【doc】一类半线性波动方程具分块常数初值的Cauchy问题

【doc】一类半线性波动方程具分块常数初值的Cauchy问题 一类半线性波动方程具分块常数初值的 Cauchy问题 泼万征/组l')袈 No~heast.]Vlath.J. 19(3)(1994).42l一426 The ,坦uationswithPiecewiseConstantsData) ZhangYongqian(张永前) (胁.n妇石面-瓦蕊200433) cI? (=)fZ AbstractInmpal~r?we~tx]ytheexi窖tenoeandregularityoftheweaksolutionto (一若一碍=l,..?)+()(I嘶l一I"I). 廿lplecewiseconstantsdatan龋rthe01'n.wealsogiveagcrg=ralizatlo~to雌serndlinearsysumas. KeyWordsandPhra~CauchyProblem;t~ecewlseConstantsDala;SomilinearWaveEquati on) WeakSolution)Singularlty ?1.Introduction In[1],chenhasconsideredtheCauchyproblemforacofsernilinear"sg/aveeqomas follow: f(哥一一碍=^0,,."), ()l.0=0, l.0=rJlQ,), where_『_lisa.functionofitsarguments.(.,,,0)=0forany(,,)?/P,川0,)= 0,?0,),(,,)?c()and = >0,>0; <0.>0; >0,<0; <0,<0. sothe?norn日 Iregularityforthesolutionwasestablished.Meanwhile.Beals&BEzardC]got theexistenceofanunnecessarilyboundedsolu~onforaclag8ofsemilinearwaveequationsin [0,7,]Xh=forne0and?3. Inthispaper,westudytheexistenceandregularityoftheweaksolutiontothepro~em f(哥一一碍一,(.,,w)-H9(?)(II一ll). (P)l=0, l.0=^0.), Rd删Ju鹏10.1993. *)Project仰?蚰byNSF埴E~inaandNEe NOR~,Mm工VoLl0 where0,tt)一0,)0.),,andafecfunctionsoftheirarguments,f(t,,,,0)一 0forany(,,,,)?,andinthefinal9ectionwegivethegeneralizationtothesemflinearsys- tems.Themethodwhichweapplyto? lvetheproblem(P)andtoestablishitsgeneralizationcan befoundinBitsadze[]andStmus.44]forobtainingsn?hsolutions. ?2.NotationsandMainResult First?,eintroducethespaceofconormaldislributiontodescribetheregularityofthes0lu— tiorlasin[5],[】]and[4].Letusset 一 {一,},一{一,),马一{:一,), 置一{一一,),墨一"一+), andlet—裙bethes吐ofallaCCeptablevectorfieldstangentialto?I,?t,?s.?',?I.Aswe know,canbespannedbyacompletebasks.whichwederK~teby(1?1)(螂[, forexample).Thenweintroduce t r(一{"?L2l…0)?L2(),?曲?,where哦?0,?z}?一j and ,(—)一{"?l妇?It'卯,forany?c()} NOW.wecanstateourmainresult. 1heom1.neexists0andthereisaweaksolution—(,,,,)to(P).defined inE0.]×,suchthatnEC([0,].H())n(E0,].()),?,尸([0,]× ),"iseon~'mal,?imrespectto蜀,马,晶.蜀.and"isC'~awayfromthesesufa~es. ?3.SolutionofSemilmearProblem(P,) BeforeWegive_恤eproofofthemainresult,wehaveto咆1withthesemilinearproblem ).Firstwestatetheresultof121~en(see[1]). Prop0si0lIF0rstroll>0,thereisaweak~lution"to(),definedin口一[0,]? ×chthat"?(E0,],L2())Nc([0,],())n,尸(口).and"isCOin,?O andC啪awayfrom?L,?2,?3,?