Study of The Existence and Blow-up For a Class of Nonlinear Damped Wave Equation
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Date
2022
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Abstract
Thisdissertationisdevotedtostudythewell-posednessandtheblow-upofsolutionsofsome
nonlinearhyperbolicproblemsinvolvingnon-classicalnonlinearities.Weprovedundersuitable
assumptionsontheexponentsofnonlinearitythelocal,globalexistenceandestablishtheresults
ofblow-upofsomewaveequations.
Itseemsthatthesourceterminhibitstheglobalexistence(intime)ofthesolutionofthe
problemistosaythattheenergyoftheproblem(orsolution)tendstoin nityforthenormof
spacewhenttendstoa nitetimeT.Obviously,thedampingtermstabilizesthesolutionofthe
problem,anditisclearthatintheabsenceofsourceterms,ifthesolutionexistslocally,wecan
alwaysexpanditintoaglobalsolution.Thisinteractionbetweensourceanddampingtermshas
beenatargetinmanystudiesandisstills-Itisimportantalsotoknowwhichtermisdominant
totheother-.
Wecansaythatourresearchisanexpansionofsomeresultsdonebypreviousresearchers.
Mainly,bymakingappropriatemodi cations,weextendedsomeknownresultsofsomenonlinear
waveequationswithconstantandvariable-exponentnonlinearitiesstudiedbyMessaoudi,and
exploitideasbyGeorgievandTodorova.
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Keywords
Blow-up,globalexistence,sourceterm,waveequation,viscosity, negativeinitialenergy,variableexponents,positiveinitialenergy,existenceanduniqueness, Faedo-Galerkin