Li T., Zhou Y. Nonlinear Wave Equations

Springer-Verlag GmbH Germany and Shanghai Scientific and Technical Publishers, 2017. — 397 p. — (Series in Contemporary Mathematics 02) — ISBN: 3662557231.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.Introduction and Overview
Linear Wave Equations
Sobolev Type Inequalities with Decay Factor
Estimates on Solutions to the Linear Wave Equations
Some Estimates on Product Functions and Composite Functions
Cauchy Problem of the Second-Order Linear Hyperbolic Equations
Reduction of Nonlinear Wave Equations to a Second-Order Quasi-linear Hyperbolic System
Cauchy Problem of One-Dimensional Nonlinear Wave Equations
Cauchy Problem of nð 3Þ-Dimensional Nonlinear Wave Equations
Cauchy Problem of Two-Dimensional Nonlinear Wave Equations
Cauchy Problem of Four-Dimensional Nonlinear Wave Equations
Null Condition and Global Classical Solutions to the Cauchy Problem of Nonlinear Wave Equations
Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term F ¼ FðDu;DxDuÞ on the Right-Hand Side Does not Depend on u Explicitly
Sharpness of Lower Bound Estimates on the Life-Span of Classical Solutions to the Cauchy Problem—The Case that the Nonlinear Term F ¼ Fðu; Du;DxDuÞ on the Right-Hand Side Depends on u Explicitly
Applications and Developments