Racke R. Lectures on Nonlinear Evolution Equations: Initial Value Problems

2nd Edition. — Springer, 2015. — 315 p. — ISBN: 978-3-319-21872-4.The presentation of the theory is made using the classical method of continuation of local
solutions with the help of a priori estimates obtained for small data. The corresponding global existence theorems have been proved mainly in the last decade, focussing on fully nonlinear systems. Related questions concerning large data problems, the existence of weak solutions or the analysis of shock waves are not discussed. Also the question of optimal regularity assumptions on the coefficients is beyond the scope of the book and is touched only in part and exemplarily.
Most of the material presented here has only been previously published in original papers, and some of the material has never been published until now. Therefore, I hope that both the interested beginner in the field and the expert will benefit from reading the book. In addition, a long list of references has been included, although it is not intended to be exhaustive. Of course the selection of the material follows personal interests and tastes.Global solutions to wave equations — existence theorems
Lp–Lq-decay estimates for the linear wave equation
Linear symmetric hyperbolic systems
Some inequalities
Local existence for quasilinear symmetric hyperbolic systems
High energy estimates
Weighted a priori estimates for small data
Global solutions to wave equations — proofs
Other methods
Development of singularities
More evolution equations
Further aspects and questions
Evolution equations in waveguides
Appendix
Interpolation
The Theorem of Cauchy–Kowalevsky
A local existence theorem for hyperbolic-parabolic systems