一类非线性粘弹性波动方程声学边界问题的研究.pdf

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5、ut

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7、u−∆utt+g(t−s)∆u(x,s)ds=

8、u

9、u(x,t)∈Ω×(0,∞),(1.1)0u=0(x,t)∈Γ1×(0,∞),(1.2)Z∂u∂ut∂u(s)tt′+−g(t−s)ds=y(x,t)∈Γ0×(0,∞),(1.3)∂ν∂ν0∂ν′ut+p(x)y+q(x)y=0(x,t)∈Γ0×(0,∞),(1.4)u(x,0)=u0(x),ut(x,0)=u1(x)x∈Ω.(1.5)!&'pgp,yy*asBZs.pB
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12、ut

13、utudx+∇ut∇udx+uydΓ+p(x)(y)dΓ,ρ+1ΩΩΓ02Γ0ZZt1ρΦ(t)=−

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15、utg(t−s)(u(t)−u(s))dsdxρ+1Ω0ZZt−∇utg(t−s)(∇u(t)−∇u(s))dsdx.Ω0Æ:bt/qC;Faedo-Galerkin
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16、;t;E)#d.i℄S}%}6AbstractThispaperisconcernedwiththeexistence,theexponentialdecayfortheso-lutionofthefollowingnonlinearviscoelasticwaveequationwithacousticbound-aryconditions,Ztργ

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18、utt−∆u−∆utt+g(t−s)∆u(x,s)ds=

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20、u(x,t)∈Ω×(0,∞),(1.1)0u=0(x,t)∈Γ1×(0,∞),(1.2)Zt∂u∂utt∂u(s)

21、′+−g(t−s)ds=y(x,t)∈Γ0×(0,∞),(1.3)∂ν∂ν0∂ν′ut+p(x)y+q(x)y=0(x,t)∈Γ0×(0,∞),(1.4)u(x,0)=u0(x),ut(x,0)=u1(x)x∈Ω.(1.5)Thisdissertationisdividedintofoursections.Inthefirstsection,weintroducetheimportanceandtheinternationalre-searchprogressofthenonlinearviscoelasticwaveequation,wealso

22、statethehypothesesfortheproblem.Inthesecondsection,welistsomepreliminariessuchastheSobolevimbed-dingtheorem,severalkindsofimportantinequalitiesandsoon.Inthethirdsection,weprovetheexistenceofthesolutiontotheproblem(1.1)-(1.5).TheproofoftheexistenceincludesFaedo-Galer