This course is an introduction to the theory of non-linear wave equations. The plan is to cover the following topics:
(1) The linear wave equation in Minkowski space: Representation formula, Fourier techniques, energy estimates, vectorfield method of Klainerman, local existence for general linear equations
(2) Local well-posedness theory non-linear wave equations.
(3) Small data global existence for semi-linear equations with null condition.
(4) Classical blow-up results for semi-linear wave equations
(5) Shock formation results for quasi-linear wave equations (in symmetry)
Some applications to the Einstein equations of general relativity will also be discussed.