^.?I.Moreover.thesolutionis伽withrespecttothese st~faces?andtheweaksingularitiesofn蛉~lutionactuallyapp~i-onthecharacter?cs. gythisproposition,t~ereexistsaconstant腰>0.suchthat l{[0.x)?<?. NextWeestabli~anestimateforthesolutionto(). LenI眦lLetbeawP..~lk~lutionto(),'?([O,],())n6,1([0,], ())flL~([0,]×)for9.me>0,andH{.?<?forSomeR>0. Then如rany>O,thereexists'>0,8uchthat ll{[0.r?]×,)?M, ? THECAUO-FYpROBLEMFOR孙?uEARWAVEEQUATIONS423 where 一训n…sup, ) and Q(TO一{(..,)?[0,T0]×l+矿?0+r)} withr=--~Uam(supp(k)). ProofLet0bethesolutionofthefinearproblemand = , sup . ThenWehavethat,foranyf?%,"isastrongsolutiontothelinearproblemwithriShthand side0,z,,)and 且一0.)一{(,-)l0一.I)+(一j)?(芒一f)} (see[6]),and.bytheenergytfmateinthebackwardlightcone(8?[6]and[7]) I(([o.]x)f))一0- Thm.wehave ()()?ll(f)llc+. ma ' xl(t)11~-() ?ll0(f)ll(+.且 IfWechooseTsmallenou【gh.theresultfoUowsimmediately. ?4.SolutionofallInitialValueProblem InthisS~CtiOrl?WeO0n蚵cIertheinitialvalueproblem ? , whereFisa8mentionedin?1.Thenwehave Lemnm2Thereexistsauniquesolutionof(I),:(ml,砌)—--矗,where(仍1.小2)is thelifespanofthesolution,andWehavethat (1)If伽>0,thenl..?>O. (2)m】<0,弛>0. ProofFortheease一0.itis州via1. Weonlyhavetosolvetheproblemforthecase>0.Wehaveknownthatthesolutionof (I)isunique,anditsufficestofindasolutionof(I). First,vehavetofindthes0【On"一(oftheproblem 一 吖 Z 一 w NORTHEASTMATH.J. f品(嚣)一一c(嚣), (I){J.一o, 1 . So 一 击』:叫一脚a私一J.叫一j婶' Wehaveknownthatisastricflyincreasingfunction.and 器=扣p(一>o.pI—J. Hence"巴()isadiffeomorphism.where":'—(l'砌),and "(0)一0.m】<0,>0. Thus.ifwedenoteitsinverseby一():(mJ,m2)—.then,veCallgetthesolutionof (I). ?5.ReductionoftheProblem Inthissection,havetofindtherelationbetw~ntheproem(P)and(P),wherewe set )=,肿1)=丛. Infact,wehave— Lenmm3If"isaweaksolutionto(P,)and"?([0.】],H'())n([0,1], ())n(.】)for?rne>0,where一[0,I]×,then{canfinda>0,such that=(")definedin0一[0.7]×isaweaksolutionto(P),and?c([0,],H'()) nc([0,明,())n,(0). n?lFirst,wehavet 0,(等一舞一)妒)=(,妒),V妒?(口1).(1) Thenbecauseof"?H.(0】)n().We龃nextendthefunctionalby' 一 ()+(,,)=(,),V?脚(.】)n,(.1).(2) ByLemma1ifwesetM=min(,1'chehnwec~tnget7~0suchthat (m?M, where0一[0,明×.ThenbyLemma2,wehave—()welldefinedinn Nextweset一(),where妒?(口).11lenwehave 一 ,)+(Vv,(,))一(,)+(Vv,) = (,(c,,,w),妒), andso 一 (0)(1l一ll,妒)一(矾,)+(,) , NO,3THECAUCHYPROBLEMFORSEM1L1NEARWAVEEQUATIONS425 一 (,(,,,,),),V?(9). By?([O,],())nc.