This book presents several recent developments in the theory of hyperbolic equations. The carefully selected invited and peer-reviewed contributions deal with questions of low regularity, critical growth, ill-posedness, decay estimates for solutions of different non-linear hyperbolic models, and introduce new approaches based on microlocal methods.
Author(s): Michael Reissig, Bert-Wolfgang Schulze
Series: Operator Theory: Advances and Applications Advances in Partial Differential Equations 159
Edition: 1
Publisher: Birkhauser
Year: 2005
Language: English
Pages: 301
Contents......Page 6
Editorial Preface......Page 11
1 Preface......Page 14
2.1 Introduction......Page 17
2.2 Harmonic maps and special harmonic maps on the sphere......Page 19
2.3 Equivariant wave maps and construction of special solutions......Page 25
3.1 Introduction......Page 30
3.2 Localization in time......Page 33
3.3 Estimates for the homogeneous problem......Page 37
3.4 Estimates for the non-homogeneous problem......Page 39
3.5 Bilinear estimates for the homogeneous problem in H[sup(s,δ)]......Page 44
3.6 Bilinear estimates in H[sup(s,δ)] for the inhomogeneous problem......Page 49
4.1 Introduction......Page 53
4.2 Construction of the solutions......Page 55
4.3 Higher regularity of the solution......Page 59
4.4 Appendix......Page 65
5.1 Introduction......Page 68
5.2 Equivariant and self-similar solutions......Page 70
5.3 Low regularity self-similar solutions......Page 73
5.4 Appendix A: The self-similar ODE......Page 79
5.5 Appendix B: Some technical lemmas......Page 85
6.1 Introduction......Page 90
6.2 Well-posedness of the Cauchy problem for semilinear wave equations......Page 92
6.3 The wave map system in stereographic projection......Page 93
6.4 Conclusion of the proof of Theorem 6.1......Page 96
6.5 Proof of Theorem 6.2......Page 99
7.1 Introduction......Page 103
7.2 Proof of Theorem 7.1......Page 105
7.3 Appendix......Page 115
References......Page 122
1 Introduction......Page 125
2 Single wave equation......Page 129
2.1 Blow-up......Page 131
2.2 Small data global existence......Page 137
2.3 Almost global existence......Page 145
2.4 Self-similar solution......Page 149
2.5 Asymptotic behavior......Page 152
3 Semilinear system, I......Page 168
3.1 Blow-up......Page 169
3.2 Small data global existence......Page 175
3.3 Self-similar solution......Page 178
3.4 Asymptotic behavior......Page 179
4 Semilinear system, II......Page 180
4.1 Small data global existence......Page 182
4.2 Self-similar solution......Page 185
4.3 Generalization......Page 188
5 Semilinear system, III......Page 190
5.1 Blow-up......Page 191
5.2 Small data global existence......Page 195
6 Nonlinear system......Page 199
6.1 Blow-up......Page 204
6.2 Null condition......Page 208
Appendix......Page 214
References......Page 217
1 Introduction......Page 224
2 Preliminaries......Page 227
3.1 Problem and result......Page 229
3.2 Proof of Theorem 3.1.......Page 230
3.3 Proof of Corollary 3.1.......Page 234
4.1 Problem and result......Page 237
4.2 Proof of Theorem 4.1......Page 238
4.3 Proof of Theorem 4.2......Page 243
5.1 Problem and result......Page 244
5.2 Proof of Proposition 5.1......Page 246
5.3 Proof of Theorems 5.1 and 5.2......Page 248
5.4 Proof of Proposition 5.2......Page 250
6.1 Problem and result......Page 252
6.2 Proof of Theorem 6.2......Page 254
7.1 Problem and result......Page 260
7.2 Proof of Theorems 7.1 and 7.2......Page 262
7.3 Proof of Theorem 7.3......Page 264
8.1 Problem and result......Page 270
8.2 Energy decay for the quasilinear equation......Page 272
8.3 Estimation of higher-order derivatives of solutions......Page 276
8.4 Proof of Theorems 8.2 and 8.3.......Page 283
9.1 Problem and result......Page 291
9.2 A basic inequality......Page 294
9.3 Proof of Theorem 9.1......Page 297
9.4 Proof of Theorems 9.2 and 9.3......Page 300
10 Some open problems......Page 304
References......Page 306
1 Introduction......Page 311
2 Counterexamples to the global existence......Page 313
3 Blow-up for the problem with large potential energy of nonlinearity......Page 330
4 Parametric resonance and wave map type equations......Page 334
5.1 Some properties of the Hill's equation......Page 337
5.2 Borg's Theorem......Page 344
5.3 Construction of an exponentially increasing solution to Hill's equation......Page 348
5.4 Construction of blow-up solutions......Page 352
6 Coefficient stabilizing to a periodic one. Parametric resonance dominates......Page 354
7 Proof of Theorem 6.1: Perturbation theory......Page 355
8 Nonexistence for equations with permanently restricted domain of influence......Page 364
9 Global existence for a model equation with a polynomially growing coefficient......Page 368
10 An example with an exponentially growing coefficient......Page 372
11 Fast oscillating coefficients: no resonance?!......Page 380
12 Linear wave equations with oscillating coefficients......Page 383
References......Page 392
1 Introduction......Page 396
2.1 Function and symbol spaces......Page 401
2.2 Levi conditions......Page 403
2.3 Factorization......Page 406
2.4 The linear problem......Page 407
2.5 Commutators......Page 412
2.6 The equivalent quasilinear system......Page 416
2.7 Local C[sup(∞)] solutions......Page 418
2.8 Analytic regularity......Page 419
3.1 The linear problem......Page 425
3.2 Gevrey-Levi conditions......Page 428
3.3 Factorization under Gevrey-Levi conditions......Page 431
3.4 Linear systems......Page 432
3.5 The equivalent quasilinear system in Gevrey spaces......Page 436
3.6 Local Gevrey solutions and propagation of the analytic regularity......Page 437
4.1 Log-Lipschitz coefficients or unbounded derivatives......Page 439
4.2 The linear problem with non-regular coefficients......Page 441
4.3 The map [equation omitted]......Page 445
5.1 Gevrey well-posedness......Page 450
5.2 From the factorization to the quasilinear system......Page 451
References......Page 454
1 Introduction......Page 458
1.1 Well-posedness of the Cauchy problem......Page 459
1.2 Degenerate differential operators......Page 460
1.3 Notation......Page 463
2.1 Motivation and plan of the paper......Page 464
2.2 Main results......Page 466
3 A model case......Page 471
3.1 Taniguchi–Tozaki's example......Page 472
3.3 Estimation of the fundamental matrix......Page 473
3.4 Function spaces: An approach via edge Sobolev spaces......Page 474
3.6 Summary of Section 3......Page 479
4.1 The symbol classes S[sup(m,η;λ)]......Page 480
4.2 The symbol classes S[sup(m,η;λ)]......Page 481
4.3 The symbol classes S[sup(m,η;λ)][sub(+)] for η ∈ C[sup(∞)][sub(b)] (R[sup(n)];R)......Page 483
4.4 Function spaces: An approach via weight functions......Page 484
5 The Cauchy problem......Page 487
5.1 Improvement of Gårding's inequality......Page 489
5.2 Symmetric-hyperbolic systems......Page 490
5.3 Symmetrizable-hyperbolic systems......Page 492
5.4 Higher-order scalar equations......Page 496
5.5 Local uniqueness......Page 498
5.6 Sharpness of energy estimates......Page 500
A Supplements......Page 507
B Open problems......Page 515
References......Page 516
