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时间:2020-05-23
《具有摩擦和记忆性的非线性波动方程一致能量衰减估计.pdf》由会员上传分享,免费在线阅读,更多相关内容在应用文档-天天文库。
1、第39卷第2期曲阜师范大学学报V0l_39No.22013年4月JournalofQufuNormalUniversityApr.2013DOI:10.3969/j.issn.1001-5337.2013.2.010UniformEnergyDecay‘‘一一T31‘stimationforNonlinearWaveEquationwithFrictionandMemoryyZHAOCui—ling,LIFu—shan(SchoolofMathematicalSciences,QufuNormal
2、University,273165,Qufu,Shangdong,PRC)Abstract:Inthispaperweconsideraclassofnonlinearviscoelasticwaveequationu“一K0Au+fg(一5)div[a(x)VM(s)]ds+b()h(M)=,(u),(,)∈12×(0,。。)J0withinitial—boundaryconditions.Underweakerassumptionsonthefunctionsg,handf,theunifor
3、menergydecayateprovedbyintroducingverysimpleLyapunovfunctionalandpreciseprioriestimates.Keywords:exponentialdecay;polynomialdecay;relaxationfunction;memoryCLCnumber:O175.27;O175.29Documentcode:AArticleID:1001—5337(2013)02-0033—101IntroductionLetQbeabo
4、undeddomainofRwithasmoothboundaryn.UniformstabilityforthewaveequationM一Au+b()h(/z。)=0,(,t)∈r×(0,∞)hasbeenstudiedforalongtimebymanyauthors.Forexample.NakaoIl7]andZuazua[]establishedtheuniformdecayofsolutionsprovidedthatthefunctionb()ispositiveinthewhol
5、edomain.Inthesamedirection,Bardos,Lebeau,andRauch⋯,basedonmicrolocalanalysis,ensuredanecessaryandsufficientconditiontoobtainexpo-nentialdecay;namely,thedampingregionsatisfiesthewell-knowngeometrycontrolcondition.Later,NakaoL18Jex—tendedtheresultsofZua
6、zua,treatingfirstthecaseofalineardegenerateequationandthecaseofanonlineardissipationP(,s),assumingthatthefunctionPhasapolynomialgrowthneartheorigin.Martinez15]improvedthepreviousresultsmentionedaboveinwhatconcernedthelinearwaveequationsubjecttoanonlin
7、eardissipationP(,s),avoidingthepolynomialgrowthofthefunctionP(,s)inzero.ItisimportanttomentionthatLasieckaandTatatustudiedthenonlinearwaveequationsubjecttoanonlinearfeedbackactingonapartoftheboundaryofthesystem,andtheywerethefirsttoprovethattheenergyd
8、ecaystozeroasfastasthesolutionofsomeassociat—eddiferentialequationandwithoutassumingthatthe~edbackhasapolynomialgrowthinzero,althoughnodecayratehasbeenshowninthegeneralcase.Largenumbersofmodelsinviscoelasticmaterialsareviscoelasticpartialdiffe
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