(C0,],())and一(),wehave lf|0一(lo)=0, . I,-0一[()t']一0,), and 一 (()(1I一l训,)一,)+(,) r 一 (,0),)一II0d日, V妒?c([0,],())withl.r一妒l=0. Thenwegetthedesiredresult. Moreover?,vehave Lemr~4Ifisaweaksolutionto(P)andEc([0.],H())nc(C0,], ()),?JL?(口.Thenthefunctiongivenby一0)isal~-.alweaksglu~nto(P)and ?([O,]..())n(C0,],())n尸(口). PrO0fFirstwehave 0,(一若一)) 一 (,(,,z,,),妒)+()(l一ll),妒).V妒?旦,(口). Thenbecauseof?'(0)nL?()we?nextendthefunctionalby 一 ()+(",?) 一 (,')+(()(.一lll,).V??(口)n(.). Ifwese旧一南,thenwehave一 (n,)+(I,妒)一(^.妒),V妒?(0) Fin~lb,by ?([O,'],H())n([O,],()). Wecanhave "l—o—0l0)=0, and I,-0=, where一istheinversemappingof. Thenthe?ffollowsimmediately. ?6.ProofMsmResultandtheGenemlizationtosy甜e吣 F~rstwestateausegtlLer~nnagivenin[1]. 咖m5If?n,()forsome?z,and一P()isafunctjonofhsargu- ment,mP(0)一O,thenwehaveF(v)?JE尸n,'(). Nextwetheproofofthenresult. NORTHEAST.MAT}LJ.VOL10 ProofofTheorem1Infact,wehavethesolu~onto(i-)bytheProposition. NextweeaRehocsoT>O,suchthat=()definedin0一[0,XI~isaweaksolu~on to(P)byLemma3andLemma4. Finally.byLemma5wecangetthedesiredresun. Athast,wegivegeneraliza~onstothesemilinearsystemsa3follows: f(若一舞一霹)一,(,,)+(')(1l一ll), (s)'l0=0, 【:lt.o=0,y)le(z,), whereU一(w,…,),1??N,,andatec抽functionsoftheirarguments,and? c(),and,(,z,.0)=0, C>0,>0; z<0,>0; ,>0,<0; ,<0,<0, where(1?j?d)arecorlstants. Forthiscase.wealsohavethefoUowingresult. Theorem2ThereexistsT>0.andthereisaweaksolutionU=(.,…,)totheprob- lem(s).definedinE0,T]XR2suchthatuec([0,T],H.())lqC(E0,T],())13 L-(E0,]×).M~'eover,U?,'(—)forany?zandk>0.i.e.,Uiseonormalwith respectto27I,,2,?3,,t,,sandisCOOawayfromtheses眦faCes. ThisresulteaRbederivedinthesameway. [1]anShuxing,MI髑曲Rie抛nnpmbt~m妇辨n嘲啪veon3,(帆 两?妇,17(1992),7l5—736. E23,&]l~!zard,,EquationsdeaNonlinmires:DesSoluficps S&l~nalroEaP.-EoolePolyechrdque-1991--1992. Born~m, E33Bi蛐..SomeCl,~ssesofPartialDiffentialEquations,GordonandSdenPul:~sher?S?, 1988. [4]Stra,A.,N0nlinearWaveEquations,cBRegionalO啊1f盯e|1腮s喇嚣inMathematics'No.73' Amer.Math.Soc.,Provideno~,RI,1989. E6]Be山.M..Vecfield~a吕s.dat.dwiththenoMJn~rin啪嘶on0fogr髑ingw.avl~,曲?埘刎k ,37(1988),637--665. [6]Clm,~zaln.J.&I~rrioo,A..Introductiontothe.n啪ryofLinearPartialDiff~tialEaIofIs?North- HolIandPubli~ainsCompany.NewYork-1982. ET]Gu0IacuI:)~ian.CllienShmdns.ZhengSongmu&TYonBji.BII|aI.删0fMathe~mtticalnI州a? ChineseH_咄erFiduaeationP1嘲.Beijing.1986. J?. ,