RESEARCH ON EVOLUTION EQUATIONS COMPENDIUM VOLUME 1 -- RESEARCH ON EVOLUTION EQUATIONS COMPENDIUM VOLUME 1 -- CONTENTS -- PREFACE -- Chapter1DOUBLESCALECONVERGENCEANDHOMOGENIZATIONOFQUASILINEARPARABOLICEQUATIONS -- Abstract -- 1.Introduction -- 2.FormulationoftheProblem -- 3.DoubleScaleConvergenceandtheMainTheorem -- References -- Chapter2SECONDGRADEFLUIDSWITHENHANCEDVISCOSITYASDYNAMICALSYSTEMS -- Abstract -- 1.Introduction -- 2.SettingoftheProblem -- 3.ExistenceoftheGlobalAttractor -- 4.ExistenceofExponentialAttractors -- Acknowledgments -- References -- Chapter3PERIODICSOLUTIONSOFSOMEEVOLUTIONEQUATIONSWITHINFINITEDELAY∗ -- Abstract -- 1.Introduction -- 2.SolutionsandTheirEstimations -- 3.ExistenceofPeriodicSolutions -- 4.AMasseraTypeTheorem -- References -- Chapter4INTERNALPOLLUTIONANDDISCRIMINATINGSENTINELINPOPULATIONDYNAMICSPROBLEM -- Abstract -- 1.Introduction -- 1.1.PopulationDynamicsProblemwithMissingData -- 1.2.EquivalencetoNull-ControllabilitywithConstraintsontheControl -- 1.3.ControllabilityProblem -- 2.Null-ControllabilitywithConstraintsontheControl -- 2.1.ObservabilityInequalityAdaptedtoConstraints -- 2.2.ExistenceofOptimalControlVariable -- 3.OptimalitySystemfortheOptimalControlVariable -- 4.DiscriminatingSentinels -- 5.Conclusion -- Aknowledgments -- References -- Chapter5CENTERMANIFOLDANDSTABILITYINCRITICALCASESFORSOMEPARTIALFUNCTIONALDIFFERENTIALEQUATIONS∗ -- Abstract -- 1.Introduction -- 2.SpectralAnalysisandVariationofConstantsFormula -- 3.GlobalExistenceoftheCenterManifold -- 4.AttractivenessoftheCenterManifold -- 5.FlowontheCenterManifold -- 6.StabilityinCriticalCases -- 7.LocalExistenceoftheCenterManifold -- 8.Application -- Acknowledgments -- References -- Chapter6STABILITYRADIIOFPOSITIVELINEARFUNCTIONALDIFFERENTIALSYSTEMSINBANACHSPACES -- Abstract -- 1.Introduction -- 2.Preliminaries.

3.StabilityRadiiofLinearAbstractFunctionalDifferentialEquationsunderMulti-perturbations -- 3.1.AbstractFunctionalDifferentialEquations -- 3.2.StabilityRadii -- 4.StabilityRadiiofPositiveLinearFunctionalDifferentialEquationsunderMulti-affinePerturbations -- References -- Chapter7EFFECTSOFNONLOCALITYANDPHASESHIFTDEFINITIONSINGENERALIZINGLEVINSON'STHEOREM -- Abstract -- 1.Introduction -- 2.MethodofApproach -- 3.ExtensiontoNonlocalPotentials -- 4.PropertiesofD+(k),D(k),andL+(k)foraNonlocalPoten-tial -- 5.Solutionsatk=0 -- 6.DerivationoftheGeneralizedLevinson'sTheoremforthePhaseShift -- 6.1.CaseofnBoundStatesandOneHalf-BoundState -- 6.2.CaseofnBoundStatesandmContinuumBoundStates -- 6.3.CaseofnBoundStatesandpSpuriousStates -- 6.4.Summary -- 7.DerivationoftheGeneralizedLevinson'sTheoremforthePhaseShift -- 7.1.CaseofnBoundStatesandOneHalf-BoundState -- 7.2.CaseofnBoundStatesandmContinuumBoundStates -- 7.3.CaseofnBoundStatesandpSpuriousStates -- 7.4.Summary -- 8.Examples -- 8.1.YamaguchiPotential -- 8.1.1.PhaseShiftCalculationUsingδL -- 8.1.2.PhaseShiftCalculationUsingδD -- 8.2.BeregiPotential -- 8.2.1.PhaseShiftCalculationUsingδL(k) -- 8.2.2.PhaseShiftCalculationUsingδL -- 9.SummaryandConclusion -- References -- Chapter8ASYMPTOTICBEHAVIOROFTHEFITZHUGH-NAGUMOSYSTEM -- Abstract -- 1.Introduction -- 2.MainResults -- 3.ASemigroupAssociatedwithProblem(2.1)-(2.3) -- 4.UniformEstimatesinTime -- 5.ExistenceofGlobalAttractors -- 6.UniformBoundsofGlobalAttractorsin -- 7.CompactnessoftheUnionofGlobalAttractors -- 8.UpperSemicontinuityofGlobalAttractors -- References -- Chapter9APARTIALDIFFERENTIALEQUATIONWITHNONAUTONOMOUSPASTDELAYINL1-PHASESPACE -- Abstract -- 1.Introduction -- 2.Preliminaries -- 3.WellposednessofPartialDifferentialEquationswithNonau-tonomousPastDelay -- 4.AsymptoticBehaviourofPartialDifferentialEquationswithNonautonomousPastDelay -- 5.Application.

5.1.ThePopulationEquationasanAbstractEquationwithNonautonomousPastDelay -- 5.2.WellposednessandAsymptoticBehaviorofthePopulationEquation -- References -- Chapter10SOLVINGTHEHYPERBOLICPROBLEMOBTAINEDBYTRANSMUTATIONOPERATOR -- Abstract -- 1.Introduction -- 2.ExistenceoftheTransmutationOperator -- 3.RiemannMethod -- References -- Chapter11ELLIPTICOPERATORSWITHVARIABLECOEFFICIENTSGENERATINGFRACTIONALRESOLVENTFAMILIES -- Abstract -- Introduction -- 1.PreliminariesonFractionalResolventFamilies -- 2.EllipticDifferentialOperatorsonL2 -- References -- Chapter12EXISTENCERESULTSFORPSEUDOALMOSTPERIODICDIFFERENTIAL,FUNCTIONAL,ANDNEUTRALINTEGRALEQUATIONS -- Abstract -- 1.Introduction -- 2.Preliminaries -- 2.1.PseudoAlmostPeriodicFunctions -- 2.2.TheZima'sFixed-PointTheorem -- 3.PseudoAlmostPeriodicSolutions -- 3.1.PseudoAlmostPeriodicSolutionsToEq.(1.1) -- 3.2.PseudoAlmostPeriodicSolutionstoEq.(1.2) -- 4.PseudoAlmostPeriodicSolutionsToSomePartialEvolutionEquations -- 4.1.PseudoAlmostPeriodicSolutionstoEq.(1.3) -- 4.2.PseudoAlmostPeriodicSolutionstoEq.(1.4) -- 5.CaseoftheLogisticEquationwithDelay -- References -- Chapter13ANASYMPTOTICMODELOFANONLINEARKELVIN-VOIGTVISCOELASTICPLATE∗ -- Abstract -- 1.Introduction -- 2.Set-upoftheProblem -- 3.ScalingsofUnknowns -- References -- Chapter14MATHEMATICALANALYSISOFABILATERALOBSTACLEPROBLEMFORACLASSOFSECOND-ORDEROPERATORS -- Abstract -- 1.Introduction -- 1.1.MathematicalFramework -- 1.2.MainNotationsandAssumptionsonData -- 1.3.TwoConceptsofWeakSolutions -- 2.TheUniquenessTheorem -- 3.TheViscosityandPenalizationMethods -- 3.1.SomeaprioriEstimates -- 3.2.ConvergencetowardanEntropyProcessSolution -- References -- Chapter15APPLICATIONOFTHEPICARDOPERATORSTOSECONDORDERODE'S -- Abstract -- 1.Introduction -- 2.Preliminaries -- 3.TheMainResult -- 4.Application -- References.

