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

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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