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 W^{s,p}(Omega), W^{s,p}_{0}(Omega), and W^{s,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 L^{2}(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 |