Lectures on Nonlinear Evolution Equations : Initial Value Problems 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 behavior 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 Introduction 1. Global solutions to wave equations existence theorems 2. L^p L^q decay estimates for the linear we equation 3. Linear symmetric hyperbolic systems 3.1 Energy estimates 3.2 A global existence theorem 3.3 Remarks on other methods 4. Some inequalities 5. Local existence for quasilinear symmetric hyperbolic 6. High energy estimates 7. Weighted a priori estimates 8. Global solutions to wave equations proofs 8.1 Proof of Theorem 1.1 8.2 Proof ot Theorem 1.2 9. Other methods 10. Development of singularities 11. More evolutions equations 11.1 Equations of elasiticity 11.1.1 Initially isotropic media in R^3 11.1.2 Initially cubic media in R^3 11.2 Heat equations 11.3 Equations of thermoelasticity 11.4 Schrödinger equations 11.5 Klein Gordon equations 11.6 Maxwell equations 11.7 Plate equations 12. Further aspects and questions 13. Evolution equations in waveguides 13.1 Nonlinear wave equations 13.1.1 Linear part 13.1.2 Nonlinear part 13.2. Schrödinger equations 13.3. Equations of elasticity and Maxwell equations 13.4 General waveguides Appendix A. Interpolation B. The Theorem of Cauchy Kowalevsky C. A local existence theorem for hyperbolic parabolic systems References Notation Index