Partial differential equations and systems not solvable with respect to the highest-order derivative

Content: 1. Spaces C, C[superscript lambda], L[subscript p] H[superscript r subscript p] 1 --
2. Averages 5 --
3. Fourier Transform 7 --
4. Multipliers 13 --
5. Laplace Transform 18 --
6. Integral Representation of Functions 25 --
7. Weak Derivatives 29 --
8. Sobolev Spaces 33 --
9. Boundary-Value Problems for Ordinary Differential Equations on the Half-Axis 38 --
10. Boundary-Value Problems for Systems of Differential Equations on the Half-Axis 43 --
Chapter 2. The Cauchy Problem for Equations not Solved Relative to the Higher-Order Derivative 47 --
1. Problems Leading to Sobolev-Type Equations 48 --
2. Classes of Equations not Solved Relative to Higher-Order Derivative 50 --
3. Equations with Invertible Operator at the Higher-Order Derivative 54 --
4. Sobolev-Type Equations without Lower-Order Terms 66 --
5. Approximate Solutions to the Cauchy Problem for Equations without Lower-Order Terms 75 --
6. Estimates for Approximate Solutions 81 --
7. Existence and Uniqueness of a Solution to the Cauchy Problem for Equations without Lower-Order Terms 98 --
8. Equations with Variable Coefficients 108 --
9. Pseudohyperbolic Equations 119 --
10. Applications of Sobolev-Type Equations to the Solution of a Hyperbolic System 136 --
Chapter 3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems 151 --
1. Examples of non-Cauchy-Kovalevskaya Type Systems 152 --
2. Classes of non-Cauchy-Kovalevskaya Type Systems 155 --
3. The Cauchy Problem for Sobolev-Type Systems 159 --
4. Approximate Solutions to Sobolev-Type Systems 169 --
5. Estimates for Approximate Solutions to Sobolev-Type Systems 172 --
6. Solvability of the Cauchy Problem for Sobolev-Type Systems 181 --
7. Solvability of the Cauchy Problem for Pseudoparabolic Systems 193 --
8. Parabolic Systems 202 --
9. Approximate Solutions to Pseudoparabolic Systems 207 --
10. Estimates for Approximate Solutions to Pseudoparabolic Systems 209 --
11. Solvability of the Cauchy Problem for Pseudoparabolic Systems 219 --
12. The Cauchy Problem for Pseudoparabolic Systems with Lower-Order Terms 229 --
13. Pseudoparabolic Systems with Variable Coefficients 233 --
Chapter 4. Mixed Problems in the Quarter of the Space 237 --
1. Statement of the Mixed Boundary-Value Problems for Simple Sobolev-Type Equations 237 --
2. Solvability of Mixed Problems for Simple Sobolev-Type Equations 240 --
3. Approximate Solutions to Simple Sobolev-Type Equations 244 --
4. Properties of the Contour Integrals 248 --
5. Estimates for Approximate Solutions to Nonhomogeneous Simple Sobolev-Type Equations 253 --
6. Estimates for Approximate Solutions to Homogeneous Simple Sobolev-Type Equations 267 --
7. Solvability of Mixed Problems 281 --
8. Necessary Solvability Conditions for Mixed Problems 289 --
9. Mixed Boundary-Value Problems for Pseudoparabolic Equations 295 --
10. Sketch of Proof of Solvability of Mixed Problems for Pseudoparabolic Equations 299 --
11. Statements of the Mixed Boundary-Value Problems for Sobolev-Type Systems 306 --
12. Approximate Solutions to the Mixed Problems for Sobolev-Type Systems 321 --
13. Solvability of Mixed Problems for Sobolev-Type Systems 331 --
14. Mixed Boundary-Value Problems for Pseudoparabolic Systems 358 --
15. Approximate Solutions to Mixed Problems for Pseudoparabolic Systems 363 --
16. Convergence of Approximate Solutions 371 --
Chapter 5. Qualitative Properties of Solutions to Sobolev-Type Equations 381 --
1. Sobolev-Wiener Spaces 381 --
2. Mixed Problems for Sobolev-Type Equations in Cylindrical Domains 387 --
3. Properties of Solutions to the First Boundary-Value Problem for the Sobolev Equation 394 --
4. Algebraic Moments of Solutions to the First Boundary-Value Problem for the Sobolev Equation 401 --
5. Asymptotic Behavior of Solutions to Some Problems in Hydrodynamics 412 --
Appendix On a New Problem in Mathematical Physics / S.L. Sobolev 437.