| In this thesis we prove global existence of solutions with small initial data to the perturbed quadratic nonlinear Klein-Gordon equation.; 62t-D+V +1u=u2 0.1 .;in n = 3 space dimensions, subject to assumptions on the potential V(x). Specifically, we require that V satisfies the decay estimate |∂alpha V(x)| ≲ Calpha⟨x⟩-2-epsilon , that the associated Schrodinger operator H = -Delta + V has no eigenvalues, and that 0 is not a resonance of H.;Energy estimates alone are sufficient to establish global existence of solutions to (0.1), but provide only exponential bounds on higher Sobolev norms of a solution. A major part of the paper is thus dedicated to proving dispersion of solutions to (0.1). Dispersive estimates control the global existence problem for cubic nonlinearities, so we employ normal forms methods due to Shatah [2] to prove the full result for (0.1).;[2] Jalal Shatah. Normal Forms and Quadratic Nonlinear Klein-Gordon Equations. Comm. Pure Appl. Math. 38 1985 685--696. |
