| Foreword |
| Introduction |
| Notation |
| | Physical notation |
| | Mathematical notation |
| Part 1. Wave Equations |
| Chapter 1. | Pseudo-Hyperbolic Model Equations |
| | 1.1 Equations of Internal Waves in Fluids |
| | 1.2 Equations of Ion-Sound Waves in Plasma |
| Chapter 2. | Blow-Up of Solutions of Pseudo-Hyperbolic Equations |
| | 2.1 Blow-Up of Internal Gravitational Waves |
| | 2.2 Blow-Up of Ion-Sound Waves in Plasma |
| | 2.3 Blow-Up of Ion-Sound Waves in Plasma with Strong Spatial-Time Dispersion |
| Chapter 3. | Nonlinear Dynamical Boundary Conditions |
| | 3.1 Blow-Up of Solutions of the Equation of Ion-Sound Waves with Nonlinear Dynamical Boundary Condition |
| | 3.2 Blow-Up of Solutions of the Equation of Gravitational-Gyroscopic Waves with Nonlinear Boundary Conditions |
| Chapter 4. | Model Systems of Pseudo-Hyperbolic Equations |
| | 4.1 System of Equations of Ion-Sound Waves in Plasma |
| | 4.2 Hydrodynamic System of Oskolkov Equations |
| Part 2. Nonlocal Equations |
| Chapter 5. | Model Nonlinear Nonlocal Equations of Sobolev Type |
| | 5.1 Equations of Quasi-Stationary Fields in Crystalline Semiconductors |
| | 5.2 Model Equations |
| | 5.3 Model Equations for the Function h(t) |
| Chapter 6. | Blow-Up of Solutions of Nonlocal Sobolev Equations |
| | 6.1 Blow-Up of Solutions of an Initial-Boundary-Value Problem for the Nonlinear Nonlocal Benjamin–Bona–Mahony–Burgers Equation with Sources |
| | 6.2 Blow-Up of Solutions of the Nonlinear Nonlocal Benjamin–Bona–Mahony–Burgers Wave Equation with a Nonlocal Source |
| | 6.3 Blow-Up of Solutions of Nonlocal Dissipative Rosenau–Burgers Equation with a Source |
| | 6.4 Blow-Up of Solutions of the Nonlinear Nonlocal Equation of Spin Waves with a Source |
| Chapter 7. | Blow-Up of Solutions of Nonlocal Sobolev Systems |
| | 7.1 Blow-Up of Solutions of One System of Nonlocal Equations with Sources |
| | 7.2 Blow-Up of Solutions of the Nonlinear Nonlocal Oskolkov System with a Source |
| Chapter 8. | Blow-Up of Solutions of the Abstract Cauchy Problem |
| | 8.1 Preliminary conditions |
| | 8.2 Auxiliary results |
| | 8.3 Local Solvability in the Strong Generalized Sense |
| | 8.4 Blow-Up of Strong Generalized Solutions |
| Chapter 9. | Blow-Up of Solutions of Problems with Nonlinear Boundary Conditions |
| | 9.1 Blow-Up of Solutions of One Problem with Nonlinear Neumann Boundary condition |
| | 9.2 Blow-Up in the Problem with Nonlinear Evolution Nonlocal Boundary Condition |
| Part 3. Wave and Nonlocal Equations |
| Chapter 10. | Model Nonlinear Nonlocal Wave Equations of Sobolev Type |
| | 10.1 General Systems of Equations of Quasi-Stationary Fields |
| | 10.2 Time Dispersion |
| | 10.3 Spatial Dispersion |
| | 10.4 Nonlinear Factors |
| | 10.5 Model Integro-Differential Equations and Sobolev-Type Equations |
| Chapter 11. | Blow-Up of Solutions of Wave Integro-Differential Equations of Sobolev Type |
| | 11.1 Blow-Up of Solutions of One Equation of Third Order |
| | 11.2 Blow-up of Solutions of One Nonlocal Equation of Fifth Order |
| | 11.3 Equations of a Tunnel Diode |
| Appendix A. | Some Results Of Nonlinear Analysis |
| | A.1 | Sobolev Spaces Ws,p(Omega), Ws,p0(Omega), and Ws,p(Gamma) |
| | A.2 | Weak and \ast -Weak Convergence |
| | A.3 | Chain Rule for Frechet Derivatives |
| | A.4 | Caratheodori Functions. Nemytsky Operators. Krasnosel'sky Theorem |
| | A.5 | Compact Continuous Operators and Completely Continuous Operators |
| | A.6 | Compactness Lemma of J.-L.Lions |
| | A.7 | Browder–Minty Theorem |
| | A.8 | On the System of Ordinary Differential Equations in the Galerkin Method |
| | A.9 | Two Equivalent Definitions of a Weak Solution in the Sense of L2(0, T; B) |
| | A.10 | Basic Integro-Differential Inequalities |
| | A.11 | Auxiliary Lemmas about Homogeneous Functionals |
| | A.12 | Dense Embeddings of Banach Spaces |
| | A.13 | Frechet Derivative of the p-Laplacian |
| Appendix |
| Index |
| Appendix |
| Bibliography |
Korpusov M.O. Maksim Olegovich Korpusov received a mathematical physics degree and a Ph.D. degree in mathematical physics from the Physics Department of Moscow State University, Russia, in 1998 and 2005, respectively. Since 2008, he has been a Full Professor at the Mathematical Division of Moscow State University, where he is heading a research team working in the area of nonlinear functional analysis and its applications. He received the 1998, 2004, 2006 Moscow State University Award for young mathematicians and the President Grant for Young Mathematicians (2005, 2007, 2009) (Russia). He is currently a Member of the Editorial Board of Differential Equations and Applications (DEA, Croatia).
Ovchinnikov A.V.