DISSERTATIONES MATHEMATICAE - Journals IM PAN
DISSERTATIONES MATHEMATICAE - Journals IM PAN
DISSERTATIONES MATHEMATICAE - Journals IM PAN
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POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY<br />
D I S S E R T A T I O N E S<br />
M A T H E M A T I C A E<br />
(ROZPRAWY MATEMATYCZNE)<br />
407<br />
JERZY A. GAWINECKI<br />
Initial-boundaryvalueproblemin<br />
nonlinearhyperbolicthermoelasticity.<br />
Someapplicationsincontinuummechanics<br />
WARSZAWA 2002
CONTENTS<br />
1.Introduction......................................................................... 5<br />
2.Basicnotationandformulae.......................................................... 8<br />
3.Themaintheorem................................................................... 16<br />
4.Energyestimate..................................................................... 18<br />
4.1.Linearizedsystemofhyperbolicthermoelasticity................................. 18<br />
4.2.Energyestimateforthelinearhyperbolicsystem................................. 18<br />
5.ProofofTheorem3.1................................................................ 24<br />
6.Applicationstononlinearmicroelasticitytheory.Formulationofthemaintheorem.... 27<br />
7.Energyestimateforthelinearizedmicroelasticitysystem............................. 29<br />
7.1.Linearizedsystemofmicroelasticitytheory....................................... 29<br />
7.2.Energyestimateforthelinearsystemofmicroelasticitytheory................... 29<br />
8.ProofofTheorem6.1................................................................ 33<br />
9.Applicationtononlinearthermodiffusioninasolidbody.............................. 36<br />
10.Energyestimateforthelinearizedsystemofthermodiffusioninasolidbody.......... 39<br />
10.1.Linearizedsystemofthermodiffusioninasolidbody............................ 39<br />
10.2.Energyestimateforthelinearsystemofthermodiffusioninasolidbody......... 39<br />
10.2.1.Energyestimateforthelinearhyperbolicsystem......................... 39<br />
10.2.2.Energyestimateforthelinearparabolicsystem.......................... 40<br />
11.ProofofTheorem9.1................................................................ 43<br />
12.Generalremarks..................................................................... 47<br />
References............................................................................... 47<br />
2000MathematicsSubjectClassification:35K60,3K05,80A10,80A20,35G20,35G25,35G30,<br />
35L45,35L70.<br />
Keywordsandphrases:nonlinearhyperbolicequations,nonlinearhyperbolic-parabolicsystems<br />
ofequations,initial-boundaryvalueproblem,thermoelasticitytheory,Sobolevspaces,fixed<br />
pointtheorem,nonlinearmicroelasticitytheory,energyestimate,nonlinearthermodifusion<br />
inasolidbody.<br />
Received14.11.2001;revised7.2.2002.
Abstract<br />
Theaimofthispaperistopresentanelementaryself-containedintroductiontosomeimportant<br />
aspectsofthetheoryoflocal(intime)solutionstotheinitial-boundaryvalueproblemfor<br />
nonlinearhyperbolicequationsofthermoelasticitytheory.<br />
TherelevantexistencetheoremisprovedusingtheapproachofKatoviasemigrouptheory<br />
fortheassociatedlinearizedproblem.Next,weproveanenergyestimateinasuitablychosen<br />
Sobolevspaceforthesolutionofthelinearizedproblem,usingstandardregularizationarguments<br />
andenergymethods.Finally,weshowthatthesolutionofournonlinearproblemcanbeobtained<br />
astheuniquefixedpointofacontractionmappinginasuitablefunctionspace.<br />
Theapproachpresentedinthispapercanbeextendedtoothernonlinearsystemsofpartial<br />
differentialequationsdescribingmediaincontinuummechanics.
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