New Trends in the Theory of Hyperbolic Equations

Editorial Preface xi

Wave Maps and Ill-posedness of their Cauchy Problem

Piero D’Ancona and Vladimir Georgiev 1

1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Variational motivation of the wave maps equations . . . . . . . . . 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Harmonic maps and special harmonic maps on the sphere . 6

2.3 Equivariant wave maps and construction of special solutions 12

3 Local existence result for equivariant wave maps . . . . . . . . . . 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Localization in time . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Estimates for the homogeneous problem . . . . . . . . . . . 24

3.4 Estimates for the non-homogeneous problem . . . . . . . . 26

3.5 Bilinear estimates for the homogeneous problem in Hs,δ . . 31

3.6 Bilinear estimates in Hs,δ for the inhomogeneous problem . 36

4 Concentration of the local energy . . . . . . . . . . . . . . . . . . . 40

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Construction of the solutions . . . . . . . . . . . . . . . . . 42

4.3 Higher regularity of the solution . . . . . . . . . . . . . . . 46

4.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Non-uniqueness result in the subcritical case . . . . . . . . . . . . . 55

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Equivariant and self-similar solutions . . . . . . . . . . . . . 57

5.3 Low regularity self-similar solutions . . . . . . . . . . . . . 60

5.4 Appendix A: The self-similar ODE . . . . . . . . . . . . . . 66

5.5 Appendix B: Some technical lemmas . . . . . . . . . . . . . 72

6 Ill-posedness in the critical case (Fourier analysis approach) . . . . 77

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Well-posedness of the Cauchy problem for semilinear wave

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

vi Contents

6.3 The wave map system in stereographic projection . . . . . . 80

6.4 Conclusion of the proof of Theorem 6.1 . . . . . . . . . . . 83

6.5 Proof of Theorem6.2 . . . . . . . . . . . . . . . . . . . . . 86

7 Ill-posedness in the critical case (fundamental solution approach) . 90

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2 Proof of Theorem7.1 . . . . . . . . . . . . . . . . . . . . . 92

7.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

On the Global Behavior of Classical Solutions to Coupled Systems of

Semilinear Wave Equations

Hideo Kubo and Masahito Ohta 113

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2 Single wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.2 Small data global existence . . . . . . . . . . . . . . . . . . 125

2.3 Almost global existence . . . . . . . . . . . . . . . . . . . . 133

2.4 Self-similar solution . . . . . . . . . . . . . . . . . . . . . . 137

2.5 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . 140

3 Semilinear system, I . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.2 Small data global existence . . . . . . . . . . . . . . . . . . 163

3.3 Self-similar solution . . . . . . . . . . . . . . . . . . . . . . 166

3.4 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . 167

4 Semilinear system, II . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.1 Small data global existence . . . . . . . . . . . . . . . . . . 170

4.2 Self-similar solution . . . . . . . . . . . . . . . . . . . . . . 173

4.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . 176

5 Semilinear system, III . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.2 Small data global existence . . . . . . . . . . . . . . . . . . 183

6 Nonlinear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.1 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.2 Null condition . . . . . . . . . . . . . . . . . . . . . . . . . 196

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Decay and Global Existence for Nonlinear Wave Equations with

