Nonlinear Wave Equations

✍ Scribed by Tatsien Li,Yi Zhou (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2017
• Tongue: English
• Leaves: 397
• Series: Series in Contemporary Mathematics 2
• Edition: 1
• Category: Library

✦ Synopsis

This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms.

Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut

ions to the corresponding linear problems, the method simply applies the contraction mapping principle.

✦ Table of Contents

Front Matter ....Pages i-xiv
Introduction and Overview (Tatsien Li, Yi Zhou)....Pages 1-16
Linear Wave Equations (Tatsien Li, Yi Zhou)....Pages 17-33
Sobolev Type Inequalities with Decay Factor (Tatsien Li, Yi Zhou)....Pages 35-64
Estimates on Solutions to the Linear Wave Equations (Tatsien Li, Yi Zhou)....Pages 65-111
Some Estimates on Product Functions and Composite Functions (Tatsien Li, Yi Zhou)....Pages 113-131
Cauchy Problem of the Second-Order Linear Hyperbolic Equations (Tatsien Li, Yi Zhou)....Pages 133-153
Reduction of Nonlinear Wave Equations to a Second-Order Quasi-linear Hyperbolic System (Tatsien Li, Yi Zhou)....Pages 155-160
Cauchy Problem of One-Dimensional Nonlinear Wave Equations (Tatsien Li, Yi Zhou)....Pages 161-181
Cauchy Problem of (n({\ge }3))-Dimensional Nonlinear Wave Equations (Tatsien Li, Yi Zhou)....Pages 183-216
Cauchy Problem of Two-Dimensional Nonlinear Wave Equations (Tatsien Li, Yi Zhou)....Pages 217-254
Cauchy Problem of Four-Dimensional Nonlinear Wave Equations (Tatsien Li, Yi Zhou)....Pages 255-262
Null Condition and Global Classical Solutions to the Cauchy Problem of Nonlinear Wave Equations (Tatsien Li, Yi Zhou)....Pages 263-301
Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term (F=F(Du, D_xDu)) on the Right-Hand Side Does not Depend on u Explicitly (Tatsien Li, Yi Zhou)....Pages 303-317
Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term (F=F(u,Du, D_xDu)) on the Right-Hand Side Depends on u Explicitly (Tatsien Li, Yi Zhou)....Pages 319-361
Applications and Developments (Tatsien Li, Yi Zhou)....Pages 363-382
Back Matter ....Pages 383-391

✦ Subjects

Partial Differential Equations