Global behaviors of defocusing semilinear wave equations

TL;DR

This paper studies the long-term behavior of solutions to defocusing semilinear wave equations in higher dimensions, establishing decay estimates and scattering results in the energy and critical Sobolev spaces.

Contribution

It extends scattering results to higher dimensions without assuming spherical symmetry and provides new decay estimates for solutions.

Findings

Proves integrated local energy decay estimates for all energy subcritical and critical powers.

Derives uniform weighted energy bounds and inverse polynomial decay for solutions when p > 1 + 2/(d-1).

Shows solutions scatter in the energy and critical Sobolev spaces, extending previous results.

Abstract

In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+\frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space for $p$ with an improved lower bound. This in particular extends the existing scattering results to higher dimensions without spherical symmetry.