Scattering of rough solutions of the nonlinear Klein-Gordon equations in 3D (Q5963184)

The purpose of this paper is to study strong solutions of the defocusing nonlinear Klein-Gordon equation NEWLINE\[NEWLINE\partial_{tt}u-\Delta u+u=-| u| ^{p-1}u, \tag{1}NEWLINE\]NEWLINE with initial data, on an interval \([0,T]\). Equation (1) is closely related to the defocusing nonlinear wave equation; because of the scaling properties of the latter equation, a so-called critical exponent \(s_c=\frac{3}{2}-\frac{2}{p-1}\) is introduced. The main theorem states that for \(5>p>3\), and certain other conditions, equation (1), with initial data, has a solution, which scatters as \(T\to\infty\). The proofs, which are divided into several steps, use Strichartz estimates, Sobolev embedding, pigeonhole principle, Young, Hölder and Paley-Littlewood inequalities.