Fractional Diffusion and Wave Equations : Well-Posedness and Inverse Problems

• Autore: Zhou Yong
• Pubblicazione: Cham : , : Springer, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (355 pages)
• Nota di contenuto: Intro -- Preface -- Introduction -- Contents -- 1 Preliminaries -- 1.1 Fractional Calculus -- 1.1.1 Definitions -- 1.1.2 Properties -- 1.2 Some Results from Analysis -- 1.2.1 Mittag-Leffler Function -- 1.2.2 Laplace and Fourier Transforms -- 1.3 Semigroups -- 1.3.1 C0-Semigroup -- 1.3.2 Analytic Semigroup -- 1.3.3 Integrated Semigroup -- 2 Well-Posedness of Fractional Diffusion Equations -- 2.1 Diffusion Equation with Exponential Growth -- 2.1.1 Introduction -- 2.1.2 Orlicz Spaces and Space-Time Estimates -- 2.1.3 Global Existence -- 2.1.4 Local Existence -- 2.2 Distributed Order Diffusion Problems -- 2.2.1 Introduction -- 2.2.2 Preliminaries -- 2.2.3 Technical Tools -- 2.2.4 Well-Posedness and Decay of Solutions -- 2.3 Space-Time Fractional Diffusion Equations -- 2.3.1 Introduction -- 2.3.2 Preliminaries -- 2.3.3 Technical Tools -- 2.3.4 Well-Posedness Results -- 2.3.5 Approximate Controllability Analysis -- 3 Inverse Problems of Fractional Diffusion Equations -- 3.1 Backward Problem -- 3.1.1 Introduction -- 3.1.2 Preliminaries -- 3.1.3 Existence and Regularity -- 3.1.4 Example -- 3.2 Terminal Value Problem -- 3.2.1 Introduction -- 3.2.2 Notations and Preliminaries -- 3.2.2.1 Functional Space -- 3.2.2.2 Mild Solutions of FVP and Unboundedness of Solution Operators -- 3.2.3 Final Value Problem with a Linear Source -- 3.2.4 Final Value Problem with a Nonlinear Source -- 3.2.5 Existence -- 4 Well-Posedness and Regularity of Fractional Wave Equations -- 4.1 Damped Wave Equations -- 4.1.1 Introduction -- 4.1.2 Preliminary Results -- 4.1.3 Linear Problems -- 4.1.3.1 Solution Representation Formula -- 4.1.3.2 Existence and Regularity -- 4.1.4 Nonlinear Problems -- 4.1.4.1 Well-Posedness -- 4.1.4.2 Continuation and Blow-Up Alternative -- 4.1.4.3 Compactness Method -- 4.1.5 An Application -- 4.2 Wave Equations on RN -- 4.2.1 Introduction. ; 4.2.2 Preliminaries -- 4.2.3 Local/Global Solutions of Linear Problems -- 4.2.3.1 Solution Representation -- 4.2.3.2 Some Properties of Solution Operators -- 4.2.3.3 The Existence Results -- 4.2.4 Results of Semilinear Problems -- 4.3 Wave Equations with Exponential Nonlinearity -- 4.3.1 Introduction -- 4.3.2 Preliminaries -- 4.3.3 Existence Analysis -- 4.3.3.1 Local Existence of Solutions -- 4.3.3.2 Global Existence of Solutions -- 5 Inverse Problems of Fractional Wave Equations -- 5.1 Backward Problem -- 5.1.1 Introduction -- 5.1.2 Preliminaries -- 5.1.3 Solution Representation -- 5.1.3.1 Definition of Mild Solution -- 5.1.3.2 Some Properties -- 5.1.4 Existence and Uniqueness -- 5.1.5 Regularization -- 5.2 Initial Inverse Problem -- 5.2.1 Introduction -- 5.2.2 Preliminaries -- 5.2.3 Regularization Method -- 5.2.4 Convergence Analysis and Error Estimate -- 5.3 Terminal Value Problem -- 5.3.1 Introduction -- 5.3.1.1 Statement of the Problem -- 5.3.1.2 Motivations -- 5.3.2 Preliminaries -- 5.3.3 Existence and Regularity -- 5.3.3.1 Mild Solutions -- 5.3.3.2 Well-Posedness of the Problem (5.55) and (5.56) Under Globally Lipschitz Case -- 5.3.3.3 Well-Posedness of the Problem (5.55)-(5.56) Under Critical Nonlinearities Case -- 5.3.4 Applications -- 5.3.4.1 Time Fractional Ginzburg-Landau Equation -- 5.3.4.2 Time Fractional Burgers Equation -- 5.3.5 Proof of Theorems -- 5.3.5.1 Proof of Theorem 5.12 -- 5.3.5.2 Proofs of Theorem 5.13 -- 5.3.5.3 Proof of Theorem 5.14 -- Appendix -- (AP.) List of Important Constants -- (AP.) List of Important Constants -- (AP.1) A Singular Integral -- (AP.2) A Useful Limit -- References -- Index.
• Titolo autorizzato: Fractional Diffusion and Wave Equations
• ISBN: 3-031-74031-9
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910903786703321
• Lo trovi qui: Univ. Federico II