σξk|

c(p)σ²|ξ₁||ξ₂|,

(30)

wherec(p)=p²2^p.Nowset

wσ:=Fx⁻¹(|"vσ|). Thenapplying(30) to(29) weobtain

|Fx[Np(vσ)](ξ)|c(p)σ²

ξ=p j=1ξj

|ξ1||ξ2||"vσ(ξ1)||"vσ(ξ2)| p j=3

|"vσ(ξj)|dξ1dξ2· · ·dξp

=c(p)σ²

ξ=p j=1ξj

|ξ1||ξ2|w"σ(ξ1)w"σ(ξ2) p j=3

|w"σ(ξj)|dξ1dξ2· · ·dξp

=c(p)σ²Fx

(|D|w_σ)²·w^p_σ⁻² (ξ).

Therefore, usingPlancherel,HölderandSobolevembeddingweobtain Np(vσ)L²_xc(p)σ²(|D|wσ)²·w^p−2_σ L²_x

c(p)σ²|D|w_σ²L⁴_xw_σ^p_L⁻∞² x

Cσ²wσ²_H²wσ^p−2_H²

=Cσ²v_σ^pH²

as desiredin(28).

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