Scattering for critical wave equations with variable coefficients

TL;DR

This paper proves that solutions to a variable coefficient quintic wave equation in three dimensions scatter to linear solutions, using decay estimates despite small, decaying, and time-dependent coefficients.

Contribution

It establishes scattering results for the critical wave equation with variable, small, decaying, and time-dependent coefficients in three spatial dimensions.

Findings

Solutions scatter to linear wave solutions as time approaches infinity.

Decay of the $L^6$ norm of solutions is proven using local energy decay estimates.

The results hold even with time-dependent coefficients that decay at infinity.

Abstract

We prove that solutions to the quintic semilinear wave equation with variable coefficients in $\mathbb R^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to\infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^6$ norm of the solution as $t\to\infty$.