Study of The Existence and Blow-up For a Class of Nonlinear Damped Wave Equation

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.