Chapter16DOUBLYNONLINEARDEGENERATEPARABOLICEQUATIONSONCARNOTGROUPS∗ -- Abstract -- 1.Introduction -- 2.NotationandPreliminaryResult -- 3.NonexistenceResults -- References -- Chapter17SOMESCALARCONSERVATIONLAWSWITHDISCONTINUOUSFLUX -- Abstract -- 1.Introduction -- 2.DefinitionofanEntropySolution -- 3.ConditionsattheInterface{x0=0} -- 4.TheUniquenessTheorem -- 5.ExistenceofanEntropySolution -- 5.1.FirstStep:u0∈C∞c( ) -- 5.2.SecondStep:u0∈L∞( ) -- 6.Generalisation -- References -- Chapter18ANEWEXACTSOLUTIONTOTHEDELAYEDDIFFUSIONEQUATION -- Abstract -- 1.Introduction -- 2.MathematicalAnalysis -- 2.1.ProblemFormulation -- 2.2.DerivationofaKnownSolution -- 2.3.CaseofZeroDelay:DiffusionEquationSolution -- 3.SolutioninTermsoftheLambertW-function -- 3.1.TheInversionIntegral -- 3.2.ExactTimeDomainSolution -- 3.3.AnalyticalResults -- 4.NumericalResults -- 5.Summary -- References -- Chapter19LEVELSETSFORREACTIONDIFFUSIONEQUATIONS -- Abstract -- 1.Introduction -- 2.TheOneDimensionalCase -- 3.Then-DimensionalCase -- References -- Chapter20ONTHEALMOSTPERIODICITYOFTHESUPERPOSITIONOFFUNCTIONS -- Abstract -- 1.Introduction -- 2.Preliminaries -- 3.MainResults -- Acknowledgment -- References -- Chapter21TIMEPERIODICSOLUTIONSFORQUASIGEOSTROPHICMOTIONANDTHEIRSTABILITY -- Abstract -- 1.Introduction -- 2.TheMainResults -- 3.Conclusion -- References -- Chapter 22 JACOBIAN FEEDBACK LOOPS ANALYSIS II:STABILITY AND INSTABILITY -- 1. Introduction -- 1.1. Jacobian Loops: De nitions and Notations. -- 2. Loops and Jacobian Spectrum -- 3. Loop Stability -- 3.1. Loop Analysis in the Plane. -- 3.2. Loop Analysis for 3-dimensional System. -- 3.3. The Lorenz System. -- 3.4. Rossler System. -- References -- Chapter23EXISTENCEOFOSCILLATINGSOLUTIONFORNONLINEARSTATE-DEPENDENTDELAYDIFFERENTIALEQUATION∗ -- Abstract -- 1.Introduction -- 2.ProofofTheorems1.1and1.2 -- 3.ProofofTheorems1.3and1.4.

4.Examples -- References -- Chapter24SEMILINEARABSTRACTDIFFERENTIALEQUATIONSWITHDEVIATEDARGUMENT -- Abstract -- 1.Introduction -- 2.BasicResults -- References -- Chapter25SINGULARSOLUTIONSOFASEMI-LINEARELLIPTICEQUATIONONNONSMOOTHDOMAINS -- Abstract -- 1.Introduction -- 2.PreliminariesandKnownResults -- 3.ProofofTheorem1.1. -- 4.TheClassesKloc(D)andK(D). -- Acknowledgment -- References -- Chapter26EXISTENCEOFWEIGHTEDPSEUDOALMOSTPERIODICSOLUTIONSTOSOMENON-AUTONOMOUSDIFFERENTIALEQUATIONS -- Abstract -- 1.Introduction -- 2.Preliminaries -- 3.PropertiesofWeightedPseudoAlmostPeriodicFunctions -- 4.WeightedPseudoAlmostPeriodicSolutions -- 5.Example -- References -- Chapter27AKRASNOSELSKII-TYPEFIXEDPOINTTHEOREMFORMULTIFUNCTIONSDEFINEDONAHYPERCONVEXSPACE -- Abstract -- 1.Introduction -- 2.Preliminaries -- 3.MainResult -- 4.Remarks -- References -- INDEX -- Blank Page.