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英文数学论文
ThispaperisconcernedwiththeCauchyproblemofnonlinearwaveequationswithpotential,strong,andnonlineardampingterms.Firstly,byusingvariationalcalculusandcompactnesslemma,theexistenceofstandingwavesofthegroundstatesisobtained.Thentheinstabilityofthestandingwaveisshownbyapplyingpotential-wellargumentsandconcavitymethods.Finally,weshowhowsmalltheinitialdataarefortheglobalsolutionstoexist.
Keywords:
waveequations;nonlineardampingterms;strongdampingterms;globalexistence;blow-up
Introduction
ConsidertheCauchyproblemfornonlinearwaveequationswithpotential,strong,andnonlineardampingterms,
{utt−Δu−ωΔut+V(x)u+|ut|m−2ut=|u|p−2u,u(0,x)=u0,ut(0,x)=u1,in [0,T)×Rn,in Rn,
(1)
where p>2, m≥2, T>0, ω>0,
u0∈H1(Rn),u1∈L2(Rn),
(2)
and
2
2,for n≤2.(3)
Withtheabsenceofthestrongdampingterm Δut,andthedampingterm ut (see [1]),(1.1)canbeviewedasaninteractionbetweenoneormorediscreteoscillatorsandafieldorcontinuousmedium [2].
Forthecaseoflineardamping( ω=0, m=2)andnonlinearsources,Levine [3]showedthatthesolutionsto(1.1)withnegativeinitialenergyblow-upfortheabstractversion.Forthenonlineardampingandsourceterms( ω=0, m>2, p>2, V(x)=0),theabstractversionhasbeenconsideredbymanyresearchers [4]–[12].Forinstance,GeorgievandTodorova [4]provethatif m≥p,aglobalweaksolutionexistsforanyinitialdata,whileif 20, m=2, V(x)=0)andnonlinearsourceterms( p>2)hasbeenstudiedbyGazzolaandSquassinain [1].Theyprovetheglobalexistenceofsolutionswithinitialdatainthepotentialwellandshowthateveryglobalsolutionisuniformlyboundedinthenaturalphasespace.Moreover,theyprovefinitetimeblow-upforsolutionswithhighenergyinitialdata.However,theydonotconsiderthecaseofanonlineardampingterm( ω>0, m>2, p>2).
Tothebestofourknowledge,littleworkhasbeencarriedoutontheexistenceandinstabilityofthestandingwavefor(1.1).Inthispaper,westudytheexistenceofastandingwavewithgroundstate(ω=1),whichistheminimalactionsolutionofthefollowingellipticequation:
−Δϕ+V(x)ϕ=|ϕ|p−2ϕ.(4)
Basedonthecharacterizationofthegroundstateandthelocalwell-posednesstheory[7],weinvestigatetheinstabilityofthestandingwavefortheCauchyproblem(1.1).Finally,wederiveasufficientconditionofglobalexistenceofsolutionstotheCauchyproblem(1.1)byusingtherelationbetweeninitialdataandthegroundstatesolutionof(1.4).Itshouldbepointedoutthattheseresultsinthepresentpaperareunknownto(1.1)before.
Forsimplicity,throughoutthispaperwedenote ∫Rn⋅dx by ∫⋅dx andarbitrarypositiveconstantsby C.
Preliminariesandstatementofmainresults
Wedefinetheenergyspace Hinthecourseofnatureas
H:
={φ∈H1(Rn),∫V(x)|φ|dx<∞}.(5)
Byitsdefinition, H isaHilbertspace,continuouslyembeddedin H1(Rn),whenendowedwiththeinnerproductasfollows:
⟨φ,ϕ⟩H:
=∫(∇φ∇ϕ¯+V(x)φϕ¯)dx,(6)
whoseassociatednormisdenotedby ∥⋅∥H.If φ∈H,then
∥φ∥H=(∫|∇φ|2dx+∫V(x)|φ|2dx)12.(7)
Throughoutthispaper,wemakethefollowingassumptionson V(x):
⎧⎩⎨⎪⎪⎪⎪infx∈RnV(x)=V¯(x)>0,V(x) isa C1 boundedmeasurablefunctionon Rn,limx→∞V(x)=∞.(8)
Accordingto [1]and [7],wehavethefollowinglocalwell-posednessfortheCauchyproblem(1.1).
Proposition2.1
If (1.2) and (1.3) hold, thenthereexistsauniquesolutionu(t,x)oftheCauchyproblem (1.1) onamaximaltimeinterval[0,T), forsomeT∈(0,∞) (maximalexistencetime) suchthat
u(t,x)∈C([0,T);H1(Rn))∩C1([0,T);L2(Rn))∩C2([0,T);H−1(Rn)),ut(t,x)∈C([0,T);H1(Rn))∩Lm([0,T)×Rn),(9)
andeither T=∞or T<∞and limt→T−∥u∥H1=∞.
