## Scattering of rough solutions of the nonlinear Klein-Gordon equations in 3D.(English)Zbl 1337.35135

The purpose of this paper is to study strong solutions of the defocusing nonlinear Klein-Gordon equation $\partial_{tt}u-\Delta u+u=-| u| ^{p-1}u, \tag{1}$ 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35D35 Strong solutions to PDEs 42B25 Maximal functions, Littlewood-Paley theory
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