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B28,205–212(2007)NotationsListofSymbolsjjAbsolutevalueorthenormofavectorxCxCDmaxfxI0gdeTheceilingfunction,i.
e.
,dxeDminfm2ZWxmgTheintegerpart,i.
e.
,xDmaxfm2ZWxmghiWhichstandsforhxiDp1Cjxj20suchthatC1gfC2gfgIflimt!
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;/InnerproductinRn.
;/HInnerproductinaHilbertspaceHjjjjBNorminaBanachspaceB,!
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e.
,D@2x1C@2x2CC@2xnF.
f/orOfFouriertransformSpringerInternationalPublishingAG2018M.
R.
Ebert,M.
Reissig,MethodsforPartialDifferentialEquations,https://doi.
org/10.
1007/978-3-319-66456-9475476NotationsFx!
orOu.
t;/PartialFouriertransforminxF1.
f/InverseFouriertransformF1!
xPartialinverseFouriertransforminxfgConvolutionSpacesofFunctionsandDistributionsCk.
Rn/SpaceofktimescontinuouslydifferentiablefunctionsC1.
Rn/SpaceofinnitelycontinuouslydifferentiablefunctionsCk.
I;B/SpaceofktimescontinuouslydifferentiablefunctionsfromanintervalIRintoaBanachspaceBC10.
Rn/SpaceofinnitelycontinuouslydifferentiablefunctionswithcompactsupportS.
Rn/SchwartzspaceofrapidlydecreasingfunctionsLp.
Rn/LebesguespacesWmp.
Rn/SobolevspacesWsp.
Rn/Sobolev-SlobodeckijspacesHs.
Rn/SobolevspacesoffractionalorderbasedonL2.
Rn/Hsp.
Rn/SobolevspacesoffractionalorderorBesselpotentialspacePHsp.
Rn/HomogeneousSobolevspacesoffractionalorderBsp;q.
Rn/BesovspacesPBsp;q.
Rn/HomogeneousBesovspacesFsp;q.
Rn/Triebel-LizorkinspacesPFsp;q.
Rn/HomogeneousTriebel-LizorkinspacesS0.
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Lp!
Lq/IndexBernstein'sinequalities,459Besovspaces,446Besselpotentialspaces,445Burgers'equationblowup,74geometricalblowup,71lifespan,71Cauchy-Kovalevskajatheoremabstractversion,40classicalversion,38characteristiccurve,59classicalsolution,32co-normalboundarycondition,27convolution,426d'Alembert'sformula,119dampedwaveequationdecaybehavioroftheenergy,198decaybehaviorunderadditionalregularityofdata,211diffusionphenomenon,207energy,156localenergies,165well-posedinHs,198withabsorbingnonlinearityglobalexistenceresult,311–313proleofsolutions,315withsourcenonlinearityglobalexistenceresult,300nonexistenceofglobalsolutions,307scale-invariantmodels,320Diracdistribution,452distribution,452distributionalsolution,33,453Duhamel'sprinciple,145elasticwaveequationenergy,151localenergy,163ellipticequation,18second-orderequation,20system,22,23energysolution,33equationKovalevskiantype,17p-evolution,19parabolic,20ultrahyperbolic,20Floquettheory,364Fouriertransformation,423fundamentalsolution,454Gagliardo-Nirenberginequality,456,457Green'sfunction,83harmonicfunction,82heatequationLpLqestimates,174–177Cauchyproblem,107energies,152exteriorDirichletproblem,112SpringerInternationalPublishingAG2018M.
R.
Ebert,M.
Reissig,MethodsforPartialDifferentialEquations,https://doi.
org/10.
1007/978-3-319-66456-9477478IndexexteriorNeumannproblem,112fundamentalsolution,103interiorDirichletproblem,111interiorNeumannproblem,112localenergies,167maximumprinciple,105,110mixedproblem,111potentialtheory,103,113smoothingeffect,109uniqueness,106,111withabsorbingnonlinearityglobalexistenceresult,284proleofsolutions,285,289,290,293withsourcenonlinearityblowupresult,277globalexistenceresult,274proleofsolutions,281self-similarsolutions,275Holmgrentheoremabstractversion,53classicalversion,49Huygens'principle,121hyperbolicequation,18second-orderequation,20strictlyforasystem,22,23strictlyforanequation,18symmetricsystem,23system,22,23hypoellipticoperator,92Kato'slemma,461Kirchhoff'sformula,122Klein-Gordonequationdomainofdependence,159energyconservation,158localenergies,164well-posedinHs,218Laplaceequationboundaryvalueproblems,93exteriorDirichletproblem,95exteriorNeumannproblem,95exteriorRobinproblem,96fundamentalsolution,79interiorDirichletproblem,93interiorNeumannproblem,94interiorRobinproblem,94maximumprinciple,87meanvaluetheorem,87potentialtheory,79,96Lax-Mizohatatheorem,19LebesguespacesLp.
Rn/,445Levicondition,389Lewy'sExample,39Littmantypelemma,245,260Littman'slemma,268locallysolvableoperator,92methodofcharacteristics,57methodofdescent,128methodofstationaryphase,241Mizohataoperator,93modiedBesselfunctions,434Modulationspaces,449multipliers,432non-characteristicCauchyproblem,63non-characteristicsurface,62Parseval-Plancherel'sformula,428plateequationLpLqestimates,266,267decaybehaviorofanenergy,231energiesofhigherorder,232energy,160energyconservation,153well-posedinHs,228withclassicaldamping,228withmass,235withmassanddissipation,236withviscoelasticdamping,232Poissonintegralformula,83Riesz-Thorininterpolationtheorem,455SchrdingerequationLpLqestimates,182,183,265defocusingnonlinearity,368energyconservation,152focusingnonlinearity,368localenergies,169smoothingeffect,185Strichartzestimates,183withpowernonlinearityproleofsolutions,381scalingproperties,368well-posedresultforH1data,376well-posedresultforL2data,370Schwartzspace,424Sobolevsolution,32Index479Sobolevspaces,432,445Sobolev-Slobodeckijspaces,445structurallydampedwaveequationLpLqestimates,326–330,416diffusionphenomenon,416withsourcenonlinearityglobalexistenceresults,330,336,343–348nonexistenceofglobalsolutions,335systemenergyinequalityforsymmetrichyperbolic,396hyperbolicwithconstantcoefcients,386strictlyhyperbolic,389symmetrichyperbolic,398symmetricwithconstantcoefcients,384testfunctionmethod,307,335Triebel-Lizorkinspaces,446viscoelasticdampedwaveequationdecaybehavioroftheenergy,214decaybehaviorunderadditionalregularityofdata,216energy,157localenergies,166well-posedinHs,213waveequationLpLqestimates,256,258Cauchyproblemindimensioneven,130odd,129one,119three,122two,127conservationofenergy,150domainofdependence,121,148nitespeedofpropagation,121inhomogeneous,122localenergy,141,148,162mixedproblem,138potentialtheory,136,139propagationofsingularities,131uniquenessformixedproblems,143uniquenessofclassicalsolutionsfortheCauchyproblem,150well-posedinHs,195withabsorbingnonlinearityenergy,359globalexistenceresult,359withsourcenonlinearityglobalexistenceresults,357Katoexponent,352lifespan,358localexistenceofSobolevsolutions,353nonexistenceofglobalsolutions,355Straussexponent,352wavefrontset,134weaklyhyperbolicsystem,386well-posedCauchyproblem,29Weyl'slemma,89Young'sinequality,456