Global Existence of Solutions to Semilinear Klein-Gordon Equations
In this thesis, we prove two main results on nonlinear Klein-Gordon equations. First, we establish global existence of solutions to general second order semilinear Klein-Gordon equations for small initial data and n=3 spatial dimensions. Then, we prove low regularity well-posedness in spatial dimensions n=2 and higher for a quadratic power-type Klein-Gordon system with different masses satisfying a suitable nonresonance condition.
For the first result, our main tool is the Normal Forms Method of Shatah. The key idea behind this approach is to decompose u into a sum of two functions, U and W, where W solves a third order system and U is written explicitly as a function of u and its first order derivatives. The explicit form of U and good behavior of solutions to higher order systems allows us to gain control of both U and W, and thus u.
For the multiple mass system, we apply a standard duality argument to reduce our proof of well-posedness to the establishment of a set of trilinear estimates. The proof of these estimates relies heavily on the special properties of our iteration spaces. In particular, using these spaces allows us to readily exploit the absence of resonant terms and extend important bilinear estimates proved for free solutions to more general functions.