Lectures on Nonlinear Evolution Equations: Initial Value Problems

Description

This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered. Front Matter....Pages i-viii Introduction....Pages 1-6 Global solutions to wave equations — existence theorems....Pages 7-14 L p –L q -decay estimates for the linear wave equation....Pages 15-20 Linear symmetric hyperbolic systems....Pages 21-33 Some inequalities....Pages 34-56 Local existence for quasilinear symmetric hyperbolic systems....Pages 57-77 High energy estimates....Pages 78-81 Weighted a priori estimates for small data....Pages 82-88 Global solutions to wave equations — proofs....Pages 89-103 Other methods....Pages 104-107 Development of singularities....Pages 108-111 More evolution equations....Pages 112-206 Further aspects and questions....Pages 207-220 Evolution equations in waveguides....Pages 221-270 Back Matter....Pages 271-306