Localized Dissipations in General Exterior Domains

Mitsuhiro Nakao 213

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Contents vii

3 Local energy decay . . . . . . . . . . . . . . . . . . . . . . . . . . 218

3.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 218

3.2 Proof of Theorem3.1. . . . . . . . . . . . . . . . . . . . . . 219

3.3 Proof of Corollary 3.1. . . . . . . . . . . . . . . . . . . . . . 223

4 Total Energy decay for the wave equation with a localized dissipation226

4.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 226

4.2 Proof of Theorem4.1 . . . . . . . . . . . . . . . . . . . . . 227

4.3 Proof of Theorem4.2 . . . . . . . . . . . . . . . . . . . . . 232

5 Linear equations with variable coefficients; Unique continuation property

and a basic inequality . . . . . . . . . . . . . . . . . . . . . . . 233

5.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 233

5.2 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . 235

5.3 Proof of Theorems 5.1 and 5.2 . . . . . . . . . . . . . . . . 237

5.4 Proof of Proposition 5.2 . . . . . . . . . . . . . . . . . . . . 239

6 Lp estimates for the wave equation in exterior domains . . . . . . . 241

6.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 241

6.2 Proof of Theorem6.2 . . . . . . . . . . . . . . . . . . . . . 243

7 Semilinear wave equations . . . . . . . . . . . . . . . . . . . . . . . 249

7.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 249

7.2 Proof of Theorems 7.1 and 7.2 . . . . . . . . . . . . . . . . 251

7.3 Proof of Theorem7.3 . . . . . . . . . . . . . . . . . . . . . 253

8 Quasilinear wave equations . . . . . . . . . . . . . . . . . . . . . . 259

8.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 259

8.2 Energy decay for the quasilinear equation . . . . . . . . . . 261

8.3 Estimation of higher-order derivatives of solutions . . . . . 265

8.4 Proof of Theorems 8.2 and 8.3. . . . . . . . . . . . . . . . . 272

9 The wave equation with a half-linear dissipation . . . . . . . . . . 280

9.1 Problemand result . . . . . . . . . . . . . . . . . . . . . . . 280

9.2 A basic inequality . . . . . . . . . . . . . . . . . . . . . . . 283

9.3 Proof of Theorem9.1 . . . . . . . . . . . . . . . . . . . . . 286

9.4 Proof of Theorems 9.2 and 9.3 . . . . . . . . . . . . . . . . 289

10 Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 293

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

Global Existence in the Cauchy Problem for Nonlinear Wave Equations with

Variable Speed of Propagation

Karen Yagdjian 301

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

2 Counterexamples to the global existence . . . . . . . . . . . . . . . 303

3 Blow-up for the problem with large potential energy of nonlinearity 320

4 Parametric resonance and wave map type equations . . . . . . . . 324

viii Contents

5 Proof of Theorem 4.1: Parametric resonance . . . . . . . . . . . . . 327

5.1 Some properties of the Hill’s equation . . . . . . . . . . . . 327

5.2 Borg’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 334

5.3 Construction of an exponentially increasing solution to Hill’s

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

5.4 Construction of blow-up solutions . . . . . . . . . . . . . . 342

6 Coefficient stabilizing to a periodic one. Parametric resonance dominates

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7 Proof of Theorem 6.1: Perturbation theory . . . . . . . . . . . . . . 345

8 Nonexistence for equations with permanently restricted domain of

influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

9 Global existence for a model equation with a polynomially growing

coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

10 An example with an exponentially growing coefficient . . . . . . . 362

11 Fast oscillating coefficients: no resonance ?! . . . . . . . . . . . . . 370

12 Linear wave equations with oscillating coefficients . . . . . . . . . . 373

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

On the Nonlinear Cauchy Problem

Massimo Cicognani and Luisa Zanghirati 387

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

2 Well-posedness in C∞ . . . . . . . . . . . . . . . . . . . . . . . . . 392

2.1 Function and symbol spaces . . . . . . . . . . . . . . . . . . 392

2.2 Levi conditions . . . . . . . . . . . . . . . . . . . . . . . . . 394

2.3 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 397

2.4 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 398

2.5 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . 403

2.6 The equivalent quasilinear system . . . . . . . . . . . . . . 407

2.7 Local C∞ solutions . . . . . . . . . . . . . . . . . . . . . . . 409

2.8 Analytic regularity . . . . . . . . . . . . . . . . . . . . . . . 410

3 Well-posedness in Gevrey classes . . . . . . . . . . . . . . . . . . . 416

3.1 The linear problem . . . . . . . . . . . . . . . . . . . . . . . 416

3.2 Gevrey-Levi conditions . . . . . . . . . . . . . . . . . . . . 419

3.3 Factorization under Gevrey-Levi conditions . . . . . . . . . 422

3.4 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 423

3.5 The equivalent quasilinear system in Gevrey spaces . . . . . 427

3.6 Local Gevrey solutions and propagation of the analytic regularity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

4 Strictly hyperbolic equations with non-Lipschitz coefficients and

C∞ solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

4.1 Log-Lipschitz coefficients or unbounded derivatives . . . . . 430

4.2 The linear problem with non-regular coefficients . . . . . . 432

4.3 The map u → v . . . . . . . . . . . . . . . . . . . . . . . . . 436

Contents ix

5 H¨older coefficients and Gevrey Solutions . . . . . . . . . . . . . . . 441

5.1 Gevrey well-posedness . . . . . . . . . . . . . . . . . . . . . 441

5.2 From the factorization to the quasilinear system . . . . . . 442

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Sharp Energy Estimates for a Class of Weakly Hyperbolic Operators

Michael Dreher and Ingo Witt 449

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

1.1 Well-posedness of the Cauchy problem . . . . . . . . . . . . 450

1.2 Degenerate differential operators . . . . . . . . . . . . . . . 451

1.3 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454

2 Formulation of the results . . . . . . . . . . . . . . . . . . . . . . . 455

2.1 Motivation and plan of the paper . . . . . . . . . . . . . . . 455

2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

3 Amodel case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

3.1 Taniguchi–Tozaki’s example . . . . . . . . . . . . . . . . . . 463

3.2 Conversion into a 2 ×2 system . . . . . . . . . . . . . . . . 464

3.3 Estimation of the fundamental matrix . . . . . . . . . . . . 464

3.4 Function spaces: An approach via edge Sobolev spaces . . . 465

3.5 Establishing energy estimates . . . . . . . . . . . . . . . . . 470

3.6 Summary of Section 3 . . . . . . . . . . . . . . . . . . . . . 470

4 Symbol classes and function spaces . . . . . . . . . . . . . . . . . . 471

4.1 The symbol classes Sm,η;λ . . . . . . . . . . . . . . . . . . . 471

4.2 The symbol classes ˜ Sm,η;λ . . . . . . . . . . . . . . . . . . . 472

4.3 The symbol classes Sm,η;λ

+ for η ∈ C∞b (Rn;R) . . . . . . . . 474

4.4 Function spaces: An approach via weight functions . . . . . 475

4.5 Summary of Section 4 . . . . . . . . . . . . . . . . . . . . . 478

5 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . 478

5.1 Improvement of G˚arding’s inequality . . . . . . . . . . . . . 480

5.2 Symmetric-hyperbolic systems . . . . . . . . . . . . . . . . 481

5.3 Symmetrizable-hyperbolic systems . . . . . . . . . . . . . . 483

5.4 Higher-order scalar equations . . . . . . . . . . . . . . . . . 487

5.5 Local uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 489

5.6 Sharpness of energy estimates . . . . . . . . . . . . . . . . . 491

A Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

B Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507