This book is the first to concentrate on the theory of nonlinear nonlocal equations. The authors solve a number of problems concerning the asymptotic behavior of solutions of nonlinear evolution equations, the blow-up of solutions, and the global in time existence of solutions. In addition, a new classification of nonlinear nonlocal equations is introduced. A large class of these equations is treated by a single method, the main features of which are apriori estimates in different integral norms and use of the Fourier transform. This book will interest specialists in partial differential equations, as well as physicists and engineers.

Readership

Specialists in partial differential equations.

Reviews

"The book is a good exposition of important results. Many examples are considered. The style of the book is systematic and readable."

-- Zentralblatt MATH

"ExcellenDt contribution to the study of wave propagation ..."

-- Mathematical Reviews

"Very readable ... will be a source of inspiration not only to those who study nonlinear wave equations, but to all who like to see hard problems solved by masterful use of basic techniques."

-- Bulletin of the AMS

Table of Contents

• Introduction
• Simplest properties of solutions of nonlinear nonlocal equations
• The Cauchy problem for the Whitham equation
• The periodic problem
• The system of equations of surface waves
• Generalized solutions
• The asymptotics as \(t\rightarrow \infty\) of solutions of the generalized Kolmogorov-Petrovskiĭ-Piskunov equation
• Asymptotics of solutions of the Whitham equation for large times
• Asymptotics as \(t\rightarrow \infty\) of solutions of the nonlinear nonlocal Schrödinger equation
• Asymptotics of solutions for a system of equations of surface waves for large times
• The step-decaying problem for the Korteweg-de Vries-Burgers equation
• References