# Scattering for critical wave equations with variable coefficients

**Authors:** Shi-Zhuo Looi, Mihai Tohaneanu  
**arXiv:** [1912.06795](https://arxiv.org/abs/1912.06795)  
**Date:** 2019-12-14  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1912.06795v1/

## Key points

- Authors investigate behavior of solutions to quintic semilinear wave equation with variable coefficients in $\mathbb{R}^{1+3}$
- Solutions scatter to a solution of the corresponding linear wave equation
- Coefficients in the semilinear wave equation are small and decay as $|x|\to\infty$, but can be time-dependent
- Scattering phenomenon refers to behavior of solutions at large times where they disperse and no longer interact with each other
- Looi and Tohaneanu use local energy decay estimates to establish their result
- These estimates analyze decay of $L^6$ norm of solution as $t\to\infty$
- Showing that this norm decays demonstrates scattering to a solution of the linear wave equation
- Paper provides insights into behavior of solutions to semilinear wave equations with variable coefficients
- Implications for understanding how nonlinear effects interact with varying coefficients in physical systems described by wave equations
- Helps gain insight into long-term behavior of such systems.

## Summary

In the paper "Scattering for critical wave equations with variable coefficients," authors Shi-Zhuo Looi and Mihai Tohaneanu investigate the behavior of solutions to the quintic semilinear wave equation with variable coefficients in $\mathbb{R}^{1+3}$. They prove that these solutions scatter to a solution of the corresponding linear wave equation. The coefficients in the semilinear wave equation are assumed to be small and decay as $|x|\to\infty$, but they are allowed to be time-dependent. The scattering phenomenon refers to the behavior of solutions at large times, where they disperse and no longer interact with each other. This is in contrast to solutions that exhibit blow-up or converge to a nontrivial steady state. To establish their result, Looi and Tohaneanu employ local energy decay estimates. These estimates allow them to analyze the decay of the $L^6$ norm of the solution as $t\to\infty$. By showing that this norm decays, they demonstrate that solutions scatter to a solution of the linear wave equation. The paper provides important insights into the behavior of solutions to semilinear wave equations with variable coefficients. It has implications for understanding how nonlinear effects interact with varying coefficients in physical systems described by wave equations. Thus, it helps us gain insight into how such systems behave over long periods of time.