Remark2.2
FromProposition 2.1,itfollowsthat m=p isthecriticalcase,namelyfor p≤m,aweaksolutionexistsgloballyintimeforanycompactlysupportedinitialdata;whilefor m
Wedefinethefunctionals
S(ϕ):
=12∫|∇ϕ|2dx+12∫V(x)|ϕ|2dx−1p∫|ϕ|pdx,(10)
R(ϕ):
=∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−∫|ϕ|pdx,(11)
for ϕ∈H1(Rn),andwedefinetheset
M:
={ϕ∈H1∖{0};R(ϕ)=0}.(12)
Weconsidertheconstrainedvariationalproblem
dM:
=inf{supλ≥0S(λϕ):
R(ϕ)<0,ϕ∈H1∖{0}}.(13)
FortheCauchyproblem(1.1),wedefineunstableandstablesets, K1and K2,asfollows:
K1≡{ϕ∈H1(Rn)∣R(ϕ)<0,S(ϕ)0,S(ϕ)Themainresultsofthispaperarethefollowing.
Theorem2.3
Thereexists Q∈Msuchthat
(a1) S(Q)=infMS(ϕ)=dM;
(a2) Qisagroundstatesolutionof (1.4).
FromTheorem 2.3,wehavethefollowing.
Lemma2.4
LetQ(x)bethegroundstateof (1.4). If (1.3) holds, then
S(Q)=minMS(ϕ).(15)
Theorem2.5
Assumethat (1.2)-(1.3) holdandtheinitialenergyE(0)satisfies
E(0)=<12∫|u1|2dx+12(∫|∇u0|2dx+∫V(x)|u0|2dx)−1p∫|u0|pdxp−22p(∫|∇Q|2dx+∫V(x)|Q|2dx).(16)
(b1) If 2(b2) If2
∥ut∥22+p−2p(∫|∇u0|2dx+∫V(x)|u0|2dx)
Variationalcharacterizationofthegroundstate
Inthissection,weproveTheorem 2.3.
Lemma3.1
Theconstrainedvariationalproblem
dM:
=inf{supλ≥0S(λϕ):
R(ϕ)<0,ϕ∈H1∖{0}},(18)
isequivalentto
d1:
=inf{supλ≥0S(λϕ):
R(ϕ)=0,ϕ∈H1∖{0}}=infϕ∈MS(ϕ),(19)
and dMprovided 2
Proof
Let ϕ∈H1.Since
S(λϕ)=λ22(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λpp∫|ϕ|pdx,(20)
itfollowsthat
ddλS(λϕ)=λ(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λp−1∫|ϕ|pdx.(21)
Thusby 2≤m
supλ≥0S(λϕ)=S(λ1ϕ)=λ21(12∫|∇ϕ|2dx+12∫V(x)|ϕ|2dx−λp−21p∫|ϕ|pdx),(22)
where λ1 uniquelydependson ϕ andsatisfies
∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−λp−21∫|ϕ|pdx=0.(23)
Since
d2dλ2S(λϕ)=∫|∇ϕ|2dx+∫V(x)|ϕ|2dx−pλp−2∫|ϕ|pdx,(24)
whichtogetherwith p>2 and(3.6)impliesthat d2dλ2S(λϕ)|λ=λ1<0,wehave
supλ≥0S(λϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)pp−2(∫|ϕ|pdx)2p−2.(25)
Therefore,theaboveestimatesleadto
dM==inf{supλ≥0S(λϕ):
R(ϕ)<0,ϕ∈H1∖{0}}inf{p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)pp−2(∫|ϕ|pdx)2p−2:
R(ϕ)<0}.(26)
Itiseasytoseethat
d1==inf{supλ≥0S(λϕ):
R(ϕ)=0,ϕ∈H1∖{0}}infϕ∈M{p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)}.(27)
From(2.3)-(2.5),on M onehas
S(ϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx).(28)
Itfollowsthat d1=infϕ∈MS(ϕ) and S(ϕ)>0 on M.
Nextweestablishtheequivalenceofthetwominimizationproblems(3.1)and(3.2).
Forany ϕ0∈H1and R(ϕ0)<0,let ϕβ(x)=βϕ0.Thereexistsa β0∈(0,1)suchthat R(ϕβ0)=0,andfrom(3.8)weget
supλ≥0S(λϕβ0)===p−22p(∫|∇ϕβ0|2dx+∫V(x)|ϕβ0|2dx)pp−2(∫|ϕβ0|pdx)2p−2p−22pβ2pp−20(∫|∇ϕ0|2dx+∫V(x)|ϕ0|2dx)pp−2β2pp−20(∫|ϕ0|pdx)2p−2p−22p(∫|∇ϕ0|2dx+∫V(x)|ϕ0|2dx)pp−2(∫|ϕ0|pdx)2p−2.(29)
Consequentlythetwominimizationproblems(3.8)and(3.9)areequivalent,thatis,(3.1)and(3.2)areequivalent.
Finally,weprove dM>0byshowing d1>0intermsoftheaboveequivalence.Since 2
∫|ϕ|pdx≤C(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)p2.(30)
From R(ϕ)=0,itfollowsthat
∫|∇ϕ|2dx+∫V(x)|ϕ|2dx=∫|ϕ|pdx≤C(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)p2,(31)
whichtogetherwith p>2 implies
∫|∇ϕ|2dx+∫V(x)|ϕ|2dx≥C>0.(32)
Therefore,from(3.10),weget
S(ϕ)≥C>0,ϕ∈M.(33)
Thusfromtheequivalenceofthetwominimizationproblems(3.1)and(3.2)oneconcludesthat dM>0 for 2
ThiscompletestheproofofLemma 3.1. □
Proposition3.2
SisboundedbelowonMand dM>0.
Proof
From(2.3)-(2.6),on Monehas
S(ϕ)=p−22p(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx).(34)
Itfollowsthat S(ϕ)>0 on M.So S isboundedbelowon M.From(2.6)wehavedM>0. □
Proposition3.3
Letϕλ(x)=λϕ(x), forϕ∈H1∖{0}andλ>0. Thenthereexistsauniqueμ>0 (dependingonϕ) suchthatR(ϕμ)=0. Moreover,
R(ϕλ)>0,for λ∈(0,μ);R(ϕλ)<0,for λ∈(μ,∞);(35)
and
S(ϕμ)≥S(ϕλ),∀λ>0.(36)
Proof
By(2.3)and(2.4),wehave
S(ϕλ)=λ22(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λpp∫|ϕ|pdx,R(ϕλ)=λ2(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−λp∫|ϕ|pdx.(37)
Fromthedefinitionof M,thereexistsaunique μ>0 suchthat R(ϕμ)=0.Moreover,
R(ϕλ)>0,for λ∈(0,μ);R(ϕλ)<0,for λ∈(μ,∞).(38)
Since
ddλS(ϕλ)=λ−1R(ϕλ),(39)
and R(ϕμ)=0,wehave S(ϕμ)≥S(ϕλ), ∀λ>0. □
Next,wesolvethevariationalproblem(2.6).
Wefirstgiveacompactnesslemmain [8].
Lemma3.4
Let 1≤pInthefollowing,weproveTheorem 2.3.
ProofofTheorem 2.3
AccordingtoProposition 3.2,welet {ϕn,n∈N}⊂Mbeaminimizingsequencefor(2.6),thatis,
R(ϕn)=0,S(ϕn)→dM.(40)
From(3.15)and(3.16),weknow ∥∇ϕn∥22 isboundedforall n∈N.Thenthereexistsasubsequence {ϕnk,k∈N}⊂{ϕn,n∈N},suchthat
{ϕnk}⇀ϕ∞weaklyin H1.(41)
Forsimplicity,westilldenote {ϕnk,k∈N} by {ϕn,n∈N}.Sowehave
ϕn⇀ϕ∞weaklyin H1.(42)
ByLemma 3.4,wehave
ϕn→ϕ∞stronglyin L2(Rn),(43)
ϕn→ϕ∞stronglyin Lp(Rn).(44)
Next,weprovethat ϕ∞≠0 bycontradiction.If ϕ∞≡0,from(3.18)and(3.19),wehave
ϕn→0stronglyin L2(Rn),(45)
ϕn→0stronglyin Lp(Rn).(46)
Since ϕn∈M, R(ϕn)=0,weobtain
(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)→0,n→∞.(47)
Ontheotherhand,from(2.3)wehave
(∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)−2p∫|ϕn|pdx→dM.(48)
From(3.20)-(3.21)weobtain (∫|∇ϕ|2dx+∫V(x)|ϕ|2dx)→dM.AccordingtoProposition 3.2, dM≥c>0.Thisisincontradictoryto(3.22).Thus ϕ∞≠0.
AccordingtoProposition 3.3,wetake Q=(ϕ∞)μwith μ>0uniquelydeterminedbythecondition R(Q)=R[(ϕ∞)μ]=0.From(3.17)-(3.19),wehave
(ϕ∞)μ→Qstronglyin L2(Rn),(49)
(ϕ∞)μ→Qstronglyin Lp(Rn),(50)
(ϕ∞)μ⇀Qweaklyin H1(Rn).(51)
Since R(ϕn)=0 andbyProposition 3.3,weget
S[(ϕ∞)μ]≤S(ϕn).(52)
From(3