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1、AppliedMathematicalSciences,Vol.2,2008,no.60,2963-2972OnaNonlinearSystemA.O.Marinho1,M.R.Clark2andO.A.Lima3AbstractInthisworkwewillprovethatexistsonlyaweaksolutionthemixedproblemassociatedtothenonlinearsystemu+Δ2u−M(u2)Δu+
2、u
3、ρu+θ=fθ−Δθ+u=g,whereMisarealfunction,ρisapositiverealnumber,fa
4、ndgareknowrealfunctions.MathematicalSubjectClassification:74H45Keywords:Mixedproblem;nonlinearsystem,weakglobalsolutions1IntroductionInthisworkweconsiderthemixedsystemu+Δ2u−M(u2)Δu+
5、u
6、ρu+θ=finQθ−Δθ+u=ginQ∂u(1)u==θ=0onΣ∂ηu(x,0)=u(x);θ(x,0)=θ(x)andu(x,0)=u(x)inΩ,001whereΩisanonempty
7、openboundedsetofRn,forn≥1,withboundaryΓsmooth,QisthecylinderΩ×(0,T)ofRn+1forT>0,
8、∇u(x,t)
9、isthenorminRnofthevector∇(x,t)andΔu(x,t)istheusualLaplaceoperatorin1UFPI-DM-CMRV-SupportedPartiallybyCnpq-UFRJ-Brasil,alexmaiver@hotmail.com2UFPI-DM,Teresina-Pi-Brazil,mclark@ufpi.br3UEPB,DME,C.Grande-PB-Brazi
10、l,osmundo@hs24.com.br2964A.O.Marinho,M.R.ClarkandO.A.Liman∂Rofthefunctionu(x,t).Wedenoteby=,Σ=Γ×(0,T)isthelateral∂tboundary.Ourgoalinthisarticleistostudytheexistenceofglobalweaksolutionsofproblem(1)withinitialconditionsu∈H2(Ω),uandθ∈L2(Ω),andalso,0010theuniquenessofthesolutions.Thedynamicalpartof
11、theabovesystemwhenθ=0isanonlinearperturbationofthebeamequation,thathasbeenextensivelystudiedbyseveralauthorsindifferentphysical-mathematicalcontexts.Amongthen,wecitethefollowingrelatedworks:Ball[1][2],Biber[3],Brito[4],Pereira[9]andMedeiros[8].MorerecentlywecancitetheworksLimacoetal[5],presentedint
12、he56o¯and57o¯SBArespectively.2NotationandmainresultForthefunctionalspacesweshalluse,throughoutthispaper,thestandardnotationofthefunctionalspacesused,forinstance,inthebooksofLions[6]orMedeiros-MillaMiranda[7].Inthissectionweshallassumethefollowinghypothesis:M(λ)isaC0realfunctionsatifyingM(λ)≥−β,0<β
13、<λ1,whereλ1isthefirstauto-valueoftheSpectralProblem:2(Δu,Δv)=λ(u,v)∀v∈H0(Ω).20<ρifn=1,2and0<ρ≤ifn≥3.n−2Definition2.1Wesaythatthepairoffunctions{u(x,t),θ(x,t)}issolutionoftheproblem(1)if∞2u∈L(0,T;H0(Ω));∞2u∈L